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Analysis of dynamics near heteroclinic networks in $\mathbb{R}^{4}$ with a projected map

David C. Groothuizen Dijkema, Vivien Kirk, Claire M. Postlethwaite

Abstract

Heteroclinic cycles and networks are structures in dynamical systems composed of invariant sets and connecting heteroclinic orbits, and can be robust in systems with invariant subspaces. The usual method for analysing the stability of heteroclinic cycles and networks is to construct return maps to cross-sections near the network. From these return maps, transition matrices can be defined, whose eigenvalues and eigenvectors can be analysed to determine stability. In this paper, we introduce an extension to this methodology, the projected map, which we define by identifying trajectories with, in a certain sense, qualitatively the same dynamics. The projected map is a discrete, piecewise-smooth map of one dimension one fewer than the rank of the transition matrix. We use these maps to describe the dynamics of trajectories near three heteroclinic networks in $\mathbb{R}^{4}$ with four equilibria. We find in all three cases that the onset of trajectories that switch between cycles of the network can be caused by a fold bifurcation or border-collision bifurcation, where fixed points of the map no longer exist in the corresponding function's domain of definition. We are able to show that trajectories near these three networks are asymptotic to only one subcycle, and cannot switch between subcycles multiple times, in contrast to examples with more than four equilibria, and resolving a 30-year-old claim by Brannath. We are also able to generalise some results to all quasi-simple networks, showing that a border-collision bifurcation of the projected map, corresponding to a condition on the eigenvectors of certain transition matrices, causes a cycle to lose stability.

Analysis of dynamics near heteroclinic networks in $\mathbb{R}^{4}$ with a projected map

Abstract

Heteroclinic cycles and networks are structures in dynamical systems composed of invariant sets and connecting heteroclinic orbits, and can be robust in systems with invariant subspaces. The usual method for analysing the stability of heteroclinic cycles and networks is to construct return maps to cross-sections near the network. From these return maps, transition matrices can be defined, whose eigenvalues and eigenvectors can be analysed to determine stability. In this paper, we introduce an extension to this methodology, the projected map, which we define by identifying trajectories with, in a certain sense, qualitatively the same dynamics. The projected map is a discrete, piecewise-smooth map of one dimension one fewer than the rank of the transition matrix. We use these maps to describe the dynamics of trajectories near three heteroclinic networks in with four equilibria. We find in all three cases that the onset of trajectories that switch between cycles of the network can be caused by a fold bifurcation or border-collision bifurcation, where fixed points of the map no longer exist in the corresponding function's domain of definition. We are able to show that trajectories near these three networks are asymptotic to only one subcycle, and cannot switch between subcycles multiple times, in contrast to examples with more than four equilibria, and resolving a 30-year-old claim by Brannath. We are also able to generalise some results to all quasi-simple networks, showing that a border-collision bifurcation of the projected map, corresponding to a condition on the eigenvectors of certain transition matrices, causes a cycle to lose stability.

Paper Structure

This paper contains 32 sections, 7 theorems, 66 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Heteroclinic cycles and networks
  4. Itineraries of trajectories near heteroclinic networks
  5. Stability of invariant sets
  6. Return maps and the projected map of the Kirk--Silber network
  7. Return maps near the Kirk--Silber network
  8. Motivating the projected map
  9. The projected map of the Kirk--Silber network
  10. Analysis of projected maps
  11. The Kirk--Silber network
  12. Fixed points and admissibility
  13. Stability
  14. Equivalence of the border-collision bifurcation and Podvigina's third stability condition
  15. Discontinuity of the projected map of the Kirk--Silber network at theta s
  16. ...and 17 more sections

Key Result

Theorem 3.1

\newlabelthm:pod_bif0 For $1\leq j\leq m$, let $M_{j}$ be the full transition matrices associated with a heteroclinic cycle $\mathcal{C}_{}$. Let $\lambda_{\max}$ denote the eigenvalue of $M_{j}$ with largest absolute value, and $w_{\max}$ an associated eigenvector. The heteroclinic cycle $\mathca

