Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay
Bernold Fiedler Isabelle Schneider



	
	

	– Dedicated to Jürgen Scheurle 
	in gratitude and friendship – 

	
	
		



Stabilized rapid oscillations in a delay equation:

 Feedback control by a small resonant delay
    
Bernold Fiedler*

Isabelle Schneider*



    version of December 30, 2023
===============================================================================================

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*

Institut für Mathematik

Freie Universität Berlin

Arnimallee 3/7
 
14195 Berlin, Germany





plain
roman



We study scalar delay equations 

    ẋ (t) = λ f(x(t-1)) + b^-1 (x(t) + x(t -p/2))

with odd nonlinearity f, real nonzero parameters λ,   b, and two positive time delays 1, p/2.
We assume supercritical Hopf bifurcation from x ≡ 0 in the well-understood single-delay case b = ∞.
Normalizing f' (0)=1, branches of constant minimal period p_k = 2π/ω_k are known to bifurcate from eigenvalues iω_k = i(k+12)π at λ_k = (-1)^k+1ω_k, for any nonnegative integer k.
The unstable dimension of these rapidly oscillating periodic solutions is k, at the local branch k.
We obtain stabilization of such branches, for arbitrarily large unstable dimension k, and for, necessarily, delicately narrow regions 𝒫 of scalar control amplitudes b < 0.


For p:= p_k the branch k of constant period p_k persists as a solution, for any b≠ 0.
Indeed the delayed feedback term controlled by b vanishes on branch k:
the feedback control is noninvasive there.
Following an idea of <cit.>, we seek parameter regions 𝒫 = (b_k,b_k) of controls b ≠ 0 such that the branch k becomes stable, locally at Hopf bifurcation.
We determine rigorous expansions for 𝒫 in the limit of large k.
Our analysis is based on a 2-scale covering lift for the slow and rapid frequencies involved.


These results complement earlier results by <cit.> which required control terms 

    b^-1 (x(t-ϑ) + x(t-ϑ -p/2))

with a third delay ϑ near 1.




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§ INTRODUCTION AND MAIN RESULT



equationsection
figuresection

In an ODE setting, delayed feedback control is frequently studied for systems like
	
    𝐱̇(t) =
    	𝐅(𝐱 (t)) +
    	β (𝐱 (t) - 𝐱(t-τ))

with 𝐱∈ℝ^N, smooth nonlinearities 𝐅, and suitable N × N matrices β mediating the feedback.
If the uncontrolled system β = 0 possesses a periodic orbit 𝐱_*(t) of (not necessarily minimal) period p > 0, then 𝐱_*(t) remains a solution of (<ref>) for time delays τ = p and any control matrix β.
In this sense, the delayed feedback control is noninvasive on 𝐱_*(t).
The linearized and nonlinear stability or instability of 𝐱_*(t), however, may well be affected by the control term β.


The above idea was first proposed by Pyragas, see <cit.>.
It has gained significant popularity in the applied literature since then, with currently around 3000 publications listed.
See <cit.> and <cit.> for more recent surveys.
In fact, no previous knowledge of the nonlinearity 𝐅 is required to attempt this procedure, or any of its many variants.


One fundamental disadvantage of the Pyragas method (<ref>), from a theoretical perspective, is the replacement of the ODE 𝐱̇ = 𝐅(𝐱) in finite-dimensional phase space X = ℝ^N by the infinite-dimensional dynamical system (<ref>) in a history phase space like 
𝐱(t+ · ) ∈ X = C^0 ([-τ,  0], ℝ^N).
On the other hand, the very existence of a periodic solution 𝐱(t), for vanishing control β = 0, requires N ≥ 2.


In <cit.> we therefore started to explore the Pyragas method of delayed feedback control, in a slightly modified form, for the very simplistic scalar case
	
    ẋ(t) =
    	λ f (x(t-1)) +
    	b^-1 (x(t-ϑ) +
    	x(t-ϑ - p/2)) .

We consider nonzero real parameters λ,  b and positive delays ϑ, p/2. The case b = 0 will only appear as a formal limit β = ±∞ of infinite feedback amplitudes. The identical cases b = ±∞ account for vanishing feedback β=0 and correspond to the scalar pure delay equation
	
    ẋ(t) =
    	λ f (x(t-1))

with |λ | normalizing the remaining delay τ to unity.
See <cit.> for an early analysis of a specific equation of this type, equivalent to the delayed logistic equation.

Throughout the paper we assume f ∈ C^3 to be odd, with normalized first derivative at f(0) = 0:
	
    f(-x) =	-f (x),   
    	f'(0) = 1,   
    	f”'(0) < 0 .

	
The characteristic equation for complex eigenvalues μ of the linearization of (<ref>) at parameter λ and the trivial equilibrium x ≡ 0 then reads
	
    μ =
    	λ e^-μ + b^-1 
    	(e^-ϑμ + e^- (ϑ+ p/2) μ) .

See <cit.> for an analysis of odd periodic solutions x_k(t) of the pure delay equation (<ref>) with constant minimal period	
	
    p_k := 2 π/ ω_k,   ω_k := (k+12) π .

The periodic solutions originate by Hopf bifurcation from imaginary eigenvalues ±  i ω_k at x = 0 for parameters	
	
    λ = λ_k :=
    	(-1)^k+1ω_k .

Here k ∈ℕ_0 is any nonnegative integer.
For k >0, these periodic solutions are called rapidly oscillating because their minimal period p_k is at most 4/3.
Slowly oscillating periodic solutions, in contrast, have minimal periods exceeding 2.
For example, p_0=4.
See <cit.> for a survey of related results.
In particular see <cit.> for secondary bifurcations from these primary branches.

The case ẋ(t) = g(x(t),  x(t-1)) of the general scalar delay equation with a single time lag has attracted considerable attention; see for example <cit.> and the many references there.
After early observations by Myshkis <cit.>, Mallet-Paret <cit.> has discovered a discrete Lyapunov functional for g with monotone delayed feedback.
The global consequences of this additional structure are enormous; see <cit.>.
For example, all rapidly oscillating periodic solutions are known to be unstable.
More recent developments study this scalar equation with state dependent delays, where the time delay 1 is not constant but depends on the history x(t+ ·) of the solution itself; see for example <cit.>. An excellent survey article on the above developments for general scalar delay equations with a single delay is <cit.>.	





But let us return to the simple setting (<ref>) – (<ref>) of a pure delay equation.
Pioneering analysis by <cit.> reduces the quest for periodic solutions near all Hopf bifurcations (<ref>) to a planar Hamiltonian ODE system.
This is due to an odd-symmetry
	
    x_k(t+p_k/2) =
    	-x_k(t)

at half minimal period p_k, for all real t.
See also <cit.> for complete details, and <cit.> for a survey on the Kaplan-Yorke idea.
Remarkably, global solution branches of constant minimal period p_k emanate from each λ = λ_k towards λ of larger absolute value, in the soft spring case of strictly decreasing secant slopes x ↦ f(x)/x, for x >0.
In particular all Hopf bifurcations are locally nondegenerate and quadratically supercritical under the sign assumption f”'(0) < 0 of (<ref>). See fig. <ref> for a bifurcation diagram. 




At supercritical Hopf bifurcation it is easy to determine the unstable dimension E, i.e. the total algebraic multiplicity of Floquet multipliers outside the complex unit circle, for the emanating local branch of periodic orbits.
It coincides with the total algebraic multiplicity
	
    E = E (λ_k) = k

of the eigenvalues μ with strictly positive real part for the characteristic equation (<ref>) at the Hopf point λ = λ_k.
See for example <cit.>.
Henceforth we skip the trivial case k=0, which leads to the bifurcation of stable, slowly oscillating solutions. Let k∈ℕ be any strictly positive integer.


The local (and global) Hopf branches (λ, x_k(t)) which bifurcate at λ = λ_k,  x=0 inherit constant period p=p_k and odd-symmetry (<ref>). 
Therefore our modified Pyragas control scheme (<ref>) with p:= p_k is noninvasive on the Hopf branches with this symmetry.
For earlier applications of Pyragas control at half minimal periods in the presence of some involutive symmetry, although not in our present delay context, we refer to <cit.>.
It is our main objective to stabilize all bifurcating periodic orbits, for arbitrarily large unstable dimensions k∈ℕ, by suitable Pyragas controls (<ref>).
For odd k, in particular, this again refutes the purported “odd number limitation” of Pyragas control <cit.>. 


We define a Pyragas region 𝒫 to be a connected component of real control parameters b ≠ 0 and ϑ≥ 0 for which the periodic solutions x_k(t) emanating by local Hopf bifurcation from λ=λ_k,  x ≡ 0 become linearly asymptotically stable for sufficiently small amplitudes.
Therefore, the boundaries of Pyragas regions are either certain curves where zero eigenvalues μ = 0 occur, or else are Hopf curves characterized by purely imaginary eigenvalues μ =iω̃ of the characteristic equation (<ref>).

With this definition we can now formulate the main result of the previous paper <cit.>.
See fig. <ref> for an illustration of Hopf curves in the cases k = 9 and k =10.






<cit.>
Consider the system (<ref>) of delayed feedback control for the scalar pure delay equation (<ref>).
Let assumptions (<ref>) of oddness and normalization hold for the soft spring nonlinearity f ∈ C^3.
Then the following assertions hold for large enough k ≥ k_0.


There exist Pyragas regions 𝒫 = 𝒫_k^+  ∪ 𝒫_k^- composed of two disjoint open sets 𝒫_k^±≠∅.
Each region 𝒫_k^ι, ι = ±, is bounded by the horizontal zero line
	
    b =b_k := -2/λ_k
    	=(-1)^k · 2/ω_k
	
and three other analytic curves γ_k^0 and γ_k, ±^ι, all mutually transverse.
The zero line (<ref>) indicates a zero eigenvalue μ of the characteristic equation (<ref>) of (<ref>) at x ≡ 0.
The other curves indicate additional purely imaginary eigenvalues μ.


Define ε:= 1/ω_k.
Then an approximation of the Pyragas regions (ϑ, b) ∈𝒫_k^ι, ι = ±, up to error terms of order ε^3, is given by two parallelograms.
One exact horizontal boundary is b=b_k := (-1)^k  2 ε; see (<ref>).
The other horizontal boundary γ_k^0 is approximated by
	
    b = (-1)^k  2 ε +
    	b_k^ι ε^2 + … .

The sides γ_k, ±^ι are given, up to order ε^3, by the parallel slanted lines through the four points at b=b_k,
	
    ϑ_k, ±^ι = 1 -
    	(π2 - ι q) ε±Θ_k+2 ε^2 + … ,

with slopes σ_k^ι.
The offsets q, Θ_k+2,  b_k^ι and the slopes σ_k^ι of the Pyragas parallelograms are given by
	
    q    =  arccos  (2/π)
    	      =   0.88 … ,
    Θ_k+2    =  π (√((π2)^2-1) - 2q)
    	      =   - 1.73 … ,
    
    	b_k^ι    =  
    	(π+2 Φ_k^ι ) cos Φ_k^ι      =   1.49 … + (-1)^k ι· 1.02 … ,
    σ_k^ι    =  
    	2(-1)^k+1ι√((π2)^2-1)      =   (-1)^k+1ι· 2.42 …

Here we have used the abbreviations
	
    Φ_k^ι = 
    	  (-1)^k ι arcsin  q

In particular the areas | 𝒫_k^± | of the Pyragas regions are of very small order ε^4.
The relative areas are approximately reciprocal,
	
    lim_k ↦∞ |𝒫_k^+|
    	/ |𝒫_k^-| =
    	lim_k ↦∞ b_k^+ / b_k^- = 
    	{
    	   5.37…       for    even k  ,
       0.19 …       for    odd k .
    	
    	.





See <cit.> for full details and proofs.


The above stabilization result for rapidly oscillating periodic solutions requires two additional delays in the control term of (<ref>):
the half-period delay p_k/2, and a joint offset ϑ near 1, in addition to the normalized delay 1 of the reference system.
In the present paper we achieve the same goal with the single delay p_k/2 in the control term, i.e. with vanishing delay offset:
	
    ϑ =0 .

From now on, and for the rest of the paper, we therefore replace (<ref>) by
	
    ẋ (t) = λ f(x(t-1)) +b^-1 (x(t)+x(t-p/2)) ,

with nonlinearities f ∈ C^3 which satisfy assumptions (<ref>).
Our main result identifies unique nonempty Pyragas intervals 𝒫_k =(b_k, b_k) of control parameters b. The k-dimensionally unstable, rapidly oscillating periodic solutions of constant minimal period p_k=4/(2k+1) are stabilized near Hopf bifurcation at λ=λ_k,x=0, for b∈𝒫_k and all sufficiently large k∈ℕ. Via ε:= 1/ω_k = p_k/2π, we also provide ε-expansions, alias k-expansions, for the Pyragas boundaries b_k and b_k.
Taylor expansions with respect to ε again amount to rapid oscillation expansions at k=∞.



Consider the system (<ref>) of delayed feedback control for the scalar pure delay equation (<ref>).
Let assumptions (<ref>) of oddness and normalization hold for the soft spring nonlinearity f∈ C^3.
Then the following assertions hold for large enough k ≥ k_0, i.e. for small enough 0<ε:= 1/ω_k= ((k+12)π)^-1≤ε_0.	


The only Pyragas region of nonzero control amplitudes b is the open interval
	
    𝒫_k:= {b_k < b< b_k } .

Up to error terms of order ε^4, the lower and upper boundaries of the Pyragas interval satisfy
	
    b_k    =
    	-12π^2ε^2 
    	-34π^3ε^3+… ,
    b_k    =
    	-12π^2ε^2 
    	+14π^3ε^3+…





Although our proofs and expansions only address Hopf bifurcations at sufficiently large unstable dimensions k, and sufficiently rapid oscillation frequencies ω_k , numerical evidence suggests a single Pyragas interval 𝒫_k , for any k≥ 1. 
We do not pursue these cases here, beyond the evidence provided in figs. <ref> and <ref>.

