Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic $\mathbf{Q}_8$ Group
Adrian C. Murza
math.DS 37C80, 37G40, 34C15, 34D06, 34C15



Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic 𝐐_8 Group]
Heteroclinic Cycles in ODEs with the Symmetry of the Quaternion 𝐐_8 Group


Adrian C. Murza, Institute of Mathematics “Simion Stoilow”
of the Romanian Academy, Calea Griviţei 21, 010702 Bucharest, Romania
adrian_murza@hotmail.com






In this paper we analyze the heteroclinic cycle and the Hopf bifurcation of a generic dynamical system with the symmetry of the group 𝐐_8, constructed via a Cayley graph. While the Hopf bifurcation is similar to that of a 𝐃_8–equivariant system, our main result comes from analyzing the system under weak coupling. We identify the conditions for heteroclinic cycle between three equilibria in the three–dimensional fixed point subspace of a certain isotropy subgroup of 𝐐_8×𝐒^1. We also analyze the stability of the heteroclinic cycle.





[2000]37C80, 37G40, 34C15, 34D06, 34C15


[
    Adrian C. Murza
    December 30, 2023
=====================







§ INTRODUCTION



Heteroclinic cycles with in systems with symmetry have been widely studied over the large decades <cit.>. During the last couple of years a special interest has received the existence of heteroclinic cycles in systems related with quaternionic symmetry, see for example the works of X. Zhang <cit.> and O. Podvigina <cit.>. This is basically due to two facts. On the one hand quaternions are involved in the study of heteroclinic cycles in ODEs with symmetry in a natural way, owing to the easy representation of the dynamics in ℝ^4 in terms of quaternions. Many of the dynamical systems giving rise to heteroclinic cycles studied so far are 𝐃_n–equivariant; the action of 𝐃_n in ℝ^2 is absolutely irreducible, so ℝ^4 is 𝐃_n–simple. Therefore, quaternionic representations in ℝ^4 turned out to be very useful. On the other hand there is the intrinsic interest in the differential equations where the variables are the quaternions. We relate the study of heteroclinic cycles with the dynamics of networks of n coupled oscillators with symmetry. Ashwin and Swift <cit.> showed that the symmetry group of the network can be considered a subgroup of 𝕊_n, as long as the oscillators taken individually have no internal symmetries. Besides these two main reasons, there are also other ones that stimulates the analysis of dynamical systems with the quaternionic symmetry, and these are related to applications to other sciences. For example we can cite the heteroclinic phenomena observed in systems with quaternionic symmetry such as nematic liquid crystals <cit.>, particle physics <cit.> and improving computational efficiency <cit.>; however, these heteroclinic behaviors in such systems have never been encountered a theoretical explanation. This is one of our major motivation, together with the intrinsic value of the mathematical theory developed around this subject.

An important step in designing oscillatory networks with the symmetry of a specific group has been developed by Stork <cit.>.
The authors have shown how to construct an oscillatory network with certain designed symmetry, by the Cayley graph of the symmetry group.

In this paper we analyze the heteroclinic cycles and Hopf bifurcation in ODEs with the symmetry of the quaternionic group 𝐐_8 of order 16.
We use the methodology developed by Ashwin and Stork <cit.> to construct a network of differential systems with 𝐐_8 symmetry. We investigate the dynamical behavior of the system under the weak coupling. In this case we reduce the asymptotic dynamics to a flow on an sixteen-dimensional torus 𝕋^16. We prove the existence of heteroclinic cycles between the three steady–states existing within a three–dimensional fixed–point subspace of one of the isotropy subgroups of 𝐐_8×𝐒^1, namely 𝐙_2. We also classify the stability of heteroclinic cycles.

The paper is organized as follows. In Section <ref> we construct the most general oscillatory system with the 𝐐_8 symmetry by using the Cayley graph of this group. In Section <ref> we analyze the Hopf bifurcation of the constructed array. In Section <ref> we prove the existence of heteroclinic cycles in some of the subspaces which are invariant under the action of certain isotropy subgroups of 𝐐_8. We also analyze their stability.









§ THE CAYLEY GRAPH OF THE 𝐐_8 GROUP

In this section we construct an oscillatory system with the 𝐐_8 symmetry and describe the elements of this group, as the relationships between them. For more details about the use of the Cayley graph in constructing the network with the prescribed symmetry see <cit.> or <cit.>. The Cayley graphs for 𝐐_8 is shown in Figure (<ref>).




