Constructing robust chaos: invariant manifolds and expanding cones
Paul A. Glendinning David J. W. Simpson


ℛ
ε
D
T

theoremTheorem[section]
lemma[theorem]Lemma
proposition[theorem]Proposition

definition
definitionDefinition[section]

















Constructing robust chaos: invariant manifolds and expanding cones.

    
P.A. Glendinning^† and D.J.W. Simpson^

^†School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK.

^Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand.

    
=============================================================================================================================================================================================================================



Keywords:
piecewise-linear; piecewise-smooth; border-collision bifurcation; Lyapunov exponent; robust chaos

MSC codes:
37G35; 
39A28 



Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049–3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we also characterise an attractor as the closure of the unstable manifold of a fixed point.









§ INTRODUCTION




A fundamental difference between smooth and piecewise-smooth dynamical systems is the possibility of robust chaos.
This refers to the existence of a chaotic attractor throughout open regions of parameter space.
This cannot happen, for instance, in typical families of smooth one-dimensional maps
because in this case periodic windows are typically dense in parameter space <cit.>.





Robust chaos is highly desirable in applications that use chaos.
In chaos-based cryptography <cit.>, for example,
robust chaos is preferred because periodic windows in `key space'
can be usurped by a hacker to decipher the encryption <cit.>.



One of the most widely studied families of piecewise-smooth maps
is the two-dimensional border-collision normal form

    [ x; y ]↦
    f(x,y) = [  τ_L    1; -δ_L    0 ][ x; y ] +
    [ 1; 0 ],     x ≤ 0, 
    [  τ_R    1; -δ_R    0 ][ x; y ] +
    [ 1; 0 ],     x ≥ 0,

where τ_L,δ_L,τ_R,δ_R ∈ℝ are parameters.
This was introduced in <cit.>,
except in (<ref>) the constant term is [1,0]^ T
instead of [μ,0]^ T, where μ∈ℝ.
Via a linear rescaling, μ 0 can be transformed to μ = ± 1,
and the choice μ = 1 can be made by interchanging the roles of x<0 and x>0.
The border-collision normal form arises by transforming and truncating a piecewise-smooth map
that has a border-collision bifurcation at μ = 0 <cit.>. 
Many groups have described non-chaotic dynamics of (<ref>) in detail,
see for instance <cit.>. 


In a highly influential paper,
Banerjee, Yorke, and Grebogi <cit.> considered (<ref>) in a certain parameter regime 
where f is orientation-preserving (i.e. δ_L > 0 and δ_R > 0).
Based on the intersections of the stable and unstable manifolds of two fixed points,
they argued heuristically that f has a unique chaotic attractor.
Their arguments apply throughout , so suggest robust chaos.
Although their arguments are incomplete, their conclusions have been well supported by numerical investigations.

In this paper we prove for the first time that f has an attractor that is chaotic, in a certain sense, throughout .
We also characterise the attractor, but subject to additional restrictions on the parameter values.
The arguments in <cit.> concern the stable and unstable manifolds of the fixed points,
so are insufficient to describe all orbits of f.
To remedy this we employ methods used by Misiurewicz <cit.> for the Lozi map
(given by (<ref>) with τ_L = -τ_R and δ_L = δ_R),
and Benedicks and Carleson <cit.> for smooth maps.

For the Lozi map, Misiurewicz <cit.> considered an orientation-reversing parameter regime
and proved the existence of a topological attractor on which f is transitive.
This shows that the Lozi map exhibits robust chaos.
Collet and Levy <cit.> subsequently showed that this attractor supports an SRB measure
(and so has many nice ergodic properties <cit.>).

For parameter values where f is non-invertible (i.e. δ_L δ_R ≤ 0),
Glendinning <cit.> identified parameter regimes
where f has a (necessarily chaotic) two-dimensional attractor
by using general results on piecewise-expanding maps. 
Also, Kowalczyk <cit.> studied chaos in the case δ_R = 0
for which one-dimensional techniques suffice.

Returning to the orientation-preserving case,
Cao and Liu <cit.> used one-dimensional techniques to extend
Misiurewicz's results to arbitrarily small δ_L = δ_R > 0.
Glendinning <cit.> used Young's theorem <cit.>
to prove that in certain subsets of  there exists an attractor with an SRB measure.

The remainder of this paper is organised as follows.
We first define  and state our main results in <ref>.
In <ref> we identify a trapping region, Ω_ trap,
that necessarily contains a topological attractor.
Then in <ref> we study the evolution of tangent vectors
and identify a cone in tangent space that is forward invariant and expanding under f.
On the invariant expanding cone, tangent vectors expand under every iteration of f.
Thus if an attractor has well-defined Lyapunov exponents,
one of these exponents must be positive, <ref>.

In subsequent sections we seek to make more precise statements,
and to this end assume that both fixed points

have an eigenvalue with absolute value greater than √(2).
In <ref> we analyse the closure of the unstable manifold of one fixed point,
and in <ref> we show that on this set f is transitive.
Finally, <ref> provides a discussion and outlook for future studies.








§ PRELIMINARIES AND MAIN RESULTS







  §.§.§ The fixed points and their invariant manifolds


Let

    A_L    = [  τ_L    1; -δ_L    0 ],    
    A_R    = [  τ_R    1; -δ_R    0 ],

denote the matrices in (<ref>).
As in <cit.>, throughout this paper we assume

    δ_L    > 0,    δ_R    > 0, 
    τ_L    > δ_L + 1,    τ_R    < -(δ_R + 1).

This is equivalent to assuming that 
A_L has eigenvalues 0 < λ_L^s < 1 < λ_L^u
and A_R has eigenvalues λ_R^u < -1 < λ_R^s < 0.