The remaining sections disentangle the elements of the proof of theorem <ref>.
We give a brief outline here.
For a summary of sections <ref> – <ref>, on a precise technical level, we refer to the proof of theorem <ref> in the concluding section <ref>.

Section <ref>, and most of the remaining sections, address the characteristic equation (<ref>), at ϑ=0, for the linearization at the original Hopf bifurcation points λ = λ_k, x≡ 0.
Elementary as this task may appear, the rapidly oscillatory terms which appear in the limit ε = 1/ω_k ↘ 0 cause substantial and worthwhile difficulties.

In section <ref> we first recall some elementary results from <cit.> which address the crossing direction of an additional simple eigenvalue 0 induced by the control term.
We also introduce a 2-scale lift, which artificially represents the large, rapidly oscillatory imaginary parts of b-induced Hopf eigenvalues μ = iω̃ by, both, ω̃ itself and a scaled slow frequency
	
    Ω̃ = εω̃ .

Note how Ω̃ =1, ω̃ = ω_k correspond to the reference Hopf bifurcation at λ = λ_k.
We observe Ω̃≠ 2m cannot be at even integer resonance. We introduce a new local scaled slow frequency
	
    Ω:= Ω̃ -Ω_m ,    Ω_m := 2m+1 ,

near each odd integer resonance Ω_m. Below, in fact, we will be able to focus on -1 < Ω≤ 0.
Since
	
    ω≡ω̃2π

rotates rapidly through S^1, for small ε, we treat the two frequencies Ω, ω as independent variables, formally.
They remain related by the hashing relation 

    Ω = ε(ω +π2 (1-(-1)^k-(-1)^m)
    	-2π j) ,    
    	-π2≤ω <32π ,

j ∈ℕ, first discussed in lemma <ref>.
We will be able to restrict attention to the case of odd k, in this setting.

The above hashing trick, first used in <cit.>, will be of central importance in our analysis.
In the limit ε↘ 0, alias h ↗∞, the hatching by the hashing lines (<ref>) fills the (Ω,ω)-cylinder, densely.
Indeed, the hashing lines define steeply slanted (non-military) “barber pole” stripes of horizontal distance ε around the cylinder.
The hashing trick (<ref>) allows us to consider Ω and ω as independent cylinder variables, temporarily. 
This eliminates ε > 0 from the characteristic equation, altogether, as follows.


For the control amplitude b we proceed with the same scaling
	
    B:= 12 b ε^-1

as in <cit.>.
Inserting the scalings (<ref>) – (<ref>) into the characteristic equation (<ref>) for μ=iω̃ at ϑ=0,
we thus arrive at the 2-scale characteristic equation
	
    0= -ie^iω+Ω_m+Ω +(-1)^mB^-1cos (π2Ω) e^iπ2Ω .
	
In lemmata <ref>-<ref> we solve the ε-independent (!) complex characteristic equation (<ref>) for the real variables
	
    ω   = ω^± (Ω) ,
    
    	B    = B^± (Ω) .

See figs. <ref>, <ref> below for illustration.


In section <ref> we observe how the imaginary parts ω̃ of unstable eigenvalues μ=μ_R+iω̃ are trapped in certain strips indexed by m,j. Instability in such a strip can be induced by Hopf bifurcation at control parameters B=B_m,j^-, and be reduced again at control parameters B^+_m,j.
This involves an analysis of the crossing directions of μ, transversely to the imaginary axis μ_R=0, as B increases through B^±_m,j. See  fig. <ref> and theorem <ref>. 
In corollary <ref> we conclude the absence of any region of Pyragas stabilization for B>0.
Corollary <ref> summarizes the results of section <ref>: we reduce the proof of theorem <ref> to the three inequalities (<ref>) – (<ref>) among the Hopf parameter values B^±_m,j.


In section <ref>, we insert the solution ω =ω^± (Ω) from (<ref>) into the hashing (<ref>).
Inverting the resulting maps Ω↦ε= ε(Ω), uniformly for bounded m ≤ m_0, j ≤ j_0, we obtain ε-expansions
	
    Ω   = Ω^±_m,j(ε) ,
    ω   = ω^±_m,j(ε) ,
    
    	B   = B^±_m,j(ε)
		
for the frequencies ω̃≡ω and the control amplitudes b=2Bε of the resulting control-induced Hopf bifurcations.
In particular the crossing directions of the induced imaginary Hopf pairs with respect to b sum up such that
	
    b = 2ε B_0,1^+(ε) ,    b =2ε B_1,1^-(ε)
		
provide the boundaries of the Pyragas region 𝒫 claimed in theorem <ref>.
We illustrate the relative location of B^±_m,j, in view of the crucial inequalities required in corollary <ref>, at the end of section <ref>; see also fig <ref>.
		
		
It remains to show, however, that the candidate interval b ∈ (b,b) does not suffer any destabilization, due to any other Hopf points B_m,j^±.
This turns out to be equivalent to the estimates
	
    B_m, j_m+1^+ < B_0,1^+    and    
    	B_m,j_m^- >B_1,1^-

at j_m:= [(m+1)/2].
See (<ref>), (<ref>) and corollary <ref> again.
For bounded m ≤ m_0, these estimates are suggested by the explicit expansions (<ref>). In section <ref> we begin to settle the delicate case of large m, Ω_m by expansions with respect to
	
    δ:= Ω_m^-1 = 1/(2m+1) .

Here our second small parameter δ > 0 expands the odd integer resonance regions around Ω = Ω_m = 2m+1, for large m, in much the same way as our first small parameter ε expanded the discrete parameter k, for large k ≥ k_0, which enumerated the original Hopf bifurcations of more and more rapidly oscillating periodic solutions with higher and higher unstable dimension.


This time, we solve the characteristic equation (<ref>) to obtain expansions
	
    Ω   = Ω^±(δ, ω) ,
    
    	B   = B^±(δ, ω)

with respect to δ, uniformly in |ω | ≤π /2.
Here Ω^-, B^- refer to the case j=j_m and Ω^+, B^+ refer to j=j_m+1.
In section <ref>, the hashing relation (<ref>) then provides a δ-expansion for
	
    ε^± = ε^± (δ, ω) .

Inserting this into the already established expansions (<ref>) for b, b, and comparing the results, for small δ, we obtain
	
    B^+(δ, ω ) < B_0,1^+(ε^+(δ, ω))
    	    and     
    	B^-(δ, ω )> B_1,1^-(ε^-(δ,ω))

as claimed in (<ref>).
Well, nontrivial differences only appear at order δ^3 and after additional linearization with respect to ω, at ω = ±π2.


The proof of theorem <ref> only involves some discussion of a characteristic equation with two exponential terms of different scales.
Nevertheless, the elementary ingredients to the proof turn out to be surprisingly involved.
Therefore we summarize the various elements of the proof, as scattered across sections <ref> – <ref>, in our final section <ref>.






Acknowledgments.
For many helpful comments and suggestions, as well as most of the figures, we are much indebted to Alejandro López Nieto.
Delightful discussions were provided by several participants of the conference in honor of Jürgen Scheurle, and in particular by P.S. Krishnaprasad.
We are particularly grateful for the lucid remarks of our referees, which helped us improve the somewhat messy presentation.
Ulrike Geiger typeset the original manuscript, with expertise and diligence.
The authors have been supported by the CRC 910 “Control of Self-Organizing Nonlinear Systems:
Theoretical Methods and Concepts of Application” of the Deutsche Forschungs­gemeinschaft.







§ THE 2-SCALE CHARACTERISTIC EQUATION




The characteristic equation (<ref>) of the delay equation (<ref>) with vanishing time shift ϑ = 0 reads
	
    μ = -(-1)^k ε^-1 e^-μ +
    	b^-1 (1+e^-πεμ) ,

at Hopf bifurcation parameter λ = λ_k=(-1)^k+1ω_k, minimal period p_k=2π / ω_k, and with the abbreviation
	
    ε = ω_k^-1 = 1/((k+12)π)

for k∈ℕ.
We decompose the eigenvalue μ =μ_R +iω̃ into real and imaginary parts and define the auxiliary slow frequency Ω̃:= εω̃; see (<ref>).
For the convenient choice of
	
    ω :≡ω̃2π ,     
    	-12π≤ω < 32π ,

we obtain the crucially important 2-scale characteristic equation
	
    0    = χ (ε, δ, ω, Ω, B, μ_R):=
       = -εμ_R + i Ω̃- (-1)^k e^-μ_R+iω̃
    	-iBsin(π2Ω)
    	e^-πεμ_R+iπ2Ω
    	+ 12B(1-e^-πεμ_R),

by some elementary arithmetic and with the abbreviations
	
    Ω̃   := εω̃ ;
    
    	B   := 12bε^-1 ;
    δ   := Ω_m^-1 = 1/(2m+1) ;
    Ω   := Ω̃-Ω_m .

In particular we have utilized the complex conjugate of (<ref>).
Evidently, the solutions of (<ref>) for even parity of k are trivially obtained from the solutions for odd parity if we replace ω by ω +π2π.
For later use we note the relation
	
    sin( π2Ω ) = -(-1)^m cos(π2Ω̃) ;

see also (<ref>).

For interpretation we recall that Ω̃=1 indicates ω̃=ω_k, i.e. a 1:1 resonance ω̃=ω_k of the imaginary part i ω̃ of μ, under feedback control, with the original Hopf eigenvalue μ=i ω_k at parameter λ=λ_k. 
Similarly, Ω̃=Ω_m=(2m+1) indicates an odd integer (2m+1):1 resonance. The parameters ε, δ make k,m look continuous, respectively, and replace them eventually.

For complex nonreal eigenvalues μ, the imaginary part ω̃ = Ω̃/ε can be taken positive, without loss, and we may assume
	
    -1 < Ω≤ 0 ,        for  m=0 ,
    
    	-2 <Ω≤ 0 ,       for  m∈ℕ .

We also note that χ is real analytic in all variables, for B  0. 
Since μ_R=0, for purely imaginary eigenvalues, the characteristic function χ of (<ref>) simplifies and becomes
	
    χ_0 (δ, ω, Ω, B):=
    	-i χ(ε, δ, ω, Ω,μ_R=0)=
    	Ω̃+(-1)^k i e^iω-
    	1Bsin (π2Ω) e^iπ2Ω ,

in the Hopf case. 
Note how ε has disappeared from the characteristic equation (<ref>), in (<ref>), at the price of a hidden hashing relation between ω and Ω; see lemma <ref> below.

	
	
In the present section we collect some elementary facts about the 2-scale characteristic equation (<ref>) – (<ref>).
Real eigenvalues μ = μ_R, i.e. the case ω̃ = Ω̃ =0, are addressed in lemma <ref>.
As a corollary we eventually obtain how B has to be negative in any Pyragas region; see corollaries <ref> and <ref>.
With a brief interlude on hashing in lemma <ref>, we embark on our discussion of purely imaginary eigenvalues μ_R =0.
In lemma <ref> we show how to eliminate any two of the three variables ω, Ω, B from the resulting ε-independent 2-scale characteristic equation
	
    0 = χ_0(δ, ω, Ω, B) .

In particular we identify a quadratic loop
	
    Q(δ, Ω, B) =0 ,

by elimination of ω, such that purely imaginary eigenvalues μ_R=0 can occur only if (<ref>) is satisfied.
In section <ref> we will observe how positive real parts, μ_R>0, can only occur inside the loop, and negative real parts, i.e. linear stability as required in Pyragas regions, are confined to the exterior.
Via hashing, this leads to the definition of crucial pairs of Hopf parameter values
		
    B_m,j^- < B_m,j^+ <0

such that the loss of stability caused at B_m,j^-, by a pair of purely imaginary eigenvalues, is recovered when the control parameter B<0 increases further to pass the matching Hopf value B_m,j^+.



The characteristic equation (<ref>) possesses a real zero eigenvalue, μ = μ_R=0, if and only if
	
    B = B_0=(-1)^k .

The zero eigenvalue μ is algebraically simple, and its continuation μ = μ(B) satisfies
	
    sign Reμ'(B_0) = (-1)^k =B_0 ,

i.e. μ(B) increases towards larger |B|.





For real eigenvalues μ = μ_R, ω=Ω̃=0, and Ω=-1, the characteristic equation (<ref>) reads
	
    εμ_R= -(-1)^k e^-μ_R+
    	12B(1+e^-πεμ_R) .

Inserting μ_R=0 proves claim (<ref>).
Partial differentiation with respect to μ_R shows simplicity of μ_R=0 at B= B_0=(-1)^k.
Implicit differentiation with respect to B at B= B_0, μ_R=0 shows
	
    εμ'_R= (-1)^k    μ'_R +12(-1)^k
    	·(-πε)μ'_R-1 ,  i.e.
       μ'_R(1-(π2+B_0)ε) = B_0 .

For k∈ℕ, ε=((k+12)π)^-1, the coefficient of μ'_R is positive, and the lemma is proved.



Of course we may solve the real characteristic equation (<ref>) for B=B(μ_R), explicitly, to obtain
	
    B=B(μ_R)=1/2·1+exp (-πεμ_R)/εμ_R+(-1)^kexp (-μ_R) .

For odd k, vanishing denominator indicates the unique positive real eigenvalue μ = μ_R of the original problem (<ref>) without control.
For even k, the denominator is positive for all real μ_R, because ε=((k+12 )π)^-1 <e for k∈ℕ_0.
Moreover, πε≤2/3 for all k ∈ℕ implies lim B(μ_R)=0 for μ_R →±∞.
Let
	
    1 < B_max := max B(μ_R) <∞

denote the maximum over μ_R ∈ℝ, for even k.
Indeed B_max> 1=B_0=B(0), by lemma <ref>.
This allows us to determine the even/odd parity of the total algebraic count E(B) of all eigenvalues μ, real or complex, with strictly positive real part. 
We write E(B) ≡ 0  (mod  2) or E(B) ≡even, for even parity of E(B), 
and analogously E(B) ≡ 1  (mod  2) ≡odd, for odd parity.