The action of the group 𝐐_8 on the cells can be written as

    [                                                  Id;         a=(1 2 3 4 9 10 11 12)(5 16 15 14 13 8 7 6);       b=(1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16);      ab=(1 16 9 8)(2 5 10 13)(3 6 11 14)(4 7 12 15); b^2=(1 9)(2 10)(2 11)(4 12)(5 13)(6 14)(7 15)(8 16);     a^2=(1 3 9 11)(2 4 10 12)(5 15 13 7)(6 16 14 8);       a^3=(1 4 11 2 9 12 3 10)(5 14 7 16 13 6 15 8);       ab^2=(1 10 3 12 9 2 11 4)(5 8 15 6 13 16 7 4);  a^2b^2=(1 11 9 3)(2 12 10 4)(5 7 13 15)(6 8 14 16);    a^3b^2=(1 12 11 10 9 4 3 2)(5 6 7 8 13 14 15 16);     ba=(1 6 9 14)(2 7 10 15)(3 8 11 16)(4 13 12 15);   ba^2=(1 7 9 15)(2 8 10 16)(3 13 11 5)(4 14 12 16);    b^3=(1 13 9 5)(2 14)(3 15 11 7)(4 16 12 8)(6 10);    ab^3=(1 8 9 16)(2 13 10 5)(3 4 15 14)(6 11 12 7);    a^3b=(1 14 9 6)(2 15 10 7)(3 16 11 8)(4 5 12 13);    a^2b=(1 15 9 7)(2 16 10 8)(3 5 11 13)(4 6 12 14) ]

with the relationship between them

    [ a^8=Id,   a^4=b^2=abab,   aba=b ]



If we assign coupling g between cells related by a and coupling h between cells related by b, from the permutations in (<ref>), we can build the following pairwise system in   with the 𝐐_8 symmetry.


    [          ẋ_1=f(x_1)+g(x_12,x_1)+h(x_5,x_9),ẋ_2=f(x_2)+g(x_1,x_2)+h(x_9,x_13),;           ẋ_3=f(x_3)+g(x_2,x_3)+h(x_13,x_1),ẋ_4=f(x_4)+g(x_3,x_4)+h(x_1,x_5),;          ẋ_5=f(x_5)+g(x_4,x_9)+h(x_2,x_6),ẋ_6=f(x_6)+g(x_9,x_10)+h(x_6,x_10),;     ẋ_7=f(x_7)+g(x_10,x_11)+h(x_10,x_14),ẋ_8=f(x_8)+g(x_11,x_12)+h(x_14,x_2),;        ẋ_9=f(x_9)+g(x_6,x_5)+h(x_3,x_7),ẋ_10=f(x_10)+g(x_5,x_16)+h(x_7,x_11),; ẋ_11=f(x_11)+g(x_16,x_15)+h(x_11,x_15),ẋ_12=f(x_12)+g(x_15,x_14)+h(x_15,x_3),;    ẋ_13=f(x_13)+g(x_14,x_13)+h(x_4,x_8),ẋ_14=f(x_14)+g(x_13,x_8)+h(x_8,x_12),;     ẋ_15=f(x_15)+g(x_8,x_7)+h(x_12,x_16),ẋ_16=f(x_16)+g(x_7,x_6)+h(x_16,x_4), ]

where f:ℝ→ℝ and g, h:ℝ^2→ℝ. As shown by Ashwin and Stork <cit.> we can think of f, g, h as being generic functions that assure that the isotropy of this vector field under the action of 𝐎_16 is generically  𝐐_8.

















§ HOPF BIFURCATION

In order to consider generic one-parameter Hopf bifurcation in systems with 𝐐_8, we need to analyze the complex irreducible representations of 𝐐_8. Based on the work of Golubitsky and Stewart <cit.>, these representations are of one or two dimensions. From their theory, we have that the linear representation of a group Γ

    α_Γ:Γ× W→ W

on the complex vector space W is irreducible if and only if Γ-invariant subspaces are trivial; it is to say, {0} or W itself. It is important to notice, that (a) there need be no faithful irreducible representations, and (b) this is typical.In addition, the amount by which the representation fails to be faithful is the kernel of the action α_Γ.


The group 𝐐_8 has five irreducible representations; four of them are one-dimensional and the remaining one is two-dimensional. The one-dimensional representations can be interpreted as Hopf bifurcation with trivial or 𝐙_2 symmetry, which correspond to a quotient group of 𝐐_8.


From <cit.> the two generators of 𝐐_8 are

    a=
    (
    [  ω  0;  0 ω̅ ]),   b=
    (
    [  0 -1;  1  0 ]),

where ω=exp(π i/4).