Then f has two fixed points:

    Y    = (Y_1,Y_2) = ( -1/τ_L - δ_L - 1, δ_L/τ_L - δ_L - 1),
    
    
    X    = (X_1,X_2) = ( 1/δ_R + 1 - τ_R, -δ_R/δ_R + 1 - τ_R),

where Y_1 < 0 and X_1 > 0.
These are saddle-type fixed points because
the eigenvalues associated with Y and X are simply those of A_L and A_R, respectively.






As with smooth maps, the stable and unstable subspaces of Y and X

are lines intersecting Y and X and with slopes matching those of the eigenvectors of A_L and A_R.

Since f is piecewise-linear, the stable and unstable manifolds of Y and X

initially coincide with their corresponding subspaces as they emanate from Y and X.
Globally, the stable and unstable manifolds
have a complicated piecewise-linear structure due to the piecewise-linear nature of f.



To understand this structure,
observe that f is continuous but non-differentiable on x=0, the switching manifold.
The image of the switching manifold is y=0.
Thus if α⊆ℝ^2 is a line segment that intersects x=0 transversally,
then f(α) is the union of two line segments that meet at a point on y=0.
Thus the unstable manifolds have `kinks' at points on y=0,
and on the forward orbits of these points.
Similarly the stable manifolds have kinks at points on x=0,
and on the backward orbits of these points.

Since the eigenvalues associated with Y are positive,
the stable and unstable manifolds of Y,
W^s(Y) and W^u(Y), each have two dynamically independent branches.
In the direction of decreasing x they simply coincide with
the stable and unstable subspaces of Y: E^s(Y) and E^u(Y).
In the direction of increasing x, let D = (D_1,0) and S = (0,S_2) denote the first kinks of
W^u(Y) and W^s(Y) as we follow these manifolds outwards from Y, see Fig. <ref>.
By using the fact that the line segments YD and YS
are contained within E^u(Y) and E^s(Y),
it is a simple exercise to obtain

    D_1    = 1/1 - λ_L^s, 
    
    S_2    = -λ_L^u/λ_L^u - 1.

Notice D_1 > 1 and S_2 < -1.








  §.§.§ The parameter regime 


As we continue to follow the stable manifold W^s(Y) outwards from Y,
the manifold has its second kink at f^-1(S).
Due to the constraints (<ref>),
the point f^-1(S) lies in the first quadrant x,y>0.
Let C = (C_1,0) denote the intersection of Sf^-1(S) with y=0.
If C_1 > D_1, that is, C lies to the right of D,
then the quadrilateral Y D C S is forward invariant under f
(see Lemma 1 of <cit.> and compare Lemma <ref> below).


If instead C_1 < D_1, then f(D) lies outside Y D C S and so this quadrilateral
is not forward invariant.
Numerical explorations suggest that f has no attractor in this case.



From (<ref>) we immediately obtain

    C_1 = -S_2/δ_R - τ_R + δ_R/S_2.

By then combining (<ref>)–(<ref>) we obtain, after much simplification,

    C_1 - D_1 = ϕ(τ_L,δ_L,τ_R,δ_R)/( τ_L - δ_L - 1 ) ( δ_R - τ_R λ_L^u ),

where

    ϕ(τ_L,δ_L,τ_R,δ_R)
    = δ_R - ( τ_R + δ_L + δ_R - (1 + τ_R) λ_L^u ) λ_L^u  .


Since the denominator of (<ref>) is positive by (<ref>),
the condition ϕ > 0 ensures that C_1 > D_1.
The parameter region  of <cit.>
is defined by the constraints (<ref>)
and ϕ > 0, see Fig. <ref>.








  §.§.§ Lyapunov exponents


Let Σ_∞⊆ℝ^2 be the set of points whose forward orbits intersect x=0.
Then the Jacobian matrix f^n (z) is well-defined for all
z ∈ℝ^2 ∖Σ_∞ and all n ≥ 1.
The Lyapunov exponent of a point z ∈ℝ^2 ∖Σ_∞
in a direction v ∈ℝ^2 is defined as

    λ(z,v) = lim_n →∞1/n ln(  f^n (z) v ),

assuming this limit exists.
Oseledets' theorem <cit.> gives conditions under which (<ref>)
is well-defined for almost all points in an invariant set.
The Lyapunov exponent represents the asymptotic rate of expansion in the direction v.
For bounded invariant sets, positive Lyapunov exponents are part of the standard definitions of chaos.
The following theorem uses Lyapunov exponents to demonstrate robust chaos throughout .



Suppose (<ref>) is satisfied and ϕ > 0.
Then (<ref>) has a topological attractor Λ
with the property that for any z ∈Λ∖Σ_∞,
if the limit (<ref>) exists with v = [ 1; 0 ],
then λ(z,v) > 0.





We have not been able to show that the conditions of Oseledets' theorem are satisfied,
or verify that the limit (<ref>) exists directly.
However, below we actually show that the infimum limit of the right hand-side of (<ref>)
is positive, thus even if the limit does not exist the dynamics must still be locally expanding.
Although the two-dimensional Lebesgue measure of Σ_∞ is zero
(because it is a countable union of measure zero sets),
we do not know that μ( Σ_∞) = 0,
where μ is the invariant probability measure associated with Λ.
Also, it is not known whether or not Λ is unique,
although numerical simulations by several authors have failed to find parameter values in 
for which f has multiple attractors.