Let k∈ℕ be odd.
Then the unstable parities (mod 2) are given by
	
    E(B) ≡{
    	   odd        -   ∞ < B <-1 ,
       even       for     -   1≤ B<0 ,
       odd          0 ≤ B<+∞ .
    	
    	.

For even k∈ℕ, the unstable parities (mod 2) are given by
	
    E(B) ≡{
    	   even        -   ∞ < B <0 ,
       odd       for       0<B<1 ,
       even          1≤ B< +∞ .
    	
    	.

Pyragas regions E(B)=0 require even parity 0  (mod 2), of course.
For even k, they also require
	
    B> B_max >1 ,

in case B >0.





Since nonreal complex eigenvalues occur in complex conjugate pairs, the real eigenvalues alone determine the parity.
For odd k and at B = ±∞, i.e. at vanishing control, instability by a simple positive real eigenvalue follows from the vanishing denominator in (<ref>).
Lemma <ref> then implies claim (<ref>).


For even k, real eigenvalues are absent if B <0 or B>B_max; see (<ref>), (<ref>).
At B=B_max>1, a pair of complex eigenvalues merges and forms a positive double real eigenvalue.
Decreasing B further, one of these two positive eigenvalues becomes negative, at B=B_0=1, and the other real eigenvalue remains positive and simple; see lemma <ref>.
This proves claims (<ref>), (<ref>), and the corollary.



We study the case of purely imaginary nonzero eigenvalues μ = iω̃≠ 0, μ_R=0 next.
The 2-scale characteristic equation (<ref>) then simplifies to (<ref>), (<ref>), i.e.
	
    χ_0 (δ, ω,Ω, B) :=
    	-i χ(ε, δ, ω,Ω, B, μ_R=0)=0 .

Strictly speaking, however, the frequency ω̃ and the slow frequency Ω̃ are still related by the linear hashing relation Ω̃ = εω̃; see (<ref>) – (<ref>).
We clarify this relation next.


Consider ω̃>0, Ω = Ω̃-Ω_m with Ω_m=2m+1, and -1<Ω≤0 for m=0, but -2<Ω≤0 for m∈ℕ.
Then the hashing relation Ω̃ = εω̃, with ε= ω_k^-1, is equivalent to
	
    Ω = ε(ω +π2(1-(-1)^k-(-1)^m)-2π j) .

Here the representative ω≡ω̃ (mod 2π) is chosen such that
	
    -π2≤ω <32π ,

j∈ℕ is chosen such that

    j= 
    	{
    	   0 
    	          for k,m  both even ,
       1
    	          otherwise ,
    	
    	.

holds at Ω =0, and
	
    ω̃ = ω+2π(km+[(k+1)/2]+[(m+1)/2]-j)
    	≡ω2π .




The hashing relations Ω̃=εω̃ and (<ref>) are both affine linear in ω̃, ω with slope ε.
To show their equivalence, via Ω̃=Ω +Ω_m in (<ref>) and definition (<ref>) of ω, we only have to check (<ref>) at εω̃=Ω̃ = Ω_m and Ω =0.
Indeed we obtain
	
    ω:
    	   = 
    	ω̃-2π(km+[(k+1)/2]+[(m+1)/2]-j)=
       =
    	ε^-1·Ω_m-π(2km+2[(k+1)/2]+2[(m+1)/2])+2π j=
       =
    	(k+12)π·(2m+1)
    		-π(2km+k+m)+2π j-
       =(k+12)π·(2m+1) 
    		-π((2[(k+1)/2]-k)+(2[(m+1)/2]-m))=
       =
    	π2+2π j-π(12(1-(-1)^k)
    	+12(1-(-1)^m))=
       =
    	2π j -π2(1-(-1)^k-(-1)^m) ,

as required by (<ref>).
The choice of j in (<ref>) ensures the ranges (<ref>) for Ω < 0 near Ω=0.
This proves the lemma.





The Hopf points B∈ℝ, where purely imaginary eigenvalues μ =iω̃>0 arise, are therefore defined by the system of the 2-scale characteristic equation (<ref>) and the hashing (<ref>), in the precise sense of lemma <ref>.
For the moment we “forget” hashing and address the complex 2-scale equation (<ref>) first, in its own right.
See also the bottom rows of figs. <ref> and <ref>.



The 2-scale characteristic equation χ_0(δ, ω, Ω, B)=0 for purely imaginary eigenvalues, i.e. equation (<ref>), (<ref>), is equivalent to the system
	
    0
    	   =H(δ, Ω,ω)
    	:=Ω̃sin (π2Ω)-(-1)^k
    	cos(ω-π2Ω)=
       =(-1)^m+1(Ω̃cos (π2Ω̃)
    	-(-1)^ksin(ω-π2Ω̃))


	
    B
    	= (-1)^k sin^2(π2Ω)/cosω 
    	= (-1)^k cos^2(π2Ω̃)/cosω

with parameter δ =1/(2m+1), m∈ℕ_0.
As always, the case of even k results from odd k by addition of π to ω2π.
Here we have also used the previous notation

	
    Ω̃ = Ω +Ω_m = Ω+(2m+1)=Ω +1/δ ;

see (<ref>).
Eliminating ω∈ (-π2, 3π2) we obtain the quadratic relation
	
    0=Q(δ,Ω,B):= (Ω̃^2-1)B^2
    	+Ω̃sin (πΩ̃)B
    	+cos^2(π2Ω̃) .

The discriminant D of (<ref>) is given by
	
    D= cos^2(π2Ω̃)
    	· (1-(Ω̃cos (π2Ω̃))^2) ,

and the explicit solutions B=B^± of (<ref>) are
	
    B^± = (Ω̃^2-1)^-1 
    	(-12Ω̃sin (πΩ̃)±√(D)) .

For m∈ℕ and Ω_m=2m+1, let Ω̃_m∈ (2m,Ω_m) and Ω̃_m^max∈ (Ω_m, 2m+2) denote the unique solutions Ω̃ of D=0, i.e. of
	
    Ω̃· (-1)^mcos (π2Ω̃)=1 ,

in the respective intervals.
In terms of Ω = Ω̃-Ω_m and 0<δ = Ω_m^-1 this defines unique solution branches (Ω, B^±) of (<ref>) with
	
    B^+<0   <B^- ,       for 
    	0=:Ω̃_0<Ω̃<    Ω_0=1 ,    m=0;
    
    	B^±<0    ,       for    Ω̃_m<Ω̃<    Ω_m ,    m≥ 1 ;
    
    	0   <B^± ,         for    Ω_m<Ω̃<    Ω̃_m^max ,    m≥ 1 .









For vanishing real part μ_R=0, the 2-scale characteristic equation (<ref>) simplifies to
	
    0=i χ_0(δ,ω,Ω,B)
    	= iΩ̃-(-1)^k e^iω
    	-iBsin(π2Ω) e^iπ2Ω ;

see (<ref>), (<ref>).
To prove (<ref>), we multiply by exp(-iπ2Ω) and take real parts.
To prove (<ref>) we take real parts directly.
To eliminate ω, as in (<ref>), we solve χ_0=0 in (<ref>) for the only term exp (iω) which contains ω, and calculate the square of the absolute values of both sides.
The remaining claims (<ref>) – (<ref>) concerning the quadratic relation (<ref>) are plain high school calculus.
This proves the lemma.



In the following analysis we will skip the third case m≥ 1, B^±>0, Ω_m<Ω̃<Ω̃_m^max of (<ref>) which is completely analogous to the second case Ω̃_m<Ω̃<Ω_m, B^± <0.
Indeed that third case will turn out to be irrelevant anyway, in section <ref>; see corollary <ref>.



The 2-scale relation (<ref>) between slow and fast frequencies Ω and ω can be solved for Ω = Ω(δ,ω), implicitly, and for the inverse function ω = ω(δ,Ω), explicitly:
	
    ω = ω^±:≡π2Ω±arccos (-Ω̃sin (π2Ω))   2π ,

for k odd.
Even k require addition of π2π.


For m=0, where δ=1 and 0<Ω̃ = Ω +1<1, both functions ω^± have strictly positive and bounded derivatives with respect to Ω̃ or Ω, equivalently, in the interior domain.
For odd k, their boundary values and ranges are, accordingly,
	
    ω^+ ∈ [0,12π] ,
    	     with ω^+=0, 12π atΩ̃ =0,1 ;
    ω^- ∈ [π,32π] ,
    	     with ω^-=π, 32π atΩ̃ =0,1 .

The ranges and boundary values are interchanged for even k.
See the top row of fig. <ref>.


Let m≥ 1, where 0<δ = 1/(2m+1)≤ 1/3, and consider Ω̃_m ≤Ω̃≤ 2m+1 = Ω_m; see (<ref>).
Then we observe ranges
	
    ω∈ [-π2,π2] ,
    	     for odd k≥ 1 ,
    ω∈ [π2,32π] ,
    	     for even k≥ 2 .

Moreover, the implicit inverse function Ω = Ω(δ,ω) is strictly piecewise monotone in ω with unique local and global minimum
	
    -1<Ω̃_m -Ω_m =:
    	Ω_m ≤Ω≤ 0

and boundary values Ω =0 at ω≡±π2.
The minimal value Ω_m occurs at
	
    ω = ω   = π2Ω_m
    	     for    k odd ,
    ω = ω   = π +π2Ω_m
    	     for    k even .

The two explicit branches ω = ω^±(δ,Ω)  possess
strictly nonzero, but only locally bounded, derivatives with respect to Ω̃ or Ω, equivalently, in the domain 
Ω_m < Ω≤ 0. 
They
merge at Ω=Ω_m, where the discriminant of the quadratic B-relation (<ref>) vanishes.
See the two upper rows of fig. <ref>.





As always, we may consider odd k, without loss.
The explicit solutions ω = ω^± of (<ref>) follow directly from the first line of the 2-scale equation (<ref>).
Recall that nonnegative discriminants D in (<ref>), (<ref>) require |Ω̃sin (π2Ω)|=|Ω̃cos (π2Ω̃)|≤1; see also (<ref>).
Hence -1≤Ω≤0 implies -1≤Ω̃sin (π2Ω)≤ 0. 
In particular arccos(- Ω̃sin (π2Ω) )  ∈ [0,π2], for all m ∈ℕ_0.
This proves claim (<ref>) and the range claims (<ref>), (<ref>).
Moreover the functions ω=ω^±(Ω) are differentiable with bounded derivatives, except for the vertical tangent at the discriminant loci Ω=Ω_m, m ≥1.

For the inverse function Ω=Ω(ω), we study the monotonicity  claims, for all m,  and the minimizer claims, for m≥1, next.
Here we suppress δ, for a while.
Recall H(Ω,ω)=0, from (<ref>).
With the abbreviations
	
    S   :=sin(π2Ω) 
    	= -(-1)^mcos (π2Ω̃) ,
    
    	C   :=cos(π2Ω) 
    	= (-1)^msin (π2Ω̃) ,
    
    	s   :=sin(ω-π2Ω)  ,    
    	c:= cos (ω-π2Ω) ,

H and its partial derivatives, for odd k, are
	
    H      =   Ω̃S+c ,
       H_Ω      =    S+π2Ω̃C+π2s ,
       H_ω      =    -s  .

Elementary  arguments show that H=0 is a regular value of H.
Let Ω̇ denote the derivative of Ω(ω).
By implicit differentiation of H(Ω, ω)=0 with respect to ω, we obtain

    -H_ΩΩ̇ =  H_ω .

Suppose Ω̇ =0 at ω=ω.
Then
	
    0= H_ω =-s=-sin (ω-π2Ω)

at Ω = Ω(ω) implies ω =ω≡π2Ωπ.
Insertion into H(Ω,ω)=0 implies
	
    ± 1= -c= Ω̃S 
    	=-(-1)^mΩ̃cos(π2Ω̃) ,

using (<ref>).
Since S<0<Ω̃, we obtain c=+1.
This implies m≥1 and
	
    Ω̃ = Ω̃_m

as defined in (<ref>).
For m=0, in fact, 0≤Ω̃≤ 1 prevents any solution of (<ref>).
This proves the strong monotonicity claims for m=0, and completes the proof of lemma <ref>.




The functions B^±=B^±(δ, Ω) for the control parameter B in (<ref>), (<ref>),(<ref>) have the following properties.

For m=0, where δ=1 and 0<Ω̃= Ω+1<1, we have
	
    B^±(δ,Ω)=(-1)^k
    	sin^2(π2Ω)/cosω^±(δ, Ω).

Here ω^± have been defined in (<ref>).
Moreover B^+ increases strictly with respect to Ω, or Ω̃,
	
    B^+<0<B^-
    	     for     0<Ω̃<1 ;
       B^+=-1,0
    	     for     Ω̃=0,1 ;
       B^-=1, π/2
    	     for     Ω̃=0,1 .

See the bottom row of fig. <ref>.


Let m≥ 1, where 0<δ <1/(2m+1)≤ 1/3 and Ω̃_m≤Ω̃ < 2m+1=Ω_m; see (<ref>).
Then
	
    B^-<B^+<0    for     Ω̃_m<Ω̃< Ω_m ;
       B^± = -12(Ω̃_m^2-1)^-1Ω_msin(πΩ̃_m) ,
    	     for     Ω̃
    	=Ω̃_m ;
       B^±=0 ,
    	     for     Ω̃= Ω_m .

In terms of (<ref>), the branches B^± are parametrized over ω, instead of Ω, as follows:
	
    B^+= B(δ,ω) ,
    	     for     ω >ω:=
    	π2Ω_m ;
       B^-= B(δ,ω) ,
    	     for     ω <ω:=
    	π2Ω_m .

Moreover B^+ is strictly increasing with respect to ω, or Ω alias Ω̃.

For B^- and m≥ 1 we encounter a zero derivative ∂_ω B^- at ω= ω_*, Ω= Ω_* if, and only if,
	
    0=d(Ω,ω):=sin^2(π2Ω)sinω+
    	π2cosωsin(ω-πΩ) .

Critical points, and in particular the minimum, of ω↦ B (and of B^-) occur at certain ω= ω_*, Ω=Ω_* which satisfy
	
    πΩ_* < ω_* < 12πΩ_* <0 .