Figures (15)

  • Figure 1: Diagrammatic representations of the three heteroclinic networks between four equilibria in $\mathop{\mathrm{\mathbb{R}}}\nolimits^{4}$. Coloured vertices represent hyperbolic saddle equilibria, and directed edges represent robust heteroclinic orbits.
  • Figure 1: Examples of the action of the return map $\Phi\colon\mathbf{H}_{2}^{\mathrm{in}}\to\mathbf{H}_{2}^{\mathrm{in}}$. Points in the same orbit have been joined for clarity only. The domain $\Gamma_{3}$ is shaded blue, $\Gamma_{4}$ orange, and the excluded cusp $\Gamma_{c}$ black. The dashed blue and orange lines are the curves $\Sigma_{3}^{*}$ and $\Sigma_{4}^{*}$ (defined in \ref{['sec:ret_proj_map:ssec:explanation']}). The switching curve $\Sigma_{s}$ is a dashed red line, and $\Sigma_{c}^{-}$ and $\Sigma_{c}^{+}$ are solid red lines.
  • Figure 1: Schematic representations of the projected map of the Kirk--Silber network as cobweb plots, for different relations between parameters, as indicated. In all examples, the domain $\Theta_{3}$ is shaded blue and the domain $\Theta_{4}$ orange. The dashed red line is $\vartheta_{s}$. The dotted black line is $f\left(\vartheta\right)=\vartheta$. Each function $f_{3}$ and $f_{4}$ is shown as solid in its domain of definition and dashed otherwise. In all four figures, $\delta_{4}>1$ and $\nu_{4}>0$. In \ref{['fig:ks_proj_map:subfig:post_bcb']} and \ref{['fig:ks_proj_map:subfig:post_res_bcb']}, the values $\mathcal{E}_{n}$ (see \ref{['eqn:theta_s_pre_image']}) are indicated by dashed black lines for $n=1,2,3,4$.
  • Figure 1: Two examples of points in $S$ that are not $\vartheta_{s}$, given by the coloured dots on $S$, the solid green line. The linear subspace through them is shown as solid for vectors in the domain of $M$, and dotted for vectors in $\mathcal{D}_{c}$. The solid lines are examples of the half-lines $L(\vartheta)$. The point $p(\vartheta)$ (see \ref{['eqn:intersection_point']}) is shown as a correspondingly coloured point on the solid red lines.
  • Figure 2: Examples of the action of the map $M\colon\mathop{\mathrm{\mathbb{R}}}\nolimits_{\hbox{[}1.0]{--}}^{2}\to\mathop{\mathrm{\mathbb{R}}}\nolimits_{\hbox{[}1.0]{--}}^{2}$, where vectors extend from the origin to the black dots, and the linear subspace containing each vector is a solid grey line. The domain of $\mathcal{D}_{3}$ is shaded blue, $\mathcal{D}_{4}$ orange, and $\mathcal{D}_{c}$ black. The dashed blue and orange lines are the eigenspaces $W_{3}^{*}$ and $W_{4}^{*}$. The vectors can be seen to converge to these eigenspaces under iteration of $M$. The switching subspace $W_{s}$ is a dashed red line, and the two affine subspaces $W_{c}^{-}$ and $W_{c}^{+}$ are solid red lines. For clarity, \ref{['fig:ks_maps_log_coords:subfig:post_BCB']} only shows the logarithm of the coordinates of the trajectory in \ref{['fig:ks_maps_normal_coords:subfig:post_BCB']} that switches from the $\mathcal{C}_{[3]}$ to $\mathcal{C}_{[4]}$ cycle.
  • ...and 10 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1: podvigina_2012
  • Theorem 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Proposition 4.4
  • Theorem 5.1
  • Proposition A.1
  • ...and 1 more