  §.§.§ A homoclinic connection and a transitive attractor






Next we describe W^s(X) and W^u(X) in more detail.
Since the eigenvalues associated with X are negative,
W^s(X) and W^u(X) each have one dynamically independent branch.
Let T = (T_1,0) denote the intersection of E^u(X) with y=0,
and let V = (0,V_2) denote the intersection of E^s(X) with x=0, see Fig. <ref>.
Then W^u(X) coincides with E^u(X) on Tf(T),
and W^s(X) coincides with E^s(X) on Vf^-1(V).

As we follow W^u(X) outwards, the first part of W^u(X) that does not coincide with E^u(X)
is the line segment Tf^2(T).
Let

    Z = Tf^2(T)∩ E^s(X),

if this point of intersection exists.
The point Z corresponds to a transverse intersection
between the stable and unstable manifolds of X and
implies there exists a chaotic orbit.
This transverse intersection exists if and only if f^2(T) lies to the left of E^s(X),
which can be equated to a condition on the parameter values of f (see Lemma 2 of <cit.>).

Assuming Z exists, let Δ_0 be the (compact filled) triangle X T Z.
Then Δ = ⋃_n=0^∞ f^n(Δ_0) is forward invariant.
Also let Δ̃ = ⋂_n=0^∞ f^n(Δ).



Suppose (<ref>) is satisfied, δ_L < 1, δ_R < 1, ϕ > 0, and

    τ_L    > δ_L + 2/√(2),    τ_R    < -δ_R + 2/√(2).

Then



  * f^2(T) lies to the left of E^s(X) (so Z exists),

  * Δ̃ =  cl( W^u(X) ), and

  * f is transitive on Δ̃.




Theorem <ref> is analogous to Theorems 2 and 5 of <cit.>
for the orientation-reversing case.
The conditions (<ref>) on the parameters of f
are equivalent to the following conditions on the eigenvalues of A_L and A_R:

    λ_L^u    > √(2),    λ_R^u    < -√(2).

Certainly the conclusions of Theorem <ref> may be false
if (<ref>) is not satisfied.
For instance f^2(T) may lie to the right of E^s(X)
(see Figure 1 of <cit.> for an example)
in which case cl( W^u(X) ) has a fundamentally different character.
The conditions δ_L < 1 and δ_R < 1
are used at one place below to show that the area of f^n(Δ_0) decreases with n,
but we believe these conditions are actually unnecessary.

Theorem <ref> tells us that
in Δ the map f has a unique chaotic attractor
equal to the closure of W^u(X).
We have not proved that the quadrilateral Y D C S doesn't contain other attractors.
Certainly Y D C S may contain other invariant sets.
As an example, Fig. <ref> shows all periodic solutions of f (except Y)
with period ≤ 20 for the parameter values

    τ_L    = 1.6,    δ_L    = 0.4,    τ_R    = -1.6,    δ_R    = 0.4.

This numerical result suggests that periodic solutions are dense in cl( W^u(X) )
and form a Cantor set bounded away from cl( W^u(X) ).
The Cantor set seems to be formed from the stable manifold of a period-3 solution (not shown).
We have observed a similar partition of the periodic solutions of f
for other parameter values including those that satisfy the conditions of Theorem <ref>.
This shows that the infinite intersection of the trapping region Ω_ trap
(defined in the next section) is not always equal to cl( W^u(X) )
which is different to the analogous situation in the orientation-reversing case <cit.>.












§ A FORWARD INVARIANT REGION AND A TRAPPING REGION




Throughout this section we study f subject to (<ref>) and ϕ > 0.
This is the parameter region  of <cit.> shown in Fig. <ref>.










As illustrated in Fig. <ref>,
let B ∈YD be such that Bf(D)
is parallel to YS.
Let Ω be the triangle B D f(D).
Below we show that Ω is forward invariant under f.





Given > 0, let

    B_ = B -  (D-Y) - ^2 (S-Y).

As illustrated in Fig. <ref>,
let D_ be the point on y=0 for which B_D_ is parallel to YD,
and let F_ be the point on x=0 for which B_F_ is parallel to YS.
Let Ω_ trap be the triangle B_ D_ F_.
Below we show that if > 0 is sufficiently small then
Ω_ trap is a trapping region for f,
i.e., Ω_ trap maps to its interior.
This ensures the existence of a topological attractor:
⋂_n=0^∞ f ( Ω_ trap) is an attracting set by definition.
In (<ref>) the (S-Y)-term is smaller
than the (D-Y)-term to ensure that D_ maps inside Ω_ trap.

Our proofs use the following elementary principle
that motivates our definitions of Ω and Ω_ trap.
If α⊆ℝ^2 is a line segment in x ≤ 0
that is parallel to either YD or YS,
then f(α) is parallel to α.
This is because the directions of YD and YS
are those of the eigenvectors of A_L.



Suppose (<ref>) is satisfied and ϕ > 0.
Then f(Ω) ⊆Ω.





We have f(D) = ( τ_R D_1 + 1, -δ_R D_1 ),
thus f(D) lies in the quadrant x,y < 0
(because D_1 > 1, τ_R < -1, and δ_R > 0).
Also from (<ref>) we have

    f(C) - f(D) = ( τ_R (C_1 - D_1) + 1, -δ_R (C_1 - D_1) ),

thus f(D) lies above and to the right of f(C)
(because C_1 > D_1 by (<ref>)).
Also f(C) ∈YS (because f^-1(S) lies in x,y > 0),
thus f(D) lies above YS.

Consequently B lies between Y and f^-1(D),
where f^-1(D) is the intersection of YD with x=0.
Let U be the intersection of Df(D) with x=0, see Fig. <ref>.