In particular ∂_ω B^-<0 for -π2<ω≤πΩ_*.
Moreover B>-1.





Again we consider odd k, without loss.
We suppress the parameter δ and differentiate (<ref>) with respect to ω, implicitly, analogously to the proof of lemma <ref>:
	
    Ḃ= -
    	(π2sin(πΩ)cosω·Ω̇+sin^2(π2Ω)sinω)/cos^2ω .

We invoke lemma <ref>.


Consider m=0, first.
Then the sign of the right hand side of (<ref>) is
	
    -sign cosω^± = ∓ 1

by (<ref>).
These signs agree with our definition (<ref>) of B^±, for m=0, because 0<Ω̃<1 implies B^+<B^-, there.
More specifically, we have shown B^+<0<B^-.


To settle monotonicity of B^+, for all m, we keep considering odd k, without loss.
We aim to show Ḃ^+>0 in the interior domain of definition.
We rewrite (<ref>) as
	
    Ḃ^+= -S (π Ccosω·Ω̇+Ssinω)/cos^2ω ,

with C>0>S for -1<Ω <0; see (<ref>) for this notation.
We also recall |ω |<π2, cosω >0, Ω̇>0 and s=sin(ω-π2Ω)>0, for B^+ and odd k; see the proof of lemma <ref>.
Therefore (<ref>) implies Ḃ^+>0, if sinω≤ 0.
It remains to show interior positivity of Ḃ^+ for sinω>0, i.e. for 0<ω <π2.

Suppose Ḃ=0.
We derive the relation (<ref>) at such a zero, first.
We differentiate χ_0=0 in (<ref>), (<ref>) with respect to ω, implicitly, to obtain
	
    0=Ω̇ +e^iω-
    	π2B e^iπΩ Ω̇ .

We have used the assumption Ḃ =0 here.
We multiply (<ref>) by the complex conjugate coefficient of Ω̇ and take imaginary parts to eliminate the derivative 
Ω̇:
	
    0    = Im(e^iω(1-π2Be^-iπΩ))=
       = sinω-π2Bsin(ω-πΩ) .

Substitution of B from (<ref>) and multiplication by the resulting denominator sin^2(π2Ω) proves claim (<ref>).

We can now prove interior positivity Ḃ^+>0 for the remaining case 0<ω<π2.
Suppose  Ḃ^+=0, indirectly. 
We then use  trigonometric addition and the abbreviations of (<ref>) to rewrite (<ref>) as 

    0=d(Ω,ω)=S^2sinω + π2cosω (-Sc+Cs).

To reach a contradiction, we check positivity of each individual term.
Our assumption 0<ω<π2 implies sinω>0, cosω>0.
Moreover, -1<Ω<0 implies S^2>0, C>0.
By subtraction, we also obtain s>0 because 0<ω-π2Ω<π.
It only remains to check positivity of -Sc.
Indeed H=0 in (<ref>), (<ref>) implies -Sc = Ω̃ S^2 >0, since S^2>0 and Ω̃>0.
Hence all terms on the right hand side of (<ref>) are strictly positive.
This contradiction establishes Ḃ^+>0 in all cases.

To prove claim (<ref>), we first observe that ω <π2Ω_m < π2Ω<0 holds all along the ω^--branch, because -π2<ω<ω = π2Ω_m defines the domain of B^-; see lemma <ref>.
To show ω_* > πΩ_* at Ḃ=0, indirectly, suppose -π2<ω_* ≤πΩ_*<0.
We then claim d(Ω_*,ω_*)<0.
Indeed
	
    d(Ω_*,ω_*)    =
    	S_*^2sinω_*+
    	π2cosω_*sin(ω_*-πΩ_*)≤
       ≤ S_*^2sinω_* <0 .

This contradiction proves (<ref>).

It remains to prove B>-1, for m≥1. 
By continuity of B, and because B^±=0 at Ω̃=Ω_m=2m+1, it is sufficient to show B^-≠-1, indirectly.
Suppose B^-=-1. 
Then S^2=cos(ω), by (<ref>).
Therefore (<ref>) implies

    0=H/S     = Ω̃+cos(ω-π2Ω)/S=
        = Ω̃+(C cosω+S sinω)/S=
        = Ω̃+CS + sinω≥Ω̃-2>0 ,

since Ω̃ > Ω_m-1=2m≥ 2.
This proves B>-1, and completes the proof of the lemma.






§ CONTROL-INDUCED HOPF BIFURCATION




In absence of control, i.e. in the limit b=2ε B→±∞, the original delay equation (<ref>) possesses a trivial simple Hopf eigenvalue
	
    μ=iω̃= iω_k=iε^-1

of the characteristic equation (<ref>), at the original parameter
	
    λ = λ_k=-(-1)^k/ε .

The (scaled) control parameter B induces further purely imaginary Hopf eigenvalues μ=μ_R+iω̃, μ_R=0, of the 2-scale characteristic equation (<ref>).
We fix and suppress δ=Ω_m^-1 in this section and rewrite (<ref>) – (<ref>) as
	
    0=
    	ψ(μ,ε,B):   =
    	-ε Bμ-(-1)^k B e^-μ+
    	12(1+e^-πεμ)=
       =-ε Bμ_R-iBΩ̃-
    	(-1)^k B e^-μ_R-iω̃+
    	12(1+e^-πεμ_R-iπΩ̃) .

As before we have abbreviated Ω̃=Ω_m+Ω = δ^-1+Ω here, and μ=μ_R+iω̃ with ω̃≡ω2π, εω̃=Ω̃.
We also recall the hashing relation (<ref>), i.e.
	
    0=
    	h(ω̃, Ω,ε)=
    	-Ω̃+εω̃=
    	-Ω+ε(ω+π2(1-(-1)^k-(-1)^m)-2π j) ,

for-π2≤ω<32π; see lemma <ref>.
In other words, Hopf bifurcation is governed by the three real equations (<ref>), (<ref>) for vanishing (Ψ,h) ∈ℂ×ℝ, in the five not quite independent real variables (μ, Ω, ε,B) ∈ℂ×ℝ^3.


In trapping lemma <ref> we observe absence of nontrivial eigenvalues μ = μ_R+iω̃ with imaginary parts ω̃≡±π22π.
This traps imaginary parts in eigenvalue strips:
an old and efficient idea already present in <cit.>.
It establishes the crucial sequences of Hopf bifurcations at scaled control parameters
	
    B=B_m,j^± ,

where m,j label specific strips of the Hopf eigenvalues μ =iω̃, with Ω_m-1 < Ω̃ = εω̃<Ω_m=2m+1.

We also observe how eigenvalues μ cannot appear from, or disappear towards, Re μ=+∞.
Proposition <ref> examines eigenvalues at vanishing control b=2Bε=±∞ to establish simplicity of eigenvalues, in each strip.
With some estimates for Jacobian determinants involving Ψ and h, in proposition <ref>, we establish the transverse crossing directions of the simple Hopf eigenvalues μ, as B increases through B_m,j^± <0; see the central crossing theorem <ref> of the present section.
In fact we observe a gain of stability, i.e.  decrease of the unstable dimensions E=E(B) by 2, at B=B_m,j^+<0, and destabilization at B_m,j^-<0.
Corollary <ref> concludes that Pyragas stabilization is impossible, for B>0.
Corollary <ref> concludes that unstable eigenvalues in the complex (m,j)-strips are present if, and only if,
	
    B_m,j^- < B< B_m,j^+ .

Corollary <ref> studies the case m=0 of slow frequencies 0<Ω̃ = εω̃<1, as well as the simplest case m=1.
It concludes stability of the strip m=0, j=1 for B_0,1^+<B<0, but instability of the strip m=1, j=1 for B_1,1^-<B<0.
With the orderings of B_m,j with respect to j, for each fixed m≤ 1, as collected in proposition <ref> we arrive at the conclusion of the present section, in corollary <ref>:
the region P:= { B | B_0,1^+< B<B_1,1^-} is a nonempty Pyragas region, provided that
	
    B_m,j_m+1^+ <B_0,1^+< B_1,1^-<B_m,j_m^-

holds for j_m:= [  (m+1)/2]  and all m≥1.
The delicate ordering (<ref>) will only be established in sections <ref> and <ref> below.




For any fixed ε >0, consider strictly complex eigenvalues μ = μ_R +iω̃, i.e. solutions μ∈ℂ∖ℝ of the characteristic equation (<ref>).


(i) Assume
	
    μ_R ≥ 0>B .

Then the only eigenvalues μ=μ_R+iω̃ such that
	
    0<ω̃ = Im μ≡π2π

are the trivial eigenvalues μ = iω_k̃=i(k̃+12)π at ε= ω_k̃^-1, where k̃∈ℕ_0 has the same parity as k.


(ii) Assume
	
    ε=ω_k^-1, B≠ 0 .

Then the only eigenvalue μ =μ_R+iω̃ such that
	
    ω̃ = ω_k=(k+12) π

is the algebraically simple eigenvalue μ=iω_k.


(iii) Fix ε= ω_k^-1, for some k∈ℕ_0, and fix any constant K >1.
Consider any sequence of (scaled) control parameters B_n and nontrivial eigenvalues μ_n=μ_R,n+iω̃_n ≠ iω_k such that
	
    0    ≤μ_R,n ,
    
    	1/K    ≤ω̃_n≤ K ,   and
    
    	B_n    → 0 .

Then |μ_n| remains bounded.
Moreover, for any convergent subsequence μ_n there exists a positive integer m such that
	
    εlimμ_n =
    	limε iω̃_n   =
    	lim i Ω̃_n = iΩ_m=i(2m+1) ,  and
    lim ω̃_n  (mod  2π)    ≡3π2  for   k+m   even,
    
    	3 π2  for   k+m  odd.





To prove claim (i), suppose μ=μ_R+iω̃ with ω̃=ω_k̃:= (k̃+12)π, for some nonnegative integer k̃.
We have to conclude μ_R =0 and ε= ω_k̃^-1.


Abbreviating εω̃ =: Ω̃, we decompose the characteristic equation (<ref>) into real and imaginary parts at ω̃ = ω_k̃ to obtain
	
    -ε Bμ_R+ 
    	12
    	(1+ e^-πεμ_Rcos(πΩ̃))=0 ;
    	
    
    	-BΩ̃ +(-1)^k+k̃Be^-μ_R-
    	12e^-πεμ_Rsin(πΩ̃)=0 .

The real part (<ref>) can be solved for Bμ_R as
	
    0≥ Bμ_R=
    	12ε^-1(1+e^-πεμ_Rcos (πΩ̃))≥ 0 .

Indeed, the right inequality follows because we have assumed μ_R ≥ 0 in (<ref>), and the left inequality follows from our assumption B<0.
In particular we conclude
	
    μ_R =0    and    cos(πΩ̃)=-1 .

Insertion of μ_R=0 and cos (πΩ̃)=-1, sin(πΩ̃)=0 in the imaginary part (<ref>) of the characteristic equation  (<ref>) then implies
	
    0> BΩ̃ =(-1)^k+k̃ B e^-μ_R=(-1)^k+k̃B .

This proves 1 = Ω̃ = εω_k̃ and k≡k̃ (mod 2), as claimed.


To prove claim (ii), we first note
	
    Ω̃ = εω̃ = εω_k=1 ,
    	    exp (-iω̃)=-(-1)^ki ,

by assumptions (<ref>), (<ref>).
For the imaginary part (<ref>) of the characteristic equation (<ref>) at μ= μ_R+iω̃ this implies
	
    0= -B+Be^-μ_R ,

i.e. μ_R=0.


To show algebraic simplicity of the resulting trivial eigenvalue μ = iω_k, we differentiate the right hand side of the characteristic equation (<ref>) with respect to μ, there.
A vanishing derivative would require
	
    0=-ε B-iB+π2ε .

This contradiction proves algebraic simplicity of the trivial Hopf eigenvalue μ = ± iω_k at ε= ω_k^-1, for any B.


To prove claim (iii), we rewrite the characteristic equation (<ref>) in the form 

    b_n(εμ_n+(-1)^kexp(-μ_n))=1 + exp(-πεμ_n).

To show |μ_n| remains bounded, indirectly, we first suppose
	
    |μ_n|→∞ ,

for some subsequence.
In (<ref>) we have assumed bounded imaginary parts ω̃_n = Im μ_n.
Therefore (<ref>), (<ref>), and (<ref>) imply
	
    μ_R,n = Re μ_n → + ∞ .

From (<ref>) we then obtain, more precisely,
	
    ε lim  b_nμ_n = 1 .

Taking imaginary parts of (<ref>) and passing to a convergent subsequence of ω̃_n, we also obtain
	
    lim ω̃_n    =
    	limμ_nε b_nμ_n Im 
    	(-(-1)^k b_ne^-μ_n+e^-πεμ_n)=
       =
    	limμ_n Im (-(-1)^k b_n e^-μ_n+e^-πεμ_n)=0 .

This contradicts our lower bound (<ref>) on ω̃_n.
Therefore the sequence |μ_n| remains uniformly bounded.

Next we divide the original unscaled characteristic equation (<ref>) for μ = μ_n by εμ_n-i≠0, at fixed ε= ω_k^-1, to obtain
	
    b_nεμ_n+(-1)^kexp (-μ_n)/εμ_n-i=
    	ε1+exp (-πεμ_n)/εμ_n-i .

Both sides extend to entire functions of μ_n.
Since (<ref>) is entire, and |μ_n| remain bounded, b_n → 0 then implies
	
    ε lim 1+exp (-πεμ_n)/εμ_n-i=0 .

The denominator cancels the simple zero εμ_n = iΩ_0=i of the numerator.
The remaining zeros εμ_n=i Ω_m=i (2m+1) of the numerator prove claim (<ref>), and the trapping lemma.



We now recall the location of eigenvalues μ of the characteristic equation (<ref>) in the limit B→±∞ of vanishing control.
This is well-known material; see e.g. <cit.>.
We include a short proof for the convenience of the reader.