Therefore the standard irreducible action of 𝐐_8 on ℂ^2 is given by

    [ a(z_+,z_-)=(  √(2)/2(1+i)z_+,√(2)/2(1-i)z_-);                                             ;                      b(z_+,z_-)=(  -z_-,z_+);                                              ]

and there is a phase shift action of 𝐒^1 given by

    R_ϕ(z_+,z_-)=(e^iϕz_+,e^iϕz_-),

for ϕ∈𝐒^1. The action of 𝐐_8×𝐒^1 is similar to the action of 𝐃_8×𝐒^1. This action is generated by

    [ κ(z_+,z_-)=(z_-,z_+),     ρ(z_+,z_-)=(iz_+,-iz_-), ]

where ρ^4=κ^2=1 and ρκ=κρ^3. In our case of the group 𝐐_8 we have a^8=b^4=1,a^4=b^2 and aba=b. The kernel of this action in the 2-cycle in 𝐃_8×𝐒^1 is generated by (ρ^2,π), while the kernel of the action of 𝐐_8×𝐒^1 is generated by (a^4=b^2=1,π). Therefore, it is possible to check that

    𝐐_8×𝐒^1/kerα_𝐐_8×𝐒^1≡𝐐_8×𝐒^1/
    kerα_𝐃_8×𝐒^1.

This means that we use the results obtained in <cit.> for Hopf bifurcation in systems with 𝐃_8 symmetry, with a re-interpretation of the branches.






There are exactly three branches of periodic solutions that bifurcate from (0,0), corresponding to the isotropy subgroups 𝐃_8×𝐒^1 with two-dimensional fixed-point subspaces.


The proof is a direct application to the group 𝐃_8 of Theorem 4.2 of <cit.>. Therefore, there are exactly three branches of periodic solutions occurring generically in Hopf bifurcation with 𝐃_8 symmetry.


From Proposition 2.1, page 372 in <cit.> we have that every smooth 𝐃_8×𝐒^1-equivariant map germ g:ℂ^2→ℂ^2 has the form

    [ g(z_1,z_2)=A[ z_1; z_2 ]+B[ z_1^2z̅_1; z_2^2z̅_2 ]+
    C[ z̅_1^3z_2^4; z_1^4z̅_2^3 ]+D[ z_1^5z̅_2^4; z̅_1^4z_2^5 ], ]

where A, B, C, D are complex-valued 𝐃_8×𝐒^1-invariant functions.
The branching equations for 𝐃_8-equivariant Hopf bifurcation may be rewritten g(z_1,z_2)=0.
These branching equations are shown in Table (<ref>).



 §.§ Bifurcating branches

We now use the information in Table (<ref>) to derive the bifurcation diagrams describing the generic 𝐃_8-equivariant Hopf bifurcation. Assume

    [ (a)  Re(A_N+B)≠0,  (b)  Re(B)≠0,  (c)  Re(2A_N+B)≠0,  (d)  Re(BC̅)≠0,  
    (c)  Re(A_λ)≠0, ]

where each term is evaluated at the origin.

Assuming nondegeneracy conditions (<ref>) and the trivial branch is stable subcritically and loses stability as bifurcation parameter λ passes through 0. We summarize these facts into the next theorem.

The following statements hold.


  (a) The 𝐙̃_8 branch is super- or subcritical according to whether Re(A_N(0)+B(0)) is positive or negative. It is stable if Re(A_N(0)+B(0))>0, Re(B(0))<0.

  (b) The 𝐙_2(κ)[⊕𝐙_2^c] is super- or subcritical according to wether Re(2A_N(0)+B(0)) is positive or negative. It is stable if Re(2A_N(0)+B(0))>0, Re(B(0))>0 and Re(2B(0)C̅(0))<0.

  (c) The 𝐙_2(κ,π)[⊕𝐙_2^c] or 𝐙_2(κ,ξ)⊕𝐙_2^c branch is super- or subcritical according to wether Re(2A_N(0)+B(0)) is positive or negative. It is stable if Re(2A_N(0)+B(0))>0, Re(B(0))>0 and Re(2B(0)C̅(0))<0.



The proof is a direct application to the case 𝐃_8 of the Theorem 3.1 page 382 in <cit.>.