Write Ω = Ω_L ∪Ω_R,
where Ω_L and Ω_R are the parts of Ω in x ≤ 0 and x ≥ 0 respectively.
Notice Ω_L is the quadrilateral U f(D) B f^-1(D),
and Ω_R is the triangle D U f^-1(D).
Then f(Ω) = f(Ω_L) ∪ f(Ω_R),
where f(Ω_L) is the quadrilateral f(U) f^2(D) f(B) D,
and f(Ω_R) is the triangle f(D) f(U) D.
Since Ω is convex, to complete the proof it suffices to show that each vertex of
f(Ω_L) and f(Ω_R) belongs to Ω.

The point f(B) lies between B and D, thus f(B) ∈Ω.
Since Bf(D) is parallel to YS,
f(B)f^2(D) is also parallel to YS.
Furthermore, since Bf(D) is located above YS,
f(B)f^2(D) is located above Bf(D) (because λ_L^u > 1).
Also f^2(D) lies below YD, and f^2(D)_2 > 0 because f(D)_1 < 0.
Thus f^2(D) ∈Ω.
Finally, U lies above the line that passes through B and f(D),
thus f(U) lies on y=0, above the line through B and f(D), and to the left of D,
thus f(U) ∈Ω.
This shows that all vertices of f(Ω_L) and f(Ω_R) belong to Ω.








Suppose (<ref>) is satisfied and ϕ > 0.
Then f ( Ω_ trap) ⊆ int( Ω_ trap),
for sufficiently small > 0.





Let G_ be the intersection of B_D_ with x=0.
Then f ( Ω_ trap) is the union of
the triangles f(B_) f(G_) f(F_) and f(G_) f(D_) f(F_).
Since Ω_ trap is convex, to complete the proof
it suffices to show that the vertices of these triangles belong to
int( Ω_ trap).

We begin with f(B_).
Assume > 0 is sufficiently small that B_ lies above YS.
Since B_D_ and B_F_ are parallel to the eigenvectors of A_L
corresponding to the eigenvalues λ_L^u > 1 and 0 < λ_L^s < 1, respectively,
the point f(B_) lies below B_D_ and above B_F_.




Also B_ lies to the left of x=0, thus f(B_) lies above y=0.
These three constraints on f(B_) ensure f(B_) ∈ int( Ω_ trap).

For similar reasons f(F_) lies above B_F_ and below B_D_.
Since f(F_) lies on y=0 to the left of D,
we have f(F_) ∈ int( Ω_ trap).
Also f(G_) lies between D and D_,
thus f(G_) ∈ int( Ω_ trap).

Finally, in view of the definition of B_ (<ref>),
the point D_ is an order ^2 distance from D.
Thus f(D_) is an order ^2 distance from f(D).
But f(D) lies above B_F_ by a distance k_1  + k_2 ^2, where k_1 > 0.
Thus, for sufficiently small > 0, f(D_) lies above B_F_,
and so f(D_) ∈ int( Ω_ trap).








§ INVARIANT EXPANDING CONES





We first define invariant expanding cones for arbitrary 2 × 2 matrices.



Let A be a real-valued 2 × 2 matrix and
let K ⊆ℝ be a closed interval.
The cone

    Ψ_K = { a [ 1; m ] |  a ∈ℝ,  m ∈ K },

is said to be



  * invariant if A v ∈Ψ_K for all v ∈Ψ_K, and

  * expanding if there exists c > 1 such that
A v ≥ c  v for all v ∈Ψ_K.





In <cit.>, Misiurewicz identified invariant expanding cones
for the Jacobian matrices of the Lozi map and its inverse.
This was done to demonstrate hyperbolicity and as part of his proof of transitivity.
Many groups have studied the linear algebra problem of
the existence of a cone that is invariant for a finite collection of matrices,
see for instance <cit.>.


Invariant expanding cones have also been used to give bounds on Lyapunov exponents for maps on
tori <cit.>.









Suppose (<ref>) is satisfied.
Let

    q_L    = -τ_L/2( 1 - √(1 - 4 δ_L/τ_L^2)),    
    q_R    = -τ_R/2( 1 - √(1 - 4 δ_R/τ_R^2)),

and let K = [q_L,q_R].
Then Ψ_K is an invariant expanding cone for both A_L and A_R.
If (<ref>) is also satisfied,
then the expansion condition is satisfied for some c > √(2).



For the remainder of this section we work towards a proof of Proposition <ref>.




Let

    A = [  τ  1; -δ  0 ],

where τ,δ∈ℝ.
Given m ∈ℝ,
the slope of v = [ 1; m ] is m,
and the slope of A v = [ τ + m;    -δ ] is

    G(m) = -δ/τ + m,

assuming m  -τ.
The fact that G is undefined at m = -τ will not be a problem below
because an infinite slope corresponds to a vector in direction [ 0; 1 ].
This vector cannot belong to an invariant expanding cone
because A [ 0; 1 ] = [ 1; 0 ],
hence the direction [ 0; 1 ] is not of interest to us.

We have chosen to characterise the direction of tangent vectors by their slope,
rather than by an angle, because slopes are easier to deal with than angles algebraically.
Indeed the fixed point equation G(m) = m is quadratic, and the fixed points are

    q(τ,δ)    = -τ/2( 1 - √(1 - 4 δ/τ^2)),
    
    
    r(τ,δ)    = -τ/2( 1 + √(1 - 4 δ/τ^2)),

assuming τ^2 > 4 δ.


Notice that q_L = q(τ_L,δ_L) and q_R = q(τ_R,δ_R), see (<ref>).
Notice also that q(τ,δ) and r(τ,δ) are the slopes of the eigenvectors of A.
If the eigenvalues of A are real and distinct, call them λ^s and λ^u,
then the slopes of the eigenvectors are -λ^u (corresponding to λ^s)
and -λ^s (corresponding to λ^u).
It follows that q_L = -λ_L^s ∈ (-1,0) and q_R = -λ_R^s ∈ (0,1).