Let ε > 0.
Consider eigenvalues μ∈ℂ at vanishing control B =±∞, i.e. solutions of
	
    0= εμ +(-1)^ke^-μ .

Then the following claims (i) – (iv) hold true.


(i) If 0< Im μ≡π2π, then
	
    Re μ_R=0 , μ=iω_k̃ =i(k̃+1/2)π  and  ε= ω_k̃^-1 ,

where k̃∈ℕ_0 has the same parity as k.


(ii) At ε=1/ω_k the eigenvalue μ=iω_k is algebraically simple.
The local continuation μ=μ(ε) satisfies
	
    d/dεμ(0) = -1/ε(1+ε^2)(1+iε) .



(iii) For 1/ω_k<ε < 1/ω_k-2 the nontrivial complex eigenvalues μ∈ℂ∖ℝ with Re μ≥ 0, Im μ>0 are given by [k/2] algebraically simple eigenvalues μ_0,j, j=1, …, [k/2], one in each strip
	
    0    < Re μ_0,j ;
    ω_k-2jπ   < Im μ_0,j < ω_k -2jπ +π2 .



(iv) For even k, there do not exist real eigenvalues μ≥ 0. 
If k is odd, the only real eigenvalue μ≥ 0 is the algebraically simple eigenvalue defined by the unique positive solution of
	
    εμ = e^-μ .








Let μ = μ_R+iω̃.
To prove claim (i), we assume ω̃= ω_k̃.
We take real parts of the complex characteristic equation (<ref>) to see that cosω̃=0 implies μ_R=0.
Taking imaginary parts,
	
    0    = εω_k̃-(-1)^k 
    	e^-μ_Rsinω_k̃=
       = εω_k̃-(-1)^k+k̃

shows the remaining claims of (i).


Claim (ii) follows by implicit differentiation of (<ref>) with respect to ε:
	
    0=μ+(ε-(-1)^ke^-μ)μ'

where μ' abbreviates the implicit derivative with respect to ε.
Inserting ε = ω_k^-1 and μ=iω_k proves claim (ii).


Claim (iii) follows by global continuation of the simple Hopf eigenvalues μ=μ(ε) with respect to decreasing ε, alias increasing |λ |.
We may proceed by induction on k.
For large ε, i.e. for small λ alias small rescaled delay, already <cit.> observed Re μ→-∞ for all complex eigenvalues.
At ε= ω_k̃^-1, with k̃ of the same parity as k and j:= [k̃/2]+1, a simple Hopf eigenvalue μ= μ_0,j:= iω_k̃ appears on the imaginary axis.
By property (ii) it progresses, locally, for decreasing ε, into the strip (<ref>).
By property (i) that simple eigenvalue μ_0,j(ε) can never leave that trapping strip again, because Re μ_0,j>0 remains bounded above for ε>0 bounded below.
By standard complex analysis, therefore, each μ_0,j(ε) remains simple and continues globally in its strip, for 0<ε<ω_k̃^-1.
The last value k̃ encountered for ε> ω_k^-1 is k̃=k-2.
This proves claim (iii).


Claim (iv) on real eigenvalues μ has been addressed in lemma <ref> already.
This proves the proposition.



The following proposition collects the partial derivatives of the characteristic function
	
    ψ (μ, ε,B):=
    	-ε B μ-(-1)^k Be^-μ +12
    	(1+e^-πεμ)

introduced in (<ref>).



The partial derivatives of ψ = ψ(μ,ε,B) satisfy
	
    ψ_μ   = 
    	-ε B +(-1)^k Be^-μ-
    	π2ε e^-πεμ ;
    	
    ψ_ε   =
    	-μ(B +π2 e^-πεμ) ;
    	
    
    	Bψ_B    =
    	-ε B μ-(-1)^kBe^-μ= 
    	ψ-12(1+e^-πεμ) .

At imaginary eigenvalues μ= iω̃, where ψ (μ,ε;B)=0, and with the abbreviation Ω̃:= εω̃, we also obtain the Jacobian determinant
	
    -B ψ_(ε, B) = ω̃· 
    	(B+π2)·cos^2(π2Ω̃) .




The calculations of (<ref>) – (<ref>) are trivial.
For ψ=0 in (<ref>) we also obtain
	
    ψ_B= -12B(1+e^-πεμ) .

Since ψ∈ℂ≅ℝ^2, the Jacobian ψ_(ε, B) can be written abstractly as
	
    ψ_(ε, B) = [ Re ψ_ε Re ψ_B; Im ψ_ε Im ψ_B ]
    	= Im (ψ_ε·ψ_B) .

Insertion of (<ref>), (<ref>), ψ=0 at imaginary eigenvalues μ =iω̃ = iε^-1Ω̃ provides the Jacobian determinant
	
    -Bψ_(ε, B)   =
    	Im (ψ_ε· (-Bψ_B)) =
       =
    	12ω̃ Im ((Bi+π2
    	i e^iπΩ̃) (1+e^-iπΩ̃))=
       =
    	12ω̃ (B+π2)
    	(1+cos (πΩ̃)) .

This proves (<ref>) and the proposition.



With these lengthy preparations we can now address transverse crossings at the relevant Hopf eigenvalues, from two viewpoints.
Fix ε= ω_k^-1 and consider nontrivial Hopf eigenvalues μ= iω̃, 0<ω̃≠ω_k=(k+12)π, at control parameter B≠ 0.
Our viewpoint above was to study, equivalently,
	
    ψ(μ,ε, B) =0

for ψ defined in (<ref>); see (<ref>).


In section <ref> our viewpoint was slightly different; see (<ref>) and lemma <ref>.
Hashing with the shifted slow frequency
	
    Ω = Ω̃ -Ω_m =
    	εω̃-(2m+1)=
    	ε(ω +π2(1-(-1)^k-(-1)^m)-2π j) ,

-π2≤ω< 32π, ω≡ω̃2π, equivalently, we wrote the characteristic equation for the eigenvalue μ=iω̃ as
	
    0= χ_0 (δ, ω, Ω, B)=
    	Ω̃+(-1)^ki e^iω-
    	1Bsin(π2Ω) e^iπ2Ω ;

see (<ref>) – (<ref>).
In lemma <ref> we have described the solutions of (<ref>) by functions
	
    Ω = Ω(ω) ,

for any fixed δ = Ω_m^-1 and nonnegative integers m.
That description was completed with B= B^± (Ω) = B^± (ω, Ω(ω)) as	
	
    B=(-1)^ksin^2(π2Ω)/cosω=
    	(-1)^kcos^2 (π2Ω̃)/cosω ;

see (<ref>) and lemmata <ref>, <ref>.


We now combine these results, for ε= ω_k^-1, with the hashing (<ref>) to define the Hopf frequencies ω_m,j^±∈ (-π2, 32π) as the intersections of (<ref>) with (<ref>), i.e.
	
    ε(ω_m,j^± + π2(1-(-1)^k-(-1)^m)-2π j)=
    	Ω(ω_m,j^±) ,

for j=1, …, j_m^max. Here and below we restrict attention to the case Ω̃_m≤Ω̃≤Ω_m, i.e. Ω_m≤Ω≤ 0. 
Indeed, the opposite case of m≥ 1 and B^± >0 in (<ref>) will turn out irrelevant in corollary <ref> below.

We have to comment on the precise meaning of (<ref>), in view of lemma <ref>.
Consider the case m=0 first; see fig. <ref>.
Since the derivatives of the two branches ω=ω^±(Ω) in (<ref>) are bounded, their intersections with the near-vertical hashing lines ω̃=Ω̃/ε of slope 1/ε are transverse, for 0<ε≤ε_0 small enough. 
This provides two intersections ω=ω_0,j^±, one pair for each j, as indicated.

The cases m≥ 1 of fig. <ref> are slightly more involved. 
The strictly decreasing lower branch ω = ω^-(Ω),  Ω_m < Ω < 0, is characterized by
	
    -π2 < ω = ω^-(Ω) < ω = π2Ω_m < 0 .

This decreasing branch provides unique, transverse intersections ω = ω_m,j^- with the increasing hashing lines.
The strictly increasing upper branch ω = ω^+(Ω),  Ω_m < Ω < 0, on the other hand, may exhibit non-transverse, and even multiple, intersections ω∈ω_m,j^+ with the near-vertical hashing lines, for near-minimal Ω≳Ω_m.
To simplify our presentation, mostly, we will think of intersection points ω = ω_m,j^+, rather than intersection sets ω∈ω_m,j^+.
Of course we will proceed with the appropriate care to address the general set case whenever necessary.
Eventually, we will be able to exclude cases where the minimal intersection ω_m,j^- of the hashing line (m,j) belongs to the upper branch ω^+(Ω);
see lemma <ref> below.

With these cautioning remarks in mind, we may proceed, mostly, with the additional requirement
	
    ω_m,j^- <ω_m,j^+ ,

for m≥ 1.
The only exception may arise by a tangency of the hashing, at maximal j=j_m, m≥ 1, where ω_m,j^-=minω_m,j^+ . 
From lemma <ref>, and in particular from (<ref>), (<ref>) we also recall the boundary values
for the functions ω^±(Ω) at Ω = Ω_m and Ω = 0.
Let us keep in mind how ω_m,j^± come with their shifted and slow variants
	
    ω̃_m,j^±, Ω̃_m,j^±,Ω_m,j^±

as an entourage; see (<ref>).
These also define the control parameters
	
    B =B_m,j^± =(-1)^kcos^2(π2Ω̃_m,j^±)/
    	cosω_m,j^±

where the nontrivial, control-induced Hopf bifurcations with eigenvalues μ = ± i ω̃_m,j^± actually occur.



With the above notation, the following holds, at ε= ω_k^-1.


(i) The values B_0,j^± of the control parameter enumerate all nontrivial Hopf bifurcations with eigenvalues μ = ± iω̃ of frequencies 0<ω̃<ω_k=(k+12)π.


(ii) The values B_m,j^± with m≥ 1 enumerate all nontrivial Hopf bifurcations with eigenvalue frequencies ω̃ > ω_k and strictly negative control parameter B<0.


(iii) All enumerated Hopf eigenvalues are algebraically simple.


(iv) The local continuations μ = μ_m,j^-(B) of all enumerated Hopf eigenvalues iω_m,j^- = μ_m,j^-(B_m,j^-) cross the imaginary axis transversely with
	
    d/dBRe μ_m,j^-(B) > 0

at B=B_m,j^-.
At B = max B_m,j^+ , generated by the frequencies ω_m,j^+ , that unstable eigenvalue recovers stability, at the latest.



The proof of claims (i) and (ii) follows from our detailed analysis of the characteristic equation χ_0=0 in section <ref>; see in particular lemma <ref>.


To prove simplicity of Hopf eigenvalues, (iii), we partially substitute the explicit expression (<ref>) for B into ψ_μ of (<ref>), and take real parts:
	
    Re ψ_μ   =
    	-ε B+(-1)^kBcosω̃-
    	π2εcos(2π2Ω̃)=
       =
    	-ε B+cos^2(π2Ω̃)-
    	π2ε(2cos^2(π2Ω̃)-1)=
       =
    	ε(π2-B)+
    	(1-πε)cos^2(π2Ω̃)>0

at πε=πω_k^-1=1/(k+12), k≥ 1, and for all enumerated B.
Indeed, by lemma <ref>, the only exception to B≤π2 arises for B=π2 at the excluded trivial eigenvalue with frequency ω̃=Ω̃/ε=1/ε =ω_k.
This proves simplicity claim (iii).






For m≥1, our proof of the remaining crossing and (de)stabilization claims (iv) will be based on the following three ingredients.
We will first invoke the implicit function theorem to show that the local continuation map
	
    (ε, B) ↦μ=μ(ε,B)

is an orientation preserving diffeomorphism, near ε=ω_k^-1 and the enumerated Hopf eigenvalues μ=iω_m,j^± at B=B_m,j^±.
In a second step we will then show
	
    ε_ω(iω_m,j^-)<0

for the partial derivatives, with respect to ω, of the local inverse function (ε,B)=(ε(μ), B(μ)) to (<ref>), at μ=iω_m,j^-. The third ingredient describes the necessary adaptations at ω_m,j^+.

We first show how claims (<ref>) and (<ref>) imply the crossing direction

    Re μ_B>0 ,

for the partial derivative of μ(ε, B) with respect to B at μ=iω_m,j^-.
Indeed consider the oriented Hopf curve  ω̃→(ε(iω̃), B(iω̃)) in the (ε,B) plane.
See fig. <ref>.
By the orientation preserving transformation (<ref>), the region Re μ<0 lies to the left of the Hopf curve.
By (<ref>), the tangent to the Hopf curve at ω=ω_m,j^- points strictly to the left of the vertical B-axis at the fixed value ε=ε_k.
Therefore, the B-axis crosses the Hopf curve transversely, at ω=ω_m,j^-, and into the unstable region Re μ>0, for increasing B.
Thus the diffeomorphism (<ref>) implies the crossing direction (<ref>).

The cases B=B_m,j^+ can be treated analogously, with a little extra care.
In the case of a single transverse crossing of the hashing line (m,j) with the upper branch ω=ω^+(Ω), at ω=ω_m,j^+, we now have ε_ω(iω_m,j^+)>0. 
Therefore, the tangent to the Hopf curve at ω=ω_m,j^+ now points strictly to the right of the vertical B-axis at the fixed value ε=ε_k.
The previous arguments then show Re μ_B<0, i.e. stabilization towards increasing B.
In case of multiple crossings, possibly involving tangents, we can prove stabilization from the last crossing (or tangency) onwards, as claimed in (iv), via generic approximation by an odd number of transverse crossings of the hashing line with the upper branch ω=ω^+(Ω). 
Put simply, destabilization occurs whenever the increasing hashing lines in the top rows of fig. <ref> enter the interior region of the 2-scale relation Ω = Ω(δ,ω), and stabilization ensues as soon as the hashing lines leave towards the exterior region; see lemma <ref>.

To prove claim (iv) for m≥1 it therefore remains to verify claims (<ref>) and (<ref>).
We will address the analogous, but simpler, case m=0 at the end of the proof.

