Heteroclinic cycles

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A heteroclinic cycle is a collection of solution trajectories that connects sequences of equilibria, periodic solutions or chaotic invariant sets via saddle-sink connections. For a more precise description of heteroclinic cycles and their stability, see Melbourne et al. (1989), Krupa and Melbourne (1995), the monograph by Field (1996), and the survey article by Krupa (1997). Such behavior is unusual in a general dynamical system. It is, however, a generic feature of dynamical systems that possess symmetry. Indeed, the presence of symmetry can lead to invariant subspaces under which a sequence of saddle-sink connections can be established, resulting in cycling behavior. As time evolves, a typical trajectory would stay for increasingly longer period of time near each solution (which could be either an equilibrium, a periodic orbit or a chaotic invariant set) before it makes a rapid excursion to the next solution. Since saddle-sink connections are robust, these cycles— called heteroclinic cycles—are robust under perturbations that preserve the symmetry of the system.

Homoclinic cycles are a specific case of heteroclinic cycles in which the sequence of connections joins invariant solutions (equilibria, periodic solutions or chaotic sets) which belong to the same group orbit.


Finding Heteroclinic Cycles

Melbourne, Chossat, and Golubitsky (1989), describe a method for finding heteroclinic cycles in symmetric systems of differential equations. Let \(\Gamma\subset O(N)\) be a Lie subgroup (where \(O(N)\) denotes the orthogonal group of order \(N\)) and let \(g:\R^N\to\R^N\) be \(\Gamma\)-equivariant, that is, \[ g(\gamma X)=\gamma g(X), \] for all \(\gamma \in \Gamma\ .\) Consider the system \[ \frac{dX}{dt}=g(X). \] Note that \(N=kn\) in an \(n\)-cell system with \(k\) state variables in each cell. Equivariance of \(g\) implies that whenever \(X(t)\) is a solution, so is \(\gamma X(t)\ .\) Using fixed-point subspaces, Melbourne et al. (1989) suggest a method for constructing heteroclinic cycles connecting equilibria. Suppose that \(\Sigma\subset\Gamma\) is a subgroup. Then the fixed-point subspace \[ \mbox{Fix}(\Sigma)=\{X\in\R^N: \sigma X=X \quad\forall \sigma\in\Sigma\} \] is a flow invariant subspace. The idea is to find a sequence of maximal subgroups \(\Sigma_j\subset\Gamma\) such that \(\dim \mbox{Fix}(\Sigma_j)=1\) and submaximal subgroups \(T_j\subset\Sigma_j\cap\Sigma_{j+1}\) such that \(\dim \mbox{Fix}(T_j)=2\ ,\) as is shown schematically in Figure 1. In addition, the equilibrium in \(\mbox{Fix}(\Sigma_j)\) must be a saddle in \(\mbox{Fix}(T_j)\) whereas the equilibrium in \(\mbox{Fix}(\Sigma_{j+1})\) must be a sink in \(\mbox{Fix}(T_j)\ .\)

Figure 1: Pattern inside lattice of subgroups that suggests the existence of heteroclinic cycles.

Such configurations of subgroups have the possibility of leading to heteroclinic cycles if saddle-sink connections between equilibria in \(\mbox{Fix}(\Sigma_j)\) and \(\mbox{Fix}(\Sigma_{j+1})\) exist in \(\mbox{Fix}(T_j)\ .\) It should be emphasized that more complicated heteroclinic cycles can exist. Generally, all that us needed to be known is that the equilibria in \(\mbox{Fix}(\Sigma_j)\) is a saddle and the equilibria in \(\mbox{Fix}(\Sigma_{j+1})\) is a sink in the fixed-point subspace \(\mbox{Fix}(T_j)\) (see Krupa and Melbourne (1995)) though the connections can not, in general, be proved. Since saddle-sink connections are robust in a plane, these heteroclinic cycles are stable to perturbations of \(g\) so long as \(\Gamma\)-equivariance is preserved by the perturbation. For a detailed discussion of asymptotic stability and nearly asymptotic stability of heteroclinic cycles, which are also very important topics, see Krupa and Melbourne (1995).