For v = [ 1; m ] we have

    v    = √(1 + m^2), 
     A v    = √((τ + m)^2 + δ^2).

Solving v  =  A v gives m = p(τ,δ) where

    p(τ,δ) = -τ^2 + δ^2 - 1/2 τ,

assuming τ 0.
We first show that p, q, and r appear as in Fig. <ref>.









Suppose δ > 0 and |τ| > δ + 1.
Then

    |q(τ,δ)| < |p(τ,δ)| < |r(τ,δ)|.











Observe:

    τ^2 √(1 - 4 δ/τ^2)   = |τ| √(τ^2 - 4 δ)
       > ( δ+1 ) √(( δ+1 )^2 - 4 δ)
       = ( δ+1 ) | δ-1 |.

Thus

    |p(τ,δ)| - |q(τ,δ)|
       = 1/2 |τ|( τ^2 + δ^2 - 1 )
    - |τ|/2( 1 - √(1 - 4 δ/τ^2)) 
       > δ+1/2 |τ|( δ - 1 + | δ-1 | ) 
       ≥ 0.

Similarly,

    |p(τ,δ)| - |r(τ,δ)|
       = 1/2 |τ|( τ^2 + δ^2 - 1 )
    - |τ|/2( 1 + √(1 - 4 δ/τ^2)) 
       < δ+1/2 |τ|( δ - 1 - | δ-1 | ) 
       ≤ 0.


















Suppose δ > 0 and |τ| > δ + 1.
Then d G/d m > 0 for all m  -τ,
and d G/d m( q(τ,δ) ) < 1.





We have

    d G/d m = δ/(τ + m)^2 ,

which is evidently positive for all m  -τ.
The function q(τ,δ) is a root of m^2 + τ m + δ = 0,
thus to evaluate d G/d m( q(τ,δ) ) we can replace
one of the (τ + m)'s in the denominator of (<ref>) with -δ/m to obtain

    d G/d m( q(τ,δ) ) = -m/τ + m ,

where m = q(τ,δ), and so

    d G/d m( q(τ,δ) ) = -1/τ/q(τ,δ) + 1 .

Notice q(τ,δ)/τ = -1/2 + √(1 - 4 δ/τ^2) > -1/2.
Thus τ/q(τ,δ) + 1 < -1, hence d G/d m (q(τ,δ)) < 1, as required.




Suppose δ > 0 and |τ| > δ + 1.
If m ∈ℝ is such that τ m > τ p(τ,δ),
then A v  >  v, where v = [ 1; m ].





We have

    A v ^2 -  v ^2
       = (τ + m)^2 + δ^2 - (1 + m^2) 
       = τ^2 + δ^2 - 1 + 2 τ m 
       > τ^2 + δ^2 - 1 + 2 τ p(τ,δ).

The last expression is zero by (<ref>),
thus A v  >  v, as required.




Suppose δ > 0, |τ| > δ + 1, and |τ| > δ + 2/√(2).
If m ∈ℝ is such that |m - τ| ≤ |q(τ,δ) - τ|,
then A v  > √(2)  v, where v = [ 1; m ].





Let

    H(m) =  A v ^2 - 2  v ^2
    = -m^2 + 2 τ m + τ^2 + δ^2 - 2.

We only need to show H ( q(τ,δ) ) > 0,
because H(m) is a concave down parabola that achieves its maximum value at m = τ.

By substituting (<ref>) into (<ref>) we obtain

    H ( q(τ,δ) ) = δ^2 + δ - 2
    + τ^2/2( -1 + 3 √(1 - 4 δ/τ^2)).

For any fixed δ > 0, this is an increasing function of |τ| because

    ∂ H ( q(τ,δ) )/∂( τ^2 ) =
    1 + 3 ( √(1 - 4 δ/τ^2) - 1 )^2/4 √(1 - 4 δ/τ^2),

which is evidently positive.
Thus H ( q(τ,δ) ) is strictly greater than its value at
|τ| = δ + 2/√(2).
From (<ref>), we obtain, after simplification,

    H ( q ( ±δ + 2/√(2),δ) )
    = 3/4 (δ+2) ( δ-2 + |δ-2| ) 
    ≥ 0.

Thus H ( q(τ,δ) ) > 0, which completes the proof.


We are now ready to prove Proposition <ref>.
Let

    G_L(m)    = -δ_L/τ_L + m,    
    G_R(m)    = -δ_R/τ_R + m,

be the `slope maps' for A_L and A_R.
Lemma (<ref>) has shown that these maps are increasing
and have stable fixed points q_L and q_R, respectively.
Consequently they appear as in Fig. <ref>,
from which we see that K is forward invariant under both G_L and G_R
(this is proved carefully below).
That Ψ_K is expanding follows from Lemmas <ref> and <ref>,
and the strong expansion (c > √(2)) follows from Lemma <ref>.







We first show that Ψ_K is expanding.
Choose any v ∈Ψ_K, and let m be its slope.
By linearity it suffices to consider v = [ 1; m ].

Since τ_L > 0, we have p(τ_L,δ_L) < q_L by Lemma <ref>.
Thus m > p(τ_L,δ_L), and so A_L v  >  v by Lemma <ref>.
Similarly, since τ_R < 0, we have p(τ_R,δ_R) > q_R.
Thus m < p(τ_R,δ_R), and so A_R v  >  v.
Since K is compact, the set
{ A_J v / v  |  J ∈{ L,R },  v ∈Ψ_K } has a minimum,
call it c, and c > 1 as required.

Next we show that Ψ_K is invariant.
To do this we show that G_J(K) ⊆ K, for both J = L and J = R.