To verify the orientation claim (<ref>) we invoke the implicit function theorem for ψ(μ,ε, B)=0; see (<ref>), (<ref>).
Indeed the Jacobian determinants, which determine the local orientations, satisfy
	
    ψ_μ·μ_(ε,B)=
    	(-ψ_(ε,B))=
    	(ψ_(ε,B)) .

Our enumeration of cases B=B_m,j^± for m≥ 1 above has skipped any positive B^± of lemma <ref>, (<ref>).
By lemma <ref> we know -1<B^-≤0.
Therefore proposition <ref>, (<ref>) asserts strict positivity of (ψ_(ε, B)) in (<ref>).
The enumerated Hopf eigenvalues μ=iω_m,j^± are simple zeros of the complex analytic characteristic function ψ∈ℂ, by claim (iii).
Therefore ψ_μ on the left is also strictly positive, by the Cauchy-Riemann equations.
This proves strict positivity of μ_(ε,B) and establishes the orientation claim (<ref>).


To determine the signs of the tangent partial derivatives ε_ω, as claimed in (<ref>), we recall the definition
	
    ε = Ω̃/ω̃ =
    	Ω̃(ω̃)/ω̃ ,

where Ω̃(ω̃)=Ω_m+Ω(ω) follows from the ε-independent characteristic equation χ_0(δ, ω, Ω, B)=0 at ω≡ω̃2π, for fixed δ= Ω_m^-1.
See (<ref>), (<ref>) and lemmata <ref>, <ref>.
Straightforward differentiation of (<ref>) with respect to ω̃>0 or ω yields
	
    ε_ω =
    	(Ω̇-Ω̃/ω̃)/ω̃
    	=(Ω̇-ε)/ω̃ ,

in the notation of lemma <ref>, where Ω̇ =dΩ/dω=dΩ̃/dω̃.
The definition of ω_m,j^- as the unique intersection of the hashing line (m,j) with the lower branch ω=ω^-(Ω) in (<ref>),  and (<ref>), (<ref>) show that (<ref>) implies the sign of ε_ω claimed in (<ref>).
See the two top rows of fig. <ref>. The case of a single transverse crossing at ω=ω_m,j^+ leads to ε_ω(iω_m,j^+)>0, analogously.
The required adaptations for multiple and/or non-transverse crossings have been described above.

It remains to address the case m=0, 0<Ω̃<1.
Consider ω_0,j^+, B_0,j^+<0 first; see lemmata <ref> – <ref>.
Here each crossing ω=ω_m,j^± is transverse and unique. 
Transformation (<ref>) remains orientation preserving, by (<ref>) and (<ref>), verbatim as for m≥ 1.
Furthermore (<ref>) implies ε_ω >0, for small enough 0<ε<ε_0, by an upper bound on the positive derivatives
	
    0<1/Ω̇=d/dΩω^± (Ω) < 1/ε_0

in lemma <ref>.
This shows claim (iv), (<ref>) at B=B_0,j^+<0, for m=0.
To show claim (iv), (<ref>) at B=B_0,j^->0, for m=0, we first note that (<ref>), (<ref>) now imply orientation reversal in (<ref>), because
	
    ψ_(ε,B) <0 .

The argument (<ref>), however, remains intact at ω^-(Ω).
This shows how the sign reversal claimed in (iv), (<ref>) remains valid for m=0, proving the lemma.



Figs. <ref> and <ref> already summarized our results, so far, separately for m=0 and for m≥ 1, B<0.
Consider the case m=0 first, i.e. slow Hopf frequencies 0<Ω̃ <Ω_0=1, alias -1<Ω=Ω̃-Ω<0.
The branch ω = ω^+(Ω) of Hopf frequencies ω≡ω̃2π provides
	
    k'=[k/2]

Hopf bifurcations at control parameters B= B_0,j^+, j=1,…,k'.
Hashing and strong monotonicity of B=B^+(Ω), lemma <ref>, imply
	
    -1<Ω_0,k'^+   < … <
    	Ω_0,2^+<Ω_0,1^+<1 ;
    
    	-1<B_0,k'^+    <… <B_0,2^+<B_0,1^+<0 .

By lemma <ref>, unstable eigenvalues μ=μ_R+iω̃ cannot cross any of the lines ω̃≡π2π, for B<0.
By proposition <ref>, each of the k'=[k/2] resulting strips
	
    Re μ>0 ,     0<ω_k-2jπ<ω̃<ω_k
    	-2jπ+π ,

j=1,…,k', contains exactly one simple eigenvalue μ_0,j inherited from B=-∞.
By theorem <ref> and analytic continuation, this simple eigenvalue persists as B increases, until it disappears into Re μ<0 by simple transverse Hopf bifurcation at
	
    B=B_0,j^+ ,    ω̃= ω_0,j^+ .

Indeed ω̃=ω_0,j^+ belongs to the same strip (<ref>), for each j=1,… , k'.
For even k=2k', this eliminates all unstable eigenvalues generated at B=-∞, once
	
    B_0,1^+<B<0 .

For odd k=2k'+1, the same statement remains true, because -1<B<0 renders the additional real eigenvalue stable; see lemma <ref> and corollary <ref>.
Note -1<B_0,1^+ here, by lemma <ref>; see also fig. <ref>.
These remarks prove the following corollary.



Let ε=1/ω_k and assume B<0.
Then μ_R<0 for any eigenvalue μ=μ_R +iω̃ with 0≤ω̃<ω_k, if and only if (<ref>) holds.




We study B>0 next.
In corollary <ref> we have already observed instability, by parity due to the presence of a real eigenvalue μ>0, in case k was odd.
Let us therefore consider even k=2k'.
At ε=1/ω_k and B=+∞ we encounter the same k' unstable simple complex eigenvalues μ_0,j , one in each of the k' strips (<ref>), as before.
This time, however, only k'-1 simple transverse Hopf bifurcations at B=B_0,j^->0 offer their assistance for stabilization by decreasing B>0.
Indeed B_0,j^- cancels the instability of μ_0,j+1, for j=1, … ,k'-1, but μ_0,k' remains unstable for all B>0.
This proves the following corollary.



Let ε =1/ω_k and assume B>0.
Then there exists an unstable eigenvalue μ, i.e. Re μ>0.
For odd k, the unstable eigenvalue can be taken to be real.
For even k ≥ 2, the unstable eigenvalue μ = μ_R +iω̃ can be taken to be strictly complex with
	
    0<ω_k-2π<ω̃<ω_k-π .

In particular, there does not exist any region of Pyragas stabilization (near Hopf bifurcation) for control parameters B>0.




Henceforth we restrict attention to the remaining case B<0.
We fix ε=1/ω_k.
All unstable real eigenvalues, or complex eigenvalues μ =μ_R +iω̃ with 0<ω̃ <ω_k come from B=-∞ and have been taken care of in corollary <ref>.
By the trapping lemma <ref> for imaginary parts, all remaining changes of stability must arise from the simple transverse Hopf bifurcations at
	
    B=B_m,j^± ,   m≥ 1 ,

as enumerated in theorem <ref>.
Note how the imaginary parts ω̃ >0 of any unstable eigenvalues μ=μ_R+iω̃ induced by these Hopf bifurcations are confined to the disjoint strips
	
    ω̃ = ω +2π(km+[(k+1)/2]+
    	[(m+1)/2]-j)

where π2<ω <3π2 for even k, and -π2<ω <π2 for odd k.
This follows from hashing (<ref>), (<ref>) at Hopf bifurcation frequencies ω̃ = ω̃_m,j^±, and persists with instability of μ, by trapping lemma <ref> of imaginary parts.
In particular, the strips are disjoint, for different (m,j), and B_m,j^± generate frequencies ω̃_m,j^± which remain in the same strip.
See also fig. <ref>.
This proves the following corollary.



Let 0<ε =1/ω_k≤ε_0 be sufficiently small, and assume B<0.
Then there exists an unstable eigenvalue μ, i.e. Re μ >0, if at least one of the following conditions holds:
	
    B<B_0,1^+    <0 ,    or
    
    	B_m,j^-<B    < min B_m,j^+≤0,

for some m≥ 1 and some j enumerated in theorem <ref>.
Here we define B_m,1^+:=0 for odd m.




We have implicitly excepted hashing tangencies Ω_m,j^-= minΩ̃_m,j^+ in equation (<ref>).
Indeed this case corresponds to a nontransverse Hopf point, at (scaled) frequency Ω̃_m,j^- = minΩ̃_m,j^+, from the stable side. 
This does not contribute to the strict unstable dimension E(B).

Hence we may assume Ω̃_m,j^- < Ω̃_m,j^+.
Then hashing (<ref>) implies B_m,j^-<B_m,j^+.
Indeed this follows from strong monotonicity of B^+(Ω) in case Ω_m,j^±= Ω(ω_m,j^±) with ω_m,j^±≥ω; see lemma <ref>.
If ω_m,j^-<ω<ω_m,j^+, then we reach the same conclusion, again by strong monotonicity of B^+(Ω̃) and lemma <ref>:
	
    B_m,j^-=B^-(Ω_m,j^-)<B^+(Ω_m,j^-)<
    	B^+(Ω_m,j^+)=B_m,j^+ .



More systematically, these arguments are collected in the following proposition.



Let ε=1/ω_k≤ε_0 be sufficiently small.
For any m ≥ 1 consider the enumeration of Hopf bifurcation parameters B_m,j^± of theorem <ref>.
Then
	
    B_m,j^- < B_m,j^+ <0 ,

except at a possible hashing tangency Ω_m,j^-= minΩ_m,j^+.
Moreover the series B_m,j^+ decreases strictly monotonically in j=1,…, j_m^max, for each fixed m≥ 1.





Claim (<ref>) has been proved in (<ref>).
Strict monotonicity of B_m,j^+= B^+(Ω_m,j^+) in j follows from strict monotonicity of Ω̃_m,j^+=ω̃_m,j^+/ε in the j-strips (<ref>) and from strict monotonicity of Ω↦ B^+(Ω) in lemma <ref>.
This proves the proposition.



We summarize the results of this section in a final corollary.
Define
	
    j_m:= [(m+1)/2] ,

for integer m≥ 1.
In the following sections we will show, for small enough 0<ε =1/ω_k≤ε_0, that B_0,1^+ is unique, and
	
    max B_m,j_m+1^+ <B_0,1^+<0    for   j_m+1≤ j_m^max ,

i.e. as long as Ω_m,j_m+1^+ exists.
The maximum is taken over all m≥ 1.
See (<ref>) for the delimiter j=1, … , j_m^max of the enumeration B_m,j^±.
On the other hand, we will also show
	
    B_0,1^+    <B_1,1^-<0 ,   and  
    
    	B_1,1^-    ≤ B_m,j^-<0 ,  for all m≥ 1  and 
    	j=1, … , 	min{ j_m,j_m^max} .

This identifies the Pyragas region 𝒫 as follows.



Let 0<ε=1/ω_k≤ε_0 be chosen small enough and assume the orderings (<ref>) – (<ref>), for all m≥ 1.
Then the nonempty Pyragas region
	
    𝒫 = { B<0 |   B_0,1^+<B<B_1,1^-}

is the only region of control parameters B=12b/ε in the delay equation (<ref>), such that Pyragas stabilization succeeds for the Hopf bifurcation at λ = λ_k=(-1)^k+1ε^-1.





By corollary <ref>, instability prevails for all B>0.
By corollary <ref>, instability holds for B<B_0,1^+, and for B_1,1^- <B<0.
It therefore remains to show that
	
    (B_m,j^-  , max B_m,j^+)∩ (B_0,1^+, B_1,1^-)= ∅ ,

for all m≥ 1 and j=1, … , j_m^max.


Strong monotonicity of B_m,j^+ with respect to j, as in proposition <ref>, and assumption (<ref>) imply
	
    max B_m,j^+ ≤max B_m,j_m+1^+ < B_0,1^+

for all m≥ 1 and j>j_m.
This establishes claim (<ref>) for j_m <j≤ j_m^max, provided that j_m <j_m^max.


It remains to show claim (<ref>) for 1≤ j≤min{ j_m, j_m^max}.
In this case we invoke assumption (<ref>) to conclude an empty intersection (<ref>), again.
Since the Pyragas region (<ref>) is nonempty, by assumption (<ref>), this proves the corollary.








§ LOCALLY UNIFORM EXPANSIONS IN Ε = 1/Ω_K




To locate and understand the Pyragas region
	
    𝒫 = { B | B_0,1^+ < B < B_1,1^-} ,

in the limit ε= 1/ω_k → 0 of Hopf bifurcations with large unstable dimensions k, it remains to establish the precise locations of the control induced Hopf bifurcations B_m,j^± relative to the gap (<ref>).
See assumptions (<ref>) – (<ref>) of corollary <ref>.
In the present section we accomplish this task, by expansions with respect to small ε, for arbitrarily bounded m=1,… , m_0, alias δ≥δ_0:= 1/Ω_m_0=1/(2m_0+1).

To be precise, we first fix any m_0∈ℕ, alias δ_0>0.
In section <ref>, we will choose δ_0 sufficiently small.
In the present section we will then consider 0<ε≤ε_0=ε_0(δ_0) small enough for certain ε-expansions of B_m,j^± to hold, uniformly for all m≤ m_0 and j≤ j_m+1, j_m=[(m+1)/2].
Note how the derivatives of ω^±(Ω) remain bounded in the relevant region; see (<ref>).
Hence ω_m,j^± and B_m,j^± are defined uniquely by transverse intersections of the 2-scale characteristic equation (<ref>) with the hashing lines ω̃ = Ω̃/ε of (<ref>), (<ref>), in the present section.



In particular, the ε-expansions of B_0,1^+ and B_1,1^- will establish the expansions (<ref>) of b_k = 2ε B_0,1^+ and b_k=2ε B_1,1^-; see (<ref>).
The limit δ→ 0 of large m →∞ requires a different approach, and will therefore be deferred to the next section.