Cycles Involving Periodic Solutions and Broken Symmetry

Near points of Hopf bifurcation, this method for constructing heteroclinic connections can be generalized to include time periodic solutions as well as equilibria. Melbourne, Chossat, and Golubitsky (1989) do this by augmenting the symmetry group of the differential equations with \(\mbox{S}^1\) --- the symmetry group of Poincare-Birkhoff normal form at points of Hopf bifurcation --- and using phase-amplitude equations in the analysis. In these cases the heteroclinic cycle exists only in the normal form equations since some of the invariant fixed-point subspaces disappear when symmetry is broken. However, when that cycle is asymptotically stable, then the cycling like behavior remains even when the equations are not in normal form. This is proved by using asymptotic stability to construct a flow invariant neighborhood about the cycle and then invoking normal hyperbolicity to preserve the flow invariant neighborhood when normal symmetry is broken. Indeed, as is shown by Melbourne (1989), normal form symmetry can be used to produce stable cycling behavior even in systems without any spatial symmetry. More generally, it also follows that if an asymptotically stable cycle can be produced in a truncated normal form equation (say truncated at third or fifth order), then cycling like behavior persists in equations with higher order terms --- even when those terms break symmetry --- and the cycling like behavior is robust.


The Guckenheimer-Holmes Cycle

Figure 2: Saddle-sink connections in phase space depict a heteroclinic cycle connecting three equilibrium points of the Guckenheimer-Holmes system. Parameters are\[\mu=1.0, a=1.0, b=0.55, c=1.5\ .\]

Figure 2 shows an example of a heteroclinic cycle involving three steady-states or equilibrium solutions of a system of ordinary differential equations (ODE's) proposed by Guckenheimer and Holmes (1988). However, this cycle was first written down, and its behaviour described, by Busse and Heikes (1980).

The group \(\Gamma \) in this example has 24 elements and is generated by the following symmetries \[ \begin{array}{l} (x,y,z)\mapsto (\pm x,\pm y,\pm z)\\ (x,y,z)\mapsto (y,z,x) \end{array} \]

Note that, in fact, this is a homoclinic cycle since the three equilibria are on the group orbit given by the cyclic generator of order 3. The actual system of ODE's can be written in the following form \[ \begin{matrix} \dot{x}_1 &=& \mu x_1 - (a x_1^2+b x_2^2+ cx_3^2)x_1 \\ \dot{x}_2 &=& \mu x_2 - (a x_2^2+b x_3^2+ cx_1^2)x_2 \\ \dot{x}_3 &=& \mu x_3 - (a x_3^2+b x_1^2+ cx_2^2)x_3\;. \end{matrix} \]

In related work that describes cycling chaos, Dellnitz et al. (1995) point out that the Guckenheimer-Holmes system can be interpreted as a coupled cell system (with three cells) in which the internal dynamics of each cell is governed by a pitchfork bifurcation of the form \[ \dot{x}_i = \mu x_i - a x_i^3, \] where \(i=1,2,3\) is the cell number. As \(\mu\) varies from negative to positive through zero, a bifurcation from the trivial equilibrium \(x_i=0\) to nontrivial equilibria \(x_i= \pm \sqrt{\mu}\) occurs. Guckenheimer and Holmes (1988) show that when the strength of the remaining terms in the system of ODE's (which can be interpreted as coupling terms) is large, an asymptotically stable heteroclinic cycle connecting these bifurcated equilibria exists. The connection between the equilibria in cell one to the equilibria in cell two occurs through a saddle-sink connection in the \(x_1 x_2-\)plane (which is forced by the internal symmetry of the cells to be an invariant plane for the dynamics). As Dellnitz et al. (1995) further indicate, the global permutation symmetry of the three-cell system guarantees connections in both the \(x_2 x_3-\)plane and the \(x_3 x_1-\)plane, leading to a heteroclinic connection between three equilibrium solutions. The Figure below shows the time-dependent trajectory of each of the individual cells or variables of the Guckenheimer-Holmes system. Observe that, as time evolves, all three trajectories spend longer periods of time near each equilibrium, as expected.