The function G_J has fixed points q_J and r_J = r(τ_J,δ_J), where r_J ∉ K by Lemma <ref>.
Thus, by Lemma <ref>, for all m ∈ K we have G_L(m) ≥ G_L(q_L) = q_L, and G_L(m) ≤ m ≤ q_R.
Similarly, for all m ∈ K we have G_R(m) ≥ m ≥ q_L, and G_R(m) ≤ G_R(q_R) = q_R.
This shows that G_J(K) ⊆ K, for both J = L and J = R.
Thus Ψ_K is an invariant expanding cone for both A_L and A_R.

Now suppose (<ref>) is also satisfied.
By Lemma <ref> and since K is compact,
to verify the strong expansion property we just need to show that for any m ∈ K we have


    |m-τ_L|    ≤ |q_L-τ_L|,
    
    and  
    |m-τ_R|    ≤ |q_R-τ_R|.

Since q_L = -λ_L^s and q_R = -λ_R^s (as explained in the text)
we have -1 < q_L < 0 < q_R < 1, and so


    q_R < 1 < 2 - q_L < 2 τ_L - q_L  .

Thus K ⊆[ q_L, 2 τ_L - q_L ], and so (<ref>) is satisfied.
For similar reasons K ⊆[ 2 τ_R - q_R, q_R ], which implies (<ref>).










§ CONSEQUENCES OF INVARIANT EXPANDING CONES












In this section we use the existence of an invariant expanding cone
(see Proposition <ref>) to prove Theorem <ref>
and show that all periodic solutions are unstable.
This includes periodic solutions with points on x=0 for which f is undefined.
The stability of such periodic solutions can be extremely complicated <cit.>,
but here a lack of stability follows simply from the definition of Lyapunov stability.



By Proposition <ref>, f has a trapping region Ω_ trap.
Thus f has a topological attractor Λ⊆Ω_ trap.

By Proposition <ref>, there exists an invariant expanding cone Ψ_K,
for both A_L and A_R, and v = v_0 = [ 1; 0 ]∈Ψ_K
(because q_L < 0 < q_R).
For all i ≥ 0, let

    v_i+1 =  f ( f^i(z) ) v_i/ f ( f^i(z) ) v_i ,

so that

    f^n (z) v  = ∏_i=0^n-1 f ( f^i(z) ) v_i .

That the v_i are well-defined is easily established inductively:
Each derivative is well-defined because z ∉Σ_∞.
Also v_i ∈Ψ_K implies that the denominator in (<ref>) is non-zero by the expansion property,
and v_i+1∈Ψ_K by invariance.

Then (<ref>) and the expansion property give
f^n (z) v ≥ c^n, for some c > 1, and so

    1/nln(  f^n (z) v ) ≥ln(c),

for all n ≥ 1.
Therefore

    lim inf_n →∞1/nln(  f^n (z) v ) > 0,

and thus λ(z,v) > 0, if the limit (<ref>) exists.








Suppose (<ref>) is satisfied.
Then all periodic solutions of f are unstable.





Let z ∈ℝ^2 be a point of a period-n solution of f.
Let ℐ be the set of all i ∈{ 0,…,n-1 } for which f^i(z) does not lie on x=0.
Let

    = min_i ∈ℐ| f^i(z)_1 |,

and = 1 if ℐ = ∅.

Choose any δ∈ (0,], and let z_δ = z + [ δ; 0 ].
For each i ≥ 0, let v_i = f^i(z_δ) - f^i(z).
Notice v_0  = δ≤, and v_0 ∈Ψ_K (the cone defined in Proposition <ref>).

For any i ≥ 0,
if v_i ≤ then f^i(z_δ) and f^i(z) do not lie on different sides of x=0
and so there exists J ∈{ L, R } such that

    f^i+1(z_δ)    = A_J f^i(z_δ) + [ 1; 0 ],    
    f^i+1(z)    = A_J f^i(z) + [ 1; 0 ].

Consequently v_i+1 = A_J v_i.
Thus if we also have v_i ∈Ψ_K, then v_i+1∈Ψ_K and v_i+1≥ c  v_i
(where c > 1).

This shows that we cannot have v_i ≤ for all i ≥ 0
because, by induction, this would imply v_i ≥ c^i δ for all i ≥ 0.
Hence v_i  > for some i ≥ 0.
That is, the forward orbit of z_δ escapes an -neighbourhood of the periodic solution.
Since we have allowed arbitrary values of δ > 0, this shows that the periodic solution is not Lyapunov stable.














§ THE UNSTABLE MANIFOLD W^U(X)




Here we prove the first two parts of Theorem <ref>.
Part (i) is proved via direct calculations.
Our proof of part (ii) mimics arguments used to prove Theorem 2 of <cit.>
and requires the assumption δ_L, δ_R < 1.



Suppose (<ref>) and (<ref>) are satisfied and ϕ > 0.
Then f^2(T) lies to the left of E^s(X).






For any z ∈ E^u(X) with z_1 ≥ 0,
we have f(z) - X = λ_R^u (z - X).
Using z = f^-1(T) and just taking the first components, we obtain

    T_1 - X_1 = | λ_R^u | X_1  .

With instead z = T we obtain

    X_1 - f(T)_1 = | λ_R^u | ( T_1 - X_1 ).

Combining these gives

    | f(T)_1 | = ( | λ_R^u | - 1/| λ_R^u |)
    ( T_1 - X_1 ).

Then by (<ref>),

    | f(T)_1 | > ( √(2) - 1/√(2)) ( T_1 - X_1 )
    = 1/√(2)( T_1 - X_1 ).


