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Asymptotic stability of heteroclinic cycles of type Y

Olga Podvigina

Abstract

We investigate stability of a new class of heteroclinic cycles that we call heteroclinic cycles of type Y. The cycles can be regarded as a generalisation of heteroclinic cycles of type Z introduced in [Podvigina, Nonlinearity 25, 2012]. The type Y cycles differ from the cycles of type Z in the following: The trajectories comprising a cycle of type Y belong to flow-invariant subspaces that can be of different dimensions. Unlike in the most studies of the stability of heteroclinic cycles, we do not require that the eigenvalues of the linearisations of the dynamical system near the equilibria are distinct. Instead of the common assumption that the cycles are robust, we prescribe flow-invariance of certain subspaces. Similarly to type Z cycles, asymptotic stability and fragmentary asymptotic stability of type Y cycles is determined by the eigenvalues and eigenvectors of transition matrices. The matrices are products of basic transition matrices that depend on the eigenvalues of linearisations and the dimensions of the contracting subspaces.

Asymptotic stability of heteroclinic cycles of type Y

Abstract

We investigate stability of a new class of heteroclinic cycles that we call heteroclinic cycles of type Y. The cycles can be regarded as a generalisation of heteroclinic cycles of type Z introduced in [Podvigina, Nonlinearity 25, 2012]. The type Y cycles differ from the cycles of type Z in the following: The trajectories comprising a cycle of type Y belong to flow-invariant subspaces that can be of different dimensions. Unlike in the most studies of the stability of heteroclinic cycles, we do not require that the eigenvalues of the linearisations of the dynamical system near the equilibria are distinct. Instead of the common assumption that the cycles are robust, we prescribe flow-invariance of certain subspaces. Similarly to type Z cycles, asymptotic stability and fragmentary asymptotic stability of type Y cycles is determined by the eigenvalues and eigenvectors of transition matrices. The matrices are products of basic transition matrices that depend on the eigenvalues of linearisations and the dimensions of the contracting subspaces.

Paper Structure

This paper contains 18 sections, 17 theorems, 121 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Stability
  4. Heteroclinic cycles
  5. Eigenvalues and eigenvectors
  6. Collection of maps associated with a heteroclinic cycle
  7. Collection of maps: definitions of stability
  8. Stability of a cycle and a collection of maps
  9. Stability of fixed points of a collection of maps
  10. Transition matrix
  11. Two types of eigenvalues of a transition matrix
  12. Properties of maps
  13. Examples of type Y cycles
  14. Example 1
  15. Example 2
  16. ...and 3 more sections

Key Result

Lemma 1

Consider a smooth mapping $g:{\mathbb R}^n\to{\mathbb R}^n$. Suppose that ${\mathbb R}^n=X\oplus Y$, where the subspace $X$ is $g$-invariant, $Y$ is one-dimensional, ${\bf x}$ and $y$ are the coordinates in $X$ and $Y$, respectively and $g_{X}$ and $g_{Y}$ denote the projections of $g$ on $X$ and $Y where $q({\bf x},y)<C_x|{\bf x}|+C_y|y|$, $m\ge1$ and generically $m=1$.

Figures (4)

  • Figure 1: Heteroclinic connections in a three-dimensional GLV system under the conditions of theorem \ref{['tlv30']} (a) and theorem \ref{['tlv3']} (b).
  • Figure 2: Time dependence of $\bf x$ for the system of section \ref{['exa1']}: $x_1$ -- red line, $x_2$ -- blue, $x_3$ -- green, $x_4$ -- cyan, $x_5$ -- violet.
  • Figure 3: Time dependence of $\bf x$ for the system of section \ref{['exa2']}: $x_1$ -- red line, $x_2$ -- blue, $x_3$ -- green, $x_4$ -- cyan, $x_5$ -- violet.
  • Figure 4: Intersection of the sets $N_1$ and $N_2$ with the plane $(0,x_2,x_3)$.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Lemma 1
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 17 more