Figure 3: Time series evolution of a typical nearby trajectory of the heteroclinic cycle of the Guckenheimer-Holmes system. Parameters are\[\mu=1.0, a=1.0, b=0.55, c=1.5\ .\]

Another example is the heteroclinic cycle in the 1:2 resonance problem, which was first identified by Jones and Proctor (1987), followed up by a longer paper in J. Fluid Mech.

A Cycle in a System of ODE's with Circular Symmetry

Melbourne et al. (1989) prove the existence of robust, asymptotically stable heteroclinic cycles involving time periodic solutions in steady-state/Hopf and Hopf/Hopf mode interactions in systems with \(O(2)\)-symmetry. In these symmetry breaking bifurcations each critical eigenvalue is doubled by symmetry --- so the center manifold for a steady-state/Hopf mode interaction is six-dimensional and for a Hopf/Hopf mode interaction it is eight-dimensional. It is well known that \(O(2)\) symmetry-breaking Hopf bifurcations at invariant equilibria lead to two types of periodic solutions: standing waves (solutions invariant under a single reflection for all time) and rotating waves (solutions whose time evolution is the same as spatial rotation). The Figure below shows a cycle connecting a steady-state with a standing wave obtained from a steady-state/Hopf mode interaction by numerically integrating a general system of ODE's with \(O(2)\)-symmetry, which has the form \[ \frac{dz}{dt}=g(z,\lambda, \mu)=(C(z),P(z))\in C\times C^2, \] where \[ \begin{matrix} C(z) &=& C^1 z_0 + C^3 \bar{z}_0 z_1 \bar{z}_2 \\[15pt] P(z) &=& P^1 \left[ \begin{matrix} z_1 \\ z_2 \end{matrix}\right] + P^2 \delta \left[ \begin{matrix} z_1 \\ -z_2 \end{matrix} \right] + P^3 \left[ \begin{matrix} z_0^2 z_2 \\ \bar{z}_0^2 z_1 \end{matrix} \right], \end{matrix} \] where \(\delta=|z_2|^2-|z_1|^2\ ,\) \(C^j=c^j+i\delta c^{j+1}\ ,\) \(c^j\) are real-valued \(O(2)\times S^1\)-invariant functions and \(P^j=p^j+q^ji\) are complex-valued \(O(2)\times S^1\)-invariant functions depending on two parameters \(\lambda\) and \(\mu\ .\) The time series in this figure are taken from three different coordinates\[x_0\] is a coordinate in the steady-state mode and \(x_1, x_2\) are coordinates in the Hopf mode. In these coordinates a standing wave is an oscillation where both coordinates oscillate equally (with just a phase shift). Other types of \(O(2)\) cycles involving only periodic solutions are obtained from Hopf/Hopf mode interactions and examples are shown by Buono, Golubitsky, and Palacios (1999). These cycles connect rotating waves with rotating waves and standing waves with standing waves.

Figure 4: Heteroclinic cycle connecting a steady-state with a standing wave in a system with \(O(2)\times S^1\) symmetry with \(S^1\) symmetry due to normal form. Parameters are\[c^1=\lambda-1.5\rho-4N, \; c^2=1.3, \; c^3=-9\ ,\] \( p^1=1.2\lambda-3\rho-N, \; p^2=4, \; p^3=4\ ,\) \( q^1=0.8\lambda+7\ ,\) and all other coefficients set to zero.

A Cycle in a System of ODE's with Dihedral Symmetry

Buono, Golubitsky, and Palacios (2000) proved the existence of heteroclinic cycles involving steady-state and time periodic solutions in differential equations with \(D_n\) symmetry. In their approach, they studied various mode interactions --- in particular, the six-dimensional steady-state/Hopf mode interaction where \(D_n\) acts by its standard representation on the critical eigenspaces. The exact cycles they discussed are found in the normal form equations which have \(D_n\times S^1\) symmetry when \(n=6\) and \(n=5\) --- though much of their discussion is relevant for a general \(D_n\) system.