Our strategy has been outlined in (<ref>) – (<ref>).
We first solve the ε-independent 2-scale characteristic equation H(Ω,ω)=0 of (<ref>) for ω = ω (Ω) explicitly:
	
    ω = ω^±(Ω)= 
    	π2Ω±arccos (-Ω̃sin (π2Ω)) ;

see lemma <ref>.
Here and below we only consider odd k, without loss of generality.
Even k add π to ω^±.
Recall |ω | ≤π2, for odd k.
We also recall Ω̃ =1/δ +Ω and
	
    0≤ -Ω̃sin (π2Ω)≤ 1 ,

by the discriminant condition (<ref>).
Equivalently
	
    0≥Ω≥Ω_m .

Note bounded derivatives of ω(Ω), locally uniformly for 0≥Ω > Ω_m.
We suppress explicit dependence on δ, viz. m, in the present section.


Next we insert (<ref>) into the hashing relation (<ref>) of lemma <ref>:
	
    Ω = ε (ω^± (Ω) -a π)

with the abbreviation
	
    a=a_m,j=2j-1+12(-1)^m .

Here we have used that k is odd; the relevant modifications for even k cancel in (<ref>) and below.
By the implicit function theorem we can solve (<ref>) for
	
    Ω = Ω(ε, a)= Ω_m,j^± (ε) ,

uniquely, for small enough 0 <ε≤ε_0 = ε_0(δ_0).
Inserting the result into ω^± of (<ref>) and B of (<ref>) we obtain expansions
	
    ω   =   ω_m,j^±(ε)    :=
    	ω^± (Ω_m,j^± (ε)) ,
    
    	B    =    B_m,j^±(ε)    :=
    	-sin^2(π2Ω_m,j^±(ε))/
    	cosω_m,j^±(ε) .

We collect these straightforward expansions in the following proposition.




For any fixed δ_0=1/Ω_m_0 >0 consider 0< ε≤ε_0 = ε_0(δ_0) small enough.
We abbreviate the coefficients

    α^±_m,j := 4j + (-1)^m-2∓1

Then the expansions for Ω_m,j^±, ω_m,^±, and B_m,j^± with respect to ε are
	
    Ω_m,j^-(ε)   =
    	α^-_m,j(-π2ε+ 2m(π2ε)^2+… ) ;
    Ω_m,j^+(ε)   =
    	α^+_m,j(-π2ε- 2(m+1)(π2ε)^2+… ) ;
    	
     
    	ω_m,j^-(ε)   =
    	π2(-1+2m α^-_m,j π2ε+
    	… ) ;
    ω_m,j^+(ε)   =
    	π2(+1-2(m+1) α^+_m,j π2ε+
    	… ) ;
    	
     
    	
    	B_m,j^-(ε)    =
    	π4m α^-_m,j (-π2ε  + 12m(4m^2-α^-_m,j)(π2ε)^2+… ) ;
    
    	B_m,j^+(ε)    =
    	π4(m+1) α^+_m,j (-π2ε  - 12(m+1)(4(m+1)^2+α^+_m,j)(π2ε)^2+… ) .

The expansions for B^±, Ω^± hold for even and odd k, alike.
The expansions for ω^± are given for odd k.
For even k, we have to add π; see lemma <ref>.




We omit the obvious and tedious calculations.




In the setting of proposition <ref> we obtain the following expansions and inequalities, for 0< ε→ 0:
	
    B_1,1^-    =
    	-(π2)^2ε+
    	(π2)^3 ε^2+… ;
    	
    
    	B_0,1^+    =
    	-(π2)^2ε-3
    	(π2)^3 ε^2+… ;
    	
    
    	0>    B_m,1^-> B_m,2^- > B_m,3^-> …

For m ≥ 1 and j_m:= [(m+1)/2] we obtain the coefficients and expansions
	
    α^-_m,j_m   = 2m,     α^+_m,j_m+1 = 2(m+1) ;
    	
    
    	B_m,j_m^-    =
    	-(π2)^2ε+
    	(2m-1)(π2)^3 ε^2+… ;
    	
    
    	B_m,j_m+1^+    =
    	-(π2)^2ε-
    	(2m+3)(π2)^3 ε^2+…




The proof follows from proposition <ref>, by explicit evaluation.




The assumptions (<ref>) – (<ref>) of corollary <ref> are satisfied for all 1 ≤ m≤ m_0 and sufficiently small 0<ε≤ε_0.
In particular, expansions (<ref>), (<ref>) determine the ε-expansions (<ref>) of the boundaries b_k, b_k of the Pyragas region, in our main theorem <ref>.





For 1 ≤ m≤ m_0, claim (<ref>) follows by comparison of the expansion (<ref>) for B_m,j_m+1^+ with expansion (<ref>) for B_0,1^+.
Likewise (<ref>), (<ref>), and (<ref>), in this order, imply
	
    B_1,1^- ≤ B_m,j_m^- < B_m,j_m-1^- < … < B_m,1^- .

This proves claim (<ref>).
The remaining claim B_0,1^+ <B_1,1^- of (<ref>) is immediate by comparison of the respective expansions (<ref>) and (<ref>).
This proves the corollary.






For m≥ 1, there is an amusing characterization of the critical index j=j_m=[(m+1)/2] as a Pyragas switch index, in our expansions.
In fact our ε-expansions (<ref>) easily imply
	
    B_m,j+1^+< B_m,j^-      for     j=1, … , j_m ,
       B_m,j+1^+> B_m,j^-      for     j>j_m .

The interpretation is easy.
In the open interval B∈ I_m,j:= (B_m,j^-, B_m,j^+), consider the Hopf-induced unstable eigenvalue μ =μ_R+iω̃, μ_R>0, with frequency ω̃ trapped in the disjoint intervals designated by m,j; see lemma <ref>, and (<ref>), (<ref>) with |ω| ≤π2.
Then (<ref>) states that successive instability intervals I_m,j and I_m,j+1 open a stability window in between, for j ≤ j_m, but overlap for j>j_m.
See fig. <ref>.
In view of (<ref>), therefore, the stability window
	
    B_m,j_m+1^+ < B < B_m,j_m^-

is the very first stability window encountered, between the instability intervals I_m,j, m≥ 1, as j decreases from j_m^max to 1, i.e. as  B<0 increases towards zero from absent control at B =-∞.


The instability inherited from B =-∞, on the other hand, is only compensated for once B_0,1^+<B<0, by the stabilizing series of Hopf bifurcations in the m=0 series at B= B_0,j^+.
The instability interval I_1,1 =(B_1,1^-, B_1,1^+) which starts from B_1,1^- >B_0,1^+, however, extends all the way to B_1,1^+=0.
Therefore any stability windows between the intervals I_m,j of instability, for m≥ 1 and j from j_m+1 down to 1, remain ineffective. 
The only exception is the first such gap (<ref>) which, somewhat miraculously, contains the Pyragas region 𝒫 of (<ref>) by (<ref>), (<ref>), as established above.


We are somewhat amazed how all these first stability windows align, simultaneously for all resonance orders m, at the same first order location
	
    B =-(π2)^2ε + … ,

to contribute to Pyragas stabilization from B=B_0,1^+ to B=B_1,1^- by a second order effect.
We will see next how such gaps also arise in the remaining limit δ→ 0 of large m→ +∞.







§ ASYMPTOTIC EXPANSIONS FOR LARGE Ω_𝐦 = 1/Δ




In the previous section we have shown how the series of destabilizing Hopf intervals B∈ I_m,j:= (B_m,j^-, B_m,j^+) skip the Pyragas candidate
	
    𝒫=(B_0,1^+, B_1,1^-) ,

for bounded values m≤ m_0, accordingly bounded j≤ j_m^max, and small enough ε=1/ω_k ≤ε_0(δ_0).
In other words, we have established assumptions (<ref>) – (<ref>) of corollary <ref>, for arbitrarily bounded m≤ m_0 and 0<ε≤ε_0(δ_0).
In the present section we will complete this analysis, for large m>m_0.

In section <ref>, we had fixed m,j, to study ε-expansions of B_m,j^±.
We also noticed the central role of
	
    j=j_m:= [(m+1)/2] ,

where the Pyragas switch (<ref>) actually occurs, between B_m,j_m+1^+ and B_m,j_m^-.
This suggests a somewhat delicate parametrization of the relevant expansions by ε and δ:= 1/Ω_m = 1/(2m+1), both tending to zero in a region δ≥δ (ε).
Instead, we choose a parametrization of the problem by a rectangular region of (δ, ω).
The ε-independent relations Ω^±= Ω(δ, ω),  B^±= B(δ, ω) will provide expansions with respect to small δ.
At j=j_m for B_m,j_m^-, and at j=j_m+1 for B_m,j_m+1^+, we will also obtain expansions for
	
    ε = ε(δ, ω) ,

from the hashing relation (<ref>).
In other words, we determine ε such that B_m,j_m^- and B_m,j_m+1^+ arise at the frequency parameter ω, for some small δ.
The ε-expansions for the Pyragas boundary B_0,1^+(ε), B_1,1^-(ε), in section <ref>, did not depend on δ. 
They will allow us to compare the resulting locations, now, uniformly for small 0<δ≤δ_0.
This will prove our main theorem, via corollary <ref>.


We address the general case in lemma <ref>.
The limits ω→±π2, for B_m,j^±, will be considered in lemma <ref>.
These results address the cases where B_m,j_m^-, B_m,j_m+1^+ actually exist, and ω_m,j_m^-<πΩ_m,j_m^-.
The remaining cases where j_m ≥ j_m^max are prepared by expansions for min B, in proposition <ref>, and are resolved in lemma <ref>.


As in section <ref>, we may restrict our attention to odd k, |ω |≤π2.
See also (<ref>), (<ref>).
We mostly replace m by δ = 1/Ω_m= 1/(2m+1) and think of 0<δ≤δ_0 and 0<ε≤ε_0(δ_0) as small continuous, rather than discrete, real variables in all expansions.
For example
	
    2j_m    := 2[(m+1)/2]=12(δ^-1-(-1)^m) ,
    	
    
    	a_m,j_m   := 2j_m-1+12(-1)^m = 12(α^±_m,j_m±1) = 12δ^-1-1 ,

for the Pyragas switch index j=j_m of (<ref>), (<ref>) and the associated shift a=a_m,j_m in the hashing (<ref>), (<ref>).
See also (<ref>).


Our δ-expansions are based on section <ref>.
From lemmata <ref>, <ref> we recall how the 2-scale characteristic equation in the form
	
    0    = (δ^-1+Ω) sin (π2Ω) +
    	cos (ω-π2Ω) ,
    	
    
    	B    = -sin^2(π2Ω)/cosω ,

of (<ref>), (<ref>) gives rise to unique functions Ω= Ω(δ, ω), B=B(δ, ω), successively.
Insertion into the hashing
	
    ε = Ω/(ω-π a_m,j_m)

with (<ref>), (<ref>) then provides ε = ε^-(δ, ω), such that we encounter
	
    B=B_m,j_m^-(δ, ω)     at    ε= ε^-(δ, ω) ,

for -π2<ω≤π2Ω_m.
Similarly, the hashing
	
    ε = Ω/(ω-π a_m,j_m+1) ,

with a_m,j_m+1 = a_m,j_m+2 in (<ref>), provides ε = ε^+(δ, ω) such that we encounter
	
    B= B_m,j_m+1^+(δ, ω)    at    ε= ε^+(δ, ω)

for π2Ω_m ≤ω <π2.
The established ε-expansions of the Pyragas boundaries B_0,1^+(ε), B_1,1^-(ε) of corollary <ref>, (<ref>) and (<ref>), finally, at ε= ε^±(δ, ω), respectively, allow us to compare these δ-expansions as required in assumptions (<ref>) and (<ref>) of corollary <ref>.

Note how any possible nonuniqueness within the sets B_m,j^+ is remedied by our parametrization: we neither claim nor require injectivity  of ω↦ε_m,j^+(δ,ω), for fixed δ.



For odd k, and uniformly in |ω|≤π2, we obtain the following expansions with respect to small δ:
	
    Ω(δ,ω)    =
    	-cπ/2(δ-sδ^2+… )
    	
    
    	B(δ,ω)   =
    	-c(δ^2-2sδ^3+… )
    	
    ε^±(δ,ω)   =
    	(π2)^-2c(δ^2+ 
    	(ωπ/2∓ 2-s)δ^3+…)
    	
    
    	B_0,1^+(δ,ω)   =
    	-c(δ^2+(ωπ/2-2-s)δ^3+… )
    	
    
    	B_1,1^-(δ, ω)   =
    	-c(δ^2+(ωπ/2+2-s)δ^3+… )

Here we use the abbreviations c=cosω, s=sinω.
Omitted terms are of the first omitted integer order in δ.





To see why we obtain a uniform expansion of Ω, we rewrite (<ref>) in the equivalent form
	
    sin (π2Ω)=
    	-ccos(π2Ω)δ / (1+δ(Ω+s)) .

For 0≤δ≤δ_0, the implicit function theorem provides Ω= Ω(δ,ω), uniformly in ω.
Note the important uniform prefactor c=cosω in
	
    Ω = (Ω/sin(π2Ω)) ·sin (π2Ω)=-cδ·(… ) ,

because Ω/sin(π2Ω) is regular nonzero.
To obtain the specific expansion (<ref>) for Ω we solve (<ref>) for δ, expand for Ω at Ω =0, and invert the resulting series for
	
    δ   =
    	-sin(π2Ω)/(cos (ω -π2Ω)
    	+Ωsin (π2Ω))
       =
    	-π/2cΩ +… .

	

To obtain expansion (<ref>) for B=-sin^2(π2Ω)/c we insert (<ref>) into (<ref>) and observe cancellation of the denominator c.
Indeed the numerator picks up a factor c^2 from expansion (<ref>).
This proves (<ref>).
Of course (<ref>) applies to all B_m,j^± = B(δ, ω), identically, at the appropriate values ω.


The expansions (<ref>) for ε^± follow from the hashings (<ref>) – (<ref>), if we replace a_m,j_m and a_m,j_m+1 = a_m,j_m+2 by their appropriate δ-dependent values 12δ^-1± 1 from (<ref>).
The prefactor c remains inherited from Ω.