§ WEAK COUPLING


The idea of studying ODEs in the weak coupling limit was introduced by Ashwin and Swift <cit.>. This situation can be uderstood as follows. In the no coupling case there is an attracting n–dimensional torus with one angle for every oscillator. The situation is completely different to the Hopf bifurcation. Instead of examining small amplitude oscillations near a Hopf bifurcation point, we make a weak coupling approximation. There is a slow evolution of the phase differences in the weak coupling. Another improvement with respect to the Hopf bifurcation is that while the Hopf bifurcation theory gives local information, the weak coupling case the yields global results on the n–dimensional torus.


System (<ref>) can be rewritten under weak coupling case as an ODE of the form:

    ẋ_i=f(x_i)+ϵ g_i(x_1,…,x_16)

for i=1,…,16, x_i∈𝒱 and commuting with the permutation action of 𝐐_8 on 𝒱^16, both f and g_i being of the class 𝒞^∞. The constant ϵ represents the coupling strength and we have ϵ≪1. As in <cit.>, or <cit.> we may assume ẋ=f(x) has an hyperbolic stable limit cycle.


It follows that if the coupling is weak, we should not just take into account the irreducible representations of 𝐐_8. Since there are 16 stable hyperbolic limit cycles in the limit of ϵ=0, it means that the asymptotic dynamics of the system factors into the asymptotic dynamics of 16 limit cycles. We assume that each limit cycle taken individually is hyperbolic for small enough values of the coupling parameter. This justifies expressing the dynamics of the system only in terms of phases, i.e. an ODE on 𝐓^16  which is 𝐐_8-equivariant.






When considering the weakly coupled system we can average it over the phases <cit.>. This is the same as introducing and phase shift symmetry by translation along the diagonal;

    R_θ(ϕ_1,…,ϕ_16):=(ϕ_1+θ,…,ϕ_16+θ),

for θ∈𝐒^1.

We obtained an ODE on that is equivariant under the action of 𝐐_8×𝐒^1,  and we have to classify the isotropy types of points under this action. This is done in Table (<ref>).
Since now on, our interest focuses in the three-dimensional space Fix(𝐙_2); it does not contain two-dimensional fixed-point subspaces. In turn, it contains several one- and zero-dimensional subspaces fixed by the isotropy subgroups 𝐐̃_8^i, 𝐙̃_8^i and 𝐙_8^i, respectively, where i={a,b,ab,a/4,b/4} as in Table (<ref>). These symmetries are not in 𝐙_2; however, they are in the normalizer of 𝐙_2.






































 §.§ Dynamics of the θ_1,θ_2 and θ_3 angles in Fix(𝐙_2)

We can define coordinates in Fix(𝐙_2) by taking a basis

    [                e_1=-1/8(1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1); e_2=-1/8(1, -1, 1, -1 ,1, -1, 1, -1 ,1, -1, 1, -1, 1, -1, 1, -1); e_3=-1/8(1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1) ]

and consider the space spanned by {e_1,e_2,e_3} parametrized by {θ_1,θ_2,θ_3}: ∑_n=1^3θ_ne_n.

By using these coordinates, we construct the following family of three-dimensional differential systems which satisfies the symmetry of Fix(𝐙_2).

    {[                      θ̇_̇1̇=usinθ_1cosθ_2+ϵsin2θ_1cos2θ_2;                                                          ;                      θ̇_̇2̇=usinθ_2cosθ_3+ϵsin2θ_2cos2θ_3;                                                          ; θ̇_̇2̇=usinθ_3cosθ_1+ϵsin2θ_3cos2θ_1
    +q(1-cosθ_1)sin2θ_3,;                                                           ].

where u,ϵ,q∈ℝ.

We will show that this vector field contains structurally stable, attracting heteroclinic cycles which may be asymptotically stable, essentially asymptotically stable or completely unstable, depending on the values of u,ϵ and q. We can assume, without loss of genericity that the space Fix(𝐙_2) is normally attracting for the dynamics and therefore the dynamics within the fixed-point space determines the stability of the full system.
In the following we will show that the planes θ_i=0 (mod π),i=1,2,3 are invariant under the flow of (<ref>).

Let 𝒳 be the vector field of system (<ref>).

We call a trigonometric invariant algebraic surface h(θ_1,θ_2,θ_3)=0, if it is invariant by the flow of (<ref>), i.e. there exists a function K(θ_1,θ_2,θ_3) such that

    𝒳h=∂ h/∂θ_1θ̇_̇1̇+∂ h/∂θ_2θ̇_̇2̇
    +∂ h/∂θ_3θ̇_̇3̇=Kh.







Functions sinθ_1, sinθ_2 and sinθ_3 are trigonometric invariant algebraic surfaces for system (<ref>).