From (<ref>) we have
T_1 = τ_L f^-1(T)_1 + f^-1(T)_2 + 1 = f^-1(T)_2 + 1, and
f^2(T)_1 = τ_L f(T)_1 + f(T)_2 + 1.
Subtracting these gives

    T_1 - f^2(T)_1    = -τ_L f(T)_1 + f^-1(T)_2 - f(T)_2 
       > -τ_L f(T)_1 
       > √(2) | f(T)_1 | 
       > T_1 - X_1  .

Thus f^2(T) lies to the left of X.
Also f^2(T) lies in y > 0 (because f(T)_2 < 0),
so certainly f^2(T) lies to the left of E^s(X).




Suppose (<ref>) and (<ref>) are satisfied,
δ_L < 1, δ_R < 1, and ϕ > 0.
Then Δ̃ =  cl( W^u(X) ).





First we show that cl( W^u(X) ) ⊆Δ̃.
Choose any z ∈ cl( W^u(X) ).
Then there exist z_k ∈ W^u(X) with z_k → z as k →∞.
For each k, the backward orbit of z_k converges to X.
The convergence eventually occurs on the unstable subspace E^u(X)
and includes points on both sides of X because λ_R^u < 0.
Thus there exists n_k ≥ 0 such that f^-n_k(z_k) ∈XT⊂Δ_0.
Thus z_k ∈ f^n_k(Δ_0), and so z_k ∈Δ.
Hence cl( W^u(X) ) ⊆Δ.
Since cl( W^u(X) ) is invariant
we must also have cl( W^u(X) ) ⊆Δ̃.

Second we show that Δ̃⊆ cl( W^u(X) ).
Choose any z ∈Δ̃.
Then z ∈ f^n(Δ) for all n ≥ 0.
Let Area(·) denote the two-dimensional Lebesgue measure
and let δ_ max = max( δ_L, δ_R ).
Then

    Area( f^n(Δ) ) ≤δ_ max^n  Area(Δ),

which converges to 0 as n →∞ because we have assumed δ_L, δ_R < 1.
Thus the distance of z to the boundary of f^n(Δ) goes to 0 as n →∞.

The boundary of Δ_0 is contained in XZ∪ W^u(X),
so the boundary of f^n(Δ_0) is contained in Xf^n(Z)∪ W^u(X).
Thus the boundary of Δ is contained in Zf(Z)∪ W^u(X),
so the boundary of f^n(Δ) is contained in f^n(Z)f^n+1(Z)∪ W^u(X).
But f^n(Z)f^n+1(Z) converges to X as n →∞.
Hence the distance of z to W^u(X) goes to 0 as n →∞.
Thus z ∈ cl( W^u(X) ) which shows that
Δ̃⊆ cl( W^u(X) ).








§ TRANSITIVITY




Here we provide three Lemmas that combine to complete the proof of Theorem <ref>.
First we use direct calculations to show that the point
U lies above the point V, as in Fig. <ref>.
This requires significant effort because the required assumption ϕ > 0
(equivalently C_1 > D_1) does not relate to the points U and V in a simple way.

Given that U lies above V,
it follows that, as in Fig. <ref>,
any line segment in f(Ω) that intersects x=0 and y=0 must also intersect E^s(X).
This is the key step to establishing transitivity and is also based on the ideas in <cit.>.
The strong expansion (c > √(2)) of Proposition <ref>
is used below in the proof of Lemma <ref>.



Suppose (<ref>) and (<ref>) are satisfied and ϕ > 0.
Then U_2 > V_2.





Similar to S, see (<ref>), the point V has y-component

    V_2 = -λ_R^u/λ_R^u - 1.

The point U is defined as the intersection of Df(D) with x=0.
From f(D) = ( τ_R D_1 + 1, -δ_R D_1 ), we obtain

    U_2 = -λ_R^s λ_R^u D_1/1 - λ_R^s - λ_R^u - 1/D_1.

Upon substituting (<ref>) into (<ref>), subtracting (<ref>),
and carefully factorising, we obtain

    U_2 - V_2 = -λ_R^u
    ( 1 - λ_L^s + λ_R^s )
    ( λ_L^s - λ_R^u )/( 1 - λ_L^s )
    ( 1 - λ_R^u )
    ( λ_L^s - λ_R^s - λ_R^u ).

Each factor in (<ref>)
is evidently positive, except possibly the middle factor in the numerator.
Thus it remains to show that 1 - λ_L^s + λ_R^s > 0.

To do this we first show that C_1 < -1/λ_R^s.
Suppose for a contradiction that C_1 ≥-1/λ_R^s.
By (<ref>) we have

    -S_2/-λ_R^s + λ_R^u ( λ_R^s - 1 + λ_R^s/S_2)≥-1/λ_R^s.

But λ_R^u < -√(2), see (<ref>), thus

    -S_2/-λ_R^s - √(2)( λ_R^s - 1 + λ_R^s/S_2)
    > -1/λ_R^s,

which is equivalent to

    S_2 + 1 + √(2) + √(2)/S_2 > √(2)/λ_R^s.

But λ_R^s > -1, thus

    S_2 + 1 + √(2) + √(2)/S_2 > -√(2),

which is equivalent to

    ( S_2 + 2 + √(2)) ( S_2 + √(2) - 1 ) > 0.

However, λ_L^u > √(2), see (<ref>),
thus by (<ref>) we have -(2 + √(2)) < S_2 < -1,
which contradicts (<ref>).

Therefore C_1 < -1/λ_R^s.
The assumption ϕ > 0 implies D_1 < C_1,
thus D_1 < -1/λ_R^s.
By (<ref>), this is equivalent to
1 - λ_L^s + λ_R^s > 0,
which completes the proof.