To prove claims (<ref>), (<ref>) it only remains to plug (<ref>) into the ε-expansions (<ref>) and (<ref>) for B_1,1^- and B_0,1^+.
Note how the new term of order δ^3, which roughly speaking corresponds to ε^3/2, arises from the leading term -(π2)^2ε^± +… alone.
This proves the lemma.




Let 0<δ≤δ_0 be small enough and consider odd k.
Let Ω, B, ε^±, B_0,1^+, B_1,1^- be expanded as in lemma <ref>.


Then we obtain the inequalities
	
    B_1,1^-(δ, ω)<B(δ,ω) < B_0,1^+(δ, ω) ,

for all 0<δ≤δ_0, |ω|<π2.
For j_m+1≤ j_m^max and at ε= ε^+(δ,ω) we conclude, in particular, 
	
    B_m,j_m+1^+ < B_0,1^+(δ,ω) ,

i.e. assumption (<ref>) of corollary <ref> holds. 
Likewise, for j_m≤ j_m^max and at ε= ε^-(δ, ω) we conclude 
	
    B_1,1^-(δ,ω)< B_m,j_m^- .



Under the additional assumption
	
    -π2 <ω_* ≤πΩ(δ,ω_*) ,

we can also assert -π2 <ω≤πΩ(δ,ω), for all -π2≤ω≤ω_*. Moreover
	
    ω↦ B(δ, ω)<0

is then strictly decreasing, for fixed δ and all -π2≤ω≤ω_*.

More specifically, suppose the additional assumption (<ref>) holds at Ω(δ,ω_*)= Ω_m,j_*^-, ω_* = ω^-(Ω_m,j_*^-), for some 1≤ j_*≤ j_m^max, m≥ 1.
Then
	
    B_m,j_*^-<… < B_m,1^-≤ 0

holds at ε= ε(δ,ω_*), for all 1≤ j≤ j_*.
For the above ranges of ε,δ, this establishes the Pyragas region 𝒫.





Claim (<ref>) may look somewhat paradoxical, at first sight.
To prove (<ref>), nevertheless, we first divide (<ref>) by c=cosω >0 and compare with expansions (<ref>), (<ref>), (<ref>) of lemma <ref>.
For |ω| <π2-η_0 bounded away from π2, claim (<ref>) becomes equivalent to the obvious inequalities
	
    ωπ/2 +2-s     >  -2s    >    ωπ/2-2-s ,
    	  i.e.
    ωπ/2+2     >  -s    >    ωπ/2-2 .

Equality holds for ω =-π2, on the left, and for ω =+π2, on the right.
Therefore η_0=0 does not seem an option for proving inequality (<ref>), uniformly for small δ, at first.
However, we obtain uniform expansions for the partial derivatives ∂_ω B, ∂_ω B_0,1^+, ∂_ω B_1,1^- as well, by differentiation of the coefficients of the δ-expansions (<ref>), (<ref>), (<ref>).
At ω =-π2 we obtain
	
    ∂_ω((B-B_1,1^-)/c)    =
    	∂_ω (s+ωπ/2+2)δ^3 +… =
    	1π/2δ^3+… >0 ,
    	
    and at ω = +π2, similarly,∂_ω ((B_0,1^+-B)/c)    =
    	∂_ω(-ωπ/2+2-s)δ^3+… =
    	-1π/2δ^3+… <0 .

This establishes positivity of the differences for all |ω|<π2, uniformly for all 0<δ≤δ_0, and settles claim (<ref>).


To prove claim (<ref>) we invoke B(δ, ω) < B_0,1^+(δ, ω) from (<ref>).
In fact
	
    B_m,j_m^- = B(δ, ω) ,    
    	B_m,j_m+1^+= B(δ,ω) ,

in our parametrization, for any m≥ 1. Note how (<ref>) refers to possibly different ε =  ε^±(δ,ω) given by (<ref>), (<ref>) and (<ref>), (<ref>), respectively.
Indeed ε has been eliminated in the 2-scale characteristic equations (<ref>), (<ref>), and only enters after the (m,j)-dependent hashings (<ref>), (<ref>).
Thus (<ref>) holds by definition of ε^+(δ, ω).
The arguments for claim (<ref>) via ε^-(δ,ω) are completely analogous.


To complete the proof we only have to establish the strict monotonicity of ω↦ B(δ, ω) and of j ↦ B_m,j^-, j=1, … ,j_* as claimed in (<ref>), (<ref>).
We invoke lemmata <ref> and <ref>.
By lemma <ref> we have strictly decreasing dependence Ω↦ω^-(δ, Ω), for fixed δ >0.
By lemma <ref>, assumption (<ref>) implies that ω↦ B(δ,ω) decreases strictly, as long as ω≤πΩ(δ,ω).
By monotonicity of Ω, we remain in this region, for all smaller ω≥ -π2, once we are ever inside.
This proves claim (<ref>).


Specifically, suppose assumption (<ref>) holds for ω_* = ω^-(δ, Ω_m,j_*^-), and hence for all -π2≤ω≤ω_*=ω^-(δ,Ω_m,j_*^-).
Then conclusion (<ref>) holds for all -π2≤ω≤ω_*=ω^-(δ,Ω_m,j_*^-).
Hashing (<ref>) implies strict monotonicity of the slow frequencies,
	
    Ω_m,j_*^-< … < Ω_m,1^-≤ 0 .

In particular (<ref>) successively implies
	
    ω_*      =  ω^-(Ω_m,j_*^-)    > …   >
    	 ω^-(Ω_m,1^-)    ≥   -π2 ,
    
    	B(δ,ω_*)     =     B_m,j_*^-    < …   <
    	   B_m,1^-    ≤      0 ,

at ε= ε(δ,ω_*). 
This proves claim (<ref>).


For the above ranges of ε,δ, the inequalities (<ref>) and (<ref>), (<ref>) also validate assumptions (<ref>), (<ref>) of corollary <ref>, respectively.
This establishes the Pyragas region 𝒫, in the above ranges of ε,δ, and the lemma is proved.



We now address the remaining segment of the piecewise strictly monotone curve ω↦Ω(δ, ω), where
	
    πΩ (δ, ω) ≤ω≤π2Ω (δ, ω)<0 .

By lemma <ref>, this segment contains
	
    B_min(δ) = min_|ω |≤π2 B(δ, ω) .




Let k be odd.
Uniformly, for ω satisfying (<ref>), we have the expansions
	
    Ω (δ, ω)    = 
    	-1π/2δ +𝒪(δ^3) ;
    	
    
    	B(δ, ω)    =
    	-δ^2 +𝒪(δ^4) .
    	
    In particular this implies
    	B_min(δ)    = 
    	-δ^2+𝒪(δ^4) .




We invoke expansions (<ref>) and (<ref>) of lemma <ref>.
Indeed (<ref>) and (<ref>) imply
	
    -2cδ + … = πΩ (δ, ω) ≤ω≤π2Ω(δ, ω)= -cδ +…

with c=cosω.
In particular ω = 𝒪(δ), uniformly in the region (<ref>).
Hence the δ-expansion (<ref>) with s= sinω implies claim (<ref>).
Similarly, (<ref>) and ω =𝒪(δ) imply (<ref>).
Claim (<ref>) follows from the uniform expansion (<ref>) by the remark preceding the proposition, and the proof is complete.




Let 0<δ≤δ_0 be small enough and consider odd k.
Let (Ω_*,ω_*) denote the unique intersection point of the line ω=πΩ with the 2-scale curve Ω = Ω(δ, ω).
Assume that the hashing line
	
    Ω = ε(ω -π a_m,j_m) ,

of j_m:= [(m+1)/2] intersects the line πΩ = ω, ω≤ 0, to the left of (Ω_*, ω_*).
Then
	
    B_1,1^-(ε) < B_min (δ) <0 .
		





We recall a_m,j_m=12δ^-1 -1; see (<ref>).
Insertion into hashing (<ref>) provides the intersection with πΩ = ω at 
	
    ε   =
    	Ω/(ω-π a_m,j_m)= 1πΩ/
    	(Ω-12δ^-1+1)=
       =
    	-1π/2δΩ/(1-2δ-2δΩ)≥
    	-1π/2δΩ(δ,ω_*)/
    	(1-2δ-2δΩ(δ, ω_*))=
       =
    	(π2)^-2δ^2/(1-2δ)+𝒪(δ^4)=
    	(π2)^-2(δ^2+2δ^3+𝒪(δ^4)) .

Here we have used intersection to the left of (Ω_*, ω_*), i.e. Ω≤Ω_* = Ω(δ,ω_*)< 0, and expansion (<ref>) at (Ω_*, ω_*).
The ε-expansion (<ref>) of corollary <ref> for B_1,1^-= B_1,1^-(ε), the estimate ε = ε^- = 𝒪(δ^2) of (<ref>), and comparison with (<ref>) via (<ref>) then yield
	
    B_1,1^- (ε) 
    	   =	-(π2)^2ε+
    	(π2)^3 ε^2    +    𝒪(ε^3)     ≤   
       ≤   - δ^2 -2δ^3    +    𝒪(δ^4)    <    B_min(δ)<0 .

This proves claim (<ref>) and the lemma.







§ PROOF OF THEOREM 1.2



Our proof of theorem <ref> is based on just a detailed stability analysis of the 2-delay characteristic equation  (<ref>), with ϑ=0, at the Hopf bifurcation points λ = λ_k := (-1)^k+1ω_k of (<ref>).
Emphasis is on control-induced eigenvalues μ = i ω̃ in the limit ε→ 0 of large frequencies 1/ε = ω_k = (k+12)π. 

We summarize the proof; see figs. <ref> and <ref> for an illustration.
We only address the case of large odd k; the case of even k is analogous.

In section <ref> we have studied the 2-scale characteristic equation (<ref>), (<ref>) which eliminates ε> 0.
Instead, imaginary eigenvalues μ = ± iω̃ have been represented by a slow Hopf frequency Ω̃ = εω̃, in addition to ω̃ itself.
The case of real eigenvalues μ, and their crossing at μ=0 due to the scaled control parameter b=2ε B, was treated in lemma <ref> and corollary <ref>.
The hashing Ω̃= εω̃ was detailed, and normalized to Ω=Ω̃-(2m+1), ω≡ω̃ (mod 2 π), in lemma <ref>.
In lemma <ref> we rewrote the characteristic equation in normalized frequencies Ω, ω instead of Ω̃, ω̃.
Lemma <ref> studied the resulting fundamental nonlinear 2-scale relation between Ω and ω.
The resulting control parameters B were addressed in lemma <ref>.
Emphasis there was on monotonicity properties with respect to ω.


Section <ref> has been devoted to the resulting nontrivial control-induced Hopf bifurcations, at B=B_m,j^±, in contrast to the spectrum inherited from the uncontrolled case.
Here m indicates the proximity of an m:1 resonance of ω̃ with ω_k.
The detailed analysis included simplicity of Hopf eigenvalues, and their transverse crossing directions; see theorem <ref>.
The analysis culminated in corollaries <ref> and <ref>.
In corollary <ref> we established the absence of any Pyragas control, for control parameters B>0.
The central corollary <ref> established the control region
	
    B∈𝒫=(B_0,1^+, B_1,1^-)

as the only Pyragas region, under assumptions (<ref>) – (<ref>) of certain inequalities among the control-induced Hopf parameter values B=B_m,j^±.


Section <ref> collected ε-expansions for B_m,j^±, and auxiliary quantities like the fast and slow normalized Hopf frequencies ω_m,j^± and Ω_m,j^±, see proposition <ref>.
These Taylor expansions in ε amount to expansions in the limit of arbitrarily large unstable dimensions k, and arbitrarily rapid oscillation frequencies ω_k=(k+12)π, of the original Hopf bifurcations of (<ref>) in absence of any control.
Corollary <ref> then established the crucial Hopf inequalities (<ref>) – (<ref>) for small enough 0<ε≤ε_0(δ_0), uniformly for bounded indices m≤ m_0 and j ≤ j_m+1 in the Hopf series B_m,j^+, as well as for B_m,j^- with j≤ j_m.
Here j_m:=[(m+1)/2].
In particular this settled assumption (<ref>).
At the end of this section, we observed how our proof hinged on a miraculous gap property of instability intervals B ∈ I_m,j=(B_m,j^-, B_m,j^+): the first gap occurs at j=j_m and contains the Pyragas region (<ref>).
See fig. <ref>.



To complete the proof, the limit of large near-resonances m>m_0, alias small δ = 1/Ω_m=1/(2m+1), was addressed in section <ref>.
For odd k, lemma <ref> collected δ-expansions for Ω, B, ε^±, and B_0,1^+, B_1,1^- in terms of the (normalized) fast Hopf frequencies ω.
The Taylor expansions with respect to δ address the limit of m:1 resonances, for m →∞.
In lemma <ref> this settled the remaining two assumptions (<ref>) and (<ref>) of corollary <ref>, up to one exceptional case.
The exceptional case was caused by our additional assumption, thus far, that all relevant Hopf points B_m,j^-,  j=1, … , j_m in (<ref>) occur in the (normalized) frequency range
	
    -π2≤ω≤πΩ <0 ,

i.e. at, or to the right of, the unique intersection (Ω_*, ω_*) of the 2-scale frequency relation Ω = Ω(δ,ω) with the straight line ω = πΩ in the (Ω, ω)-plane.
See (<ref>), (<ref>), (<ref>).
Since Hopf points themselves originate from intersections of straight hashing lines with that 2-scale frequency relation, the only remaining case was that the hashing line at j=j_m=[(m+1)/2] intersects ω = πΩ to the left of (Ω_*, ω_*).
This case was addressed, and settled, in lemma <ref>.
Indeed the minimal possible value B_min of control parameters induced by the 2-scale frequency relation then satisfies
	
    B_1,1^-(ε)< B_min(δ)<0 .

Since B_min≤ B_m,j^- for all j, this also established the remaining assumption (<ref>) in this one remaining configuration.
This completes the proof of our main theorem <ref>.




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