We can write the system (<ref>) in the form

    {[                     θ̇_̇1̇=sinθ_1(ucosθ_2+2ϵcosθ_1cos2θ_2);                                                           ;                     θ̇_̇2̇=sinθ_2(ucosθ_3+2ϵcosθ_2cos2θ_3);                                                           ; θ̇_̇3̇=sinθ_3(ucosθ_1+2ϵcos2θ_1cosθ_3
    +2q(1-cosθ_1)cosθ_3);                                                            ].



Now if we choose h_1=sinθ_1, then 𝒳h_1=cosθ_1sinθ_1(ucosθ_2+2ϵcosθ_1cos2θ_2)
so K_1=cosθ_1(ucosθ_2+2ϵcosθ_1cos2θ_2). The remaining cases follow similarly.


Since the planes θ_i=0(mod π) are invariant under the flow of (<ref>), it is clear that (0,0,0), (π,0,0), (0,π,0), and (0,0,π) are equilibria for (<ref>). To check the possibility of heteroclinic cycles in system (<ref>), we linearize about the equilibria (i.e. the zero-dimensional fixed points). The idea is proving that there are three-dimensional fixed-point spaces Fix(𝐙_2) and Fix(𝐙̃_2) which connect these fixed points, allowing the existence of such a heteroclinic network between the equilibria.

Let's assume

    [ |ϵ|<u/2    and    
    |ϵ+2q|<u/2. ]





We use the criteria of Krupa and Melbourne <cit.> to study the stability of the heteroclinic cycle.

In the following we will prove that there exists the possibility of a heteroclinic cycle in the following way:

    [ ⋯Fix(𝐐̃_8^a)Fix(𝐐̃_8^b)Fix(𝐐̃_8^ab)
    ⋯ ]

The stability of the heteroclinic cycle is:


  (a) asymptotically stable if

    [ u<0 and q<3u/4-ϵ/2, ]


  (b) unstable but essentially asymptotically stable if

    [ u<0 and 3u/4-ϵ/2<q<u/2-(u+2ϵ)^3/(-u+2ϵ)^2. ]


  (c) completely unstable if u>0.




The stability is expressed by

    ρ=∏_i=1^3ρ_i,   where   ρ_i=min{c_i/e_i,1-t_i/e_i}.


In equation (<ref>), e_i is the expanding eigenvalue at the ith point of the cycle, -c_i is the contracting eigenvalue and t_i is the transverse eigenvalue of the linearization.
For the heteroclinic cycle we have

    ρ_1=
    {[ 2u-4q/u+2ϵ if q<3u/4-ϵ/2,;                          ; -u+2ϵ/u+2ϵ if q>3u/4-ϵ/2, ]        ρ_2=ρ_3=-u+2ϵ/u+2ϵ,
    .


so from equations (<ref>) and (<ref>) we obtain

    ρ=
    {[ (-u+2ϵ)^2(2u-4q)/(u+2ϵ)^3 if u<0 and q<3u/4-ϵ/2,;                                                 ;        (-u+2ϵ)^3/(u+2ϵ)^3 if u<0 and q>3u/4-ϵ/2. ].

Then the proof follows by applying Theorem 2.4 in <cit.>.


For any u<0 we have 3u/4-ϵ/2<q<u/2-(u+2ϵ)^3/(-u+2ϵ)^2 and therefore there exist values of q for which there exist essentially asymptotic stable heteroclinic connections. In consequence, there exists an attracting heteroclinic cycle even though the linear stability of Fix(𝐐̃_8^a) has an expanding transverse eigenvalue.



§ CONCLUSIONS

We prove the existence of stable heteroclinic cycles in the most general coupled ordinary differential equations with quaternionic symmetry 𝐐_8. Our approach is generic and offers for the first time as far as we know, evidence of these phenomena in systems with this symmetry. While the results stands on its own from a mathematical point of view, it might also contribute to a better understanding of these intermittent behaviors experimentally observed in nematic liquid crystals <cit.> and particle physics <cit.>.



§ ACKNOWLEDGEMENTS

In first place I would like to address many thanks to the Referee, whose helpful indications and comments greatly improved the presentation of the paper. I acknowledge a BITDEFENDER
postdoctoral fellowship from the Institute of Mathematics Simion
Stoilow of the Romanian Academy, Contract of Sponsorship No.
262/2016 as well as economical support from a grant of the Romanian
National Authority for Scientific Research and Innovation,
CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0657.

10


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