Consider for instance a system of differential equations with the symmetries of a hexagon, which are described by the dihedral group \(D_6\ .\) Reflectional symmetries of a hexagon come in two (nonconjugate) types: those whose line of reflection connects opposite vertices of the hexagon (\(\kappa\)) and those whose line of symmetry connects midpoints of opposite sides (\(\gamma\kappa\)). It is known that \(D_6\) symmetry-breaking steady-state bifurcations produce two nontrivial equilibria --- one with each type of reflectional symmetry --- and \(D_6\) symmetry-breaking Hopf bifurcations produce two standing waves --- one with each type of reflectional symmetry. In normal form the symmetry groups of these four solutions are \(Z_2(\kappa)\times S^1\ ,\) \(Z_2(\gamma\kappa)\times S^1\ ,\) \(Z_2(\kappa)\times Z_2^c\ ,\) and \(Z_2(\gamma\kappa)\times Z_2^c\) where \(Z_2^c=Z_2(\pi,\pi)\ .\) Using the ideas described by Melbourne et al. (1989), the Figure below suggests that robust, asymptotically stable heteroclinic cycles can appear in unfoldings of \(D_6\) normal form symmetry-breaking steady-state/Hopf mode interactions.

Figure 5: Subgroups in \(D_6\times S^1\) lattice that permit the existence of heteroclinic cycles.

The cycle would connect the first steady-state with the first standing wave with the second steady-state with the second standing wave and back to the first steady-state. A general system of ODE's with \(D_6\times S^1\)-symmetry has the form \[ \frac{dz}{dt}=g(z,\lambda, \mu)=(C(z),Q(z))\in C\times C^2, \] where \[ \begin{matrix} C(z) &=& C^1 z_0 + C^3\bar{z}_0 z_1 \bar{z}_2 + C^5 \bar{z}_0^5 + C^7\bar{z}_0 (\bar{z}_1 z_2)^2 + C^9\bar{z}_0^3\bar{z}_1 z_2 + C^{11} z_0 (z_1\bar{z}_2)^3 \\[15pt] Q(z) &=& Q^1 \left[ \begin{matrix} z_1 \\ z_2 \end{matrix}\right] + Q^2 \delta \left[ \begin{matrix} z_1 \\ -z_2 \end{matrix} \right] + Q^3 \left[ \begin{matrix} z_0^2 z_2 \\ \bar{z}_0^2 z_1 \end{matrix} \right] + Q^4 \delta \left[ \begin{matrix} z_0^2 z_2 \\ -\bar{z}_0^2 z_1 \end{matrix} \right] + Q^5 \left[ \begin{matrix} \bar{z}_0^4 z_2 \\ z_0^4 z_1 \end{matrix} \right] + Q^6 \delta \left[ \begin{matrix} \bar{z}_0^4 z_2 \\ -z_0^4 z_1 \end{matrix} \right] + \\[15pt] && Q^7 \left[ \begin{matrix} \bar{z}_0^2 \bar{z}_1 z_2^2 \\ z_0^2 z_1^2 \bar{z}_2 \end{matrix} \right] + Q^{8} \delta \left[ \begin{matrix} \bar{z}_0^2 \bar{z}_1 z_2^2 \\ -z_0^2 z_1^2 \bar{z}_2 \end{matrix} \right] + Q^{9} \left[ \begin{matrix} (\bar{z}_1 z_2)^2 z_2 \\ (z_1 \bar{z}_2)^2 z_1 \end{matrix} \right] + Q^{10} \delta \left[ \begin{matrix} (\bar{z}_1 z_2)^2 z_2 \\ -(z_1 \bar{z}_2)^2 z_1 \end{matrix} \right], \end{matrix} \] where \(\delta=|z_2|^2-|z_1|^2\ ,\) \(C^j=c^j+i\delta c^{j+1}\ ,\) \(c^j\) are real-valued \(D_6\times S^1\)-invariant functions and \(Q^j=p^j+q^ji\) are complex-valued \(D_6\times S^1\)-invariant functions depending on two parameters \(\lambda\) and \(\mu\ .\) Numerical integration of this \(D_6\times S^1\)-equivariant system of ODE's (in normal form) yields the cycle shown in the Figure below.