Suppose (<ref>) and (<ref>) are satisfied and ϕ > 0.
Let α⊂Ω be a line segment with slope m ∈ K = [q_L,q_R].
Then there exists n ≥ 1 and points P on x=0 and Q on y=0
such that PQ⊆ f^n(α).





Let α_0 = α.
We iteratively construct a sequence of line segments {α_i } in Ω with slopes in K and lengths a_i, as follows.
For each i ≥ 0 suppose α_i and f(α_i) do not both intersect x=0.
Then f^2(α_i) is a union of at most two line segments
(and belongs to Ω because Ω is forward invariant, Lemma <ref>).

The line segments comprising f^2(α_i) have slopes in K because Ψ_K is invariant (see Proposition <ref>).
Also Ψ_K is expanding with some c > √(2),
thus the length of f^2(α_i) is at least c^2 a_i.
Thus f^2(α_i) contains a line segment, α_i+1, with a_i+1≥c^2 a_i/2.

This gives a_n ≥c^2 n a_0/2→∞ as n →∞ because c^2 > 2.
But Ω is bounded, so this is not possible.
Thus there exists k ≥ 0 such that α_k and f(α_k) both intersect x=0.
Notice f(α_k) is a union of at most two line segments, both of which intersect y=0.
Thus there exists a line segment PQ⊆ f(α_k) ⊆ f^2 k + 1(α)
with P on x=0 and Q on y=0.




Suppose (<ref>) and (<ref>) are satisfied and ϕ > 0.
For any open M, N ⊆ℝ^2 that have non-empty intersections with cl( W^u(X) ),
there exists n ≥ 0 such that f^n(M) ∩ N ∅.






Let α⊆ M ∩Ω be a line segment with slope in K = [q_L,q_R].
By Lemma <ref>, there exists n_1 ≥ 1 such that
f^n_1(α) contains a line segment PQ with P on x=0 and Q on y=0.

Notice PQ⊆ f(Ω) because n_1 ≥ 1 and f(Ω) is forward invariant.
Thus P lies on or above U, see Fig. <ref>.
Since V_2 < U_2 (see Lemma <ref>), P lies above E^s(X).
Also, Q lies on or to the right of f(U).
Since f(V)_1 < f(U)_1, Q lies below E^s(X).
Thus PQ intersects E^s(X) transversally.

Let z ∈ N ∩ W^u(X).
Since f^-n(z) → X as n →∞,
there exists n_2 ≥ 0 such that f^-n(z) lies in x > 0 for all n ≥ n_2.
Then there exists open N_0 ⊆ N, with z ∈ N_0, such that f^-n_2(N_0) lies in x > 0.
Iteratively define N_k ⊆ N_k-1 as the maximal open set for which
f^-(n_2 + k)(N_k) lies in x > 0.
Since f^-1 is affine in x > 0 with saddle-type fixed point X,
as k →∞ the sets f^-(n_2 + k)(N_k) approach E^s(X)
and stretch across Ω for sufficiently large values of k.
Thus there exists n_3 ≥ 0 such that f^-(n_2+n_3)(N_k) intersects PQ.
Thus there exists w ∈ M such that f^n_1(w) ∈ f^-(n_2+n_3)(N_n_3).
Thus f^n_1 + n_2 + n_3(w) ∈ N,
and so f^n_1 + n_2 + n_3(M) ∩ N ∅ as required.
(This also completes the proof of Theorem <ref>.)








§ DISCUSSION








We have used invariant expanding cones to
prove that, throughout the parameter region  of <cit.>,
no invariant set of (<ref>) can have only negative Lyapunov exponents, Theorem <ref>.
In fact we have actually proved that for any n ≥ 1
the average expansion after n iterations is at least ln(c) for some c > 1,
see (<ref>).
Thus ln(c) may be used as a lower bound on the maximal Lyapunov exponent,
assuming the Lyapunov exponents are well-defined.
One could also identify an invariant expanding cone for f^-1,
as done in <cit.> for the Lozi map, to obtain an upper bound on the minimal Lyapunov exponent.

Subject to additional constraints on the parameter values,
we have shown that (<ref>) is transitive on cl(W^u(X)), Theorem <ref>.
We have also identified a forward invariant set Δ⊆Ω_ trap
with the property that ⋂_n=0^∞ f^n(Δ) =  cl(W^u(X)).
We have not proved that there do not exist other attractors in Ω_ trap;
certainly there may be other invariant sets as in Fig. <ref>.

It remains to extend Theorems <ref> and <ref>
to larger regions of parameter space.
For instance we believe the constraint in Theorem <ref>
that both pieces of f are area-contracting is unnecessary.
It also remains to extend the ergodic theory results of <cit.> for the Lozi map
to the more general border-collision normal form, and extend results to higher dimensions.



Finally we discuss consequences for border-collision bifurcations.
The border-collision normal form contains the leading order terms of a piecewise-smooth map
in the neighbourhood of a border-collision bifurcation.
Assuming the bifurcation occurs when a parameter μ is zero,
and with μ > 0 a scaling has been done such that the constant term [μ,0]^ T
is transformed to [1,0]^ T, then the nonlinear terms that have been neglected to produce (<ref>)
are order μ (assuming the map is piecewise-C^2).
In this way the effect of the nonlinear terms increases
as the value of μ increases to move away from the border-collision bifurcation at μ = 0.
We believe that the features we have used to construct robust chaos
are also robust to these nonlinear terms.
This is because small nonlinear terms will not destroy
transverse intersections of invariant manifolds,
the existence of trapping region,
or the existence of an invariant expanding cone.







§ ACKNOWLEDGEMENTS


The authors were supported by Marsden Fund contract MAU1809,
managed by Royal Society Te Apārangi.








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