Figure 6: Heteroclinic cycle connecting steady states \(Z_2(\kappa)\times S^1\) and \(Z_2(\gamma\kappa)\times S^1\) with standing waves \(Z_2(\kappa)\times Z_2^c\) and \(Z_2(\gamma\kappa)\times Z_2^c\) in a system with \(D_6 \times S^1\) symmetry. Standing waves have different amplitudes. Parameters are\[c^1=\lambda-1.5\rho-4N, \; c^2=1.3, \; c^3=-9, \; c^5=0.5\ ,\] \( p^1=1.2\lambda-3\rho-N, \; p^9=4, \; p^2=4, \; p^3=4\ ,\) \( q^1=0.8\lambda+1, \; q^9=8\ ,\) and all other coefficients set to zero.

The heteroclinic cycle in the actual six-cell ring system with \( D_6 \) symmetry is now shown in the Figure below. Up to third order, the center manifold flow for this coupled cell system (after scaling) is the same as the flow in the previous Figure. For illustrating purposes, the second component of each cell is shown.

Figure 7: Heteroclinic cycle in a six-cell ring system with \( D_6 \) symmetry. For illustrating purposes, the second component of each cell is shown.

A Cycle in a PDE with Circular Symmetry

A generic example of a cellular-pattern-forming dynamical system is described by the Kuramoto-Sivashinsky (KS) equation, which can be written in the form \[ {\partial u \over \partial t} = \eta_{1} u - (1+\nabla^2)^2 u - \eta_{2} (\nabla u)^2 - \eta_{3} u^3 + \xi(\vec{x},t), \] where \(u=u(\vec{x},t)\) represents the perturbation of a planar front (normally assumed to be a flame front) in the direction of propagation, \(\eta_{1}\) measures the strength of the perturbation force, \(\eta_{2}\) is a parameter associated with growth in the direction normal to the domain (burner) of the front, \(\eta_{3} u^3\) is a term that has been added to help stabilize its numerical integration, and \(\xi(\vec{x},t)\) represents Gaussian white noise, which models thermal fluctuations, dimensionless in space and time. The KS equation describes the perturbations of a uniform wave front by thermo-diffusive instabilities. It has been studied in different contexts, including the existence of heteroclinic connections, by Cross and Hohenberg (1993), Armbruster, Guckenheimer, and Holmes (1988), Holmes, Lumley, and Berkooz (1996), and by Hyman and Nicolaenko (1986). Gassner, Blomgren, and Palacios (2007) have also conducted numerical explorations of the effects of noise on the KS equation in various regions of parameter space. The figure below shows a phase-space diagram of a low-dimensional system of ODE's derived from the KS equation in a region of parameter space where a heteroclinic cycle exists near a 1:2 mode interaction.

Figure 8: Phase-space depiction of a heteroclinic cycle found in the Kuramoto-Sivashinsky equation. Parameters are\[R=4.285\] (radius of domain of integration), \(D=0.0008\) (noise intensity), and \((\eta_1, \; \eta_2, \;\eta_3)=(0.32,\, 1.0,\, 0.17)\ .\)

In physical space, the 1:2 heteroclinic cycle represents repetitive excursions between a one-cell pattern and a two-cell pattern, as is shown in the figure below.

Figure 9: Heteroclinic solution \(u=u(\vec{x},t)\)</math> found in the Kuramoto-Sivashinsky equation. Parameters are\[R=4.285\] (radius of domain of integration), \(D=0.0008\) (noise intensity), and \((\eta_1, \; \eta_2, \;\eta_3)=(0.32,\, 1.0,\, 0.17)\ .\)


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See Also

Andronov-Hopf Bifurcation, Equivariant Bifurcation Theory, Equivariant Dynamical Systems, Heteroclinic Bifurcation, Normal Form

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