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Robust chaos in $\mathbb{R}^n$

Indranil Ghosh, David J. W. Simpson

Abstract

We treat $n$-dimensional piecewise-linear continuous maps with two pieces, each of which has exactly one unstable direction, and identify an explicit set of sufficient conditions for the existence of a chaotic attractor. The conditions correspond to an open set within the space of all such maps, allow all $n \ge 2$, and allow all possible values for the unstable eigenvalues in the limit that all stable eigenvalues tend to zero. To prove an attractor exists we use the stable manifold of a fixed point to construct a trapping region; to prove the attractor is chaotic we use the unstable directions to construct an invariant expanding cone for the derivatives of the pieces of the map. We also show the chaotic attractor is persistent under nonlinear perturbations, thus when such an attractor is created locally in a border-collision bifurcation of a general piecewise-smooth system, it persists and is chaotic for an interval of parameter values beyond the bifurcation.

Robust chaos in $\mathbb{R}^n$

Abstract

We treat -dimensional piecewise-linear continuous maps with two pieces, each of which has exactly one unstable direction, and identify an explicit set of sufficient conditions for the existence of a chaotic attractor. The conditions correspond to an open set within the space of all such maps, allow all , and allow all possible values for the unstable eigenvalues in the limit that all stable eigenvalues tend to zero. To prove an attractor exists we use the stable manifold of a fixed point to construct a trapping region; to prove the attractor is chaotic we use the unstable directions to construct an invariant expanding cone for the derivatives of the pieces of the map. We also show the chaotic attractor is persistent under nonlinear perturbations, thus when such an attractor is created locally in a border-collision bifurcation of a general piecewise-smooth system, it persists and is chaotic for an interval of parameter values beyond the bifurcation.

Paper Structure

This paper contains 15 sections, 5 theorems, 42 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Notation
  4. Sufficient conditions for robust chaos
  5. One direction of expansion
  6. Skew tent maps
  7. Main result
  8. Comparison to numerical simulations
  9. Geometric aspects of discrete dynamics
  10. Key definitions
  11. Trapping region construction
  12. Cone construction
  13. Preliminary calculations and estimates
  14. Formulas for points on $\Omega$
  15. Characteristic polynomial coefficients

Key Result

Theorem 2.1

If Assumption as:main holds and then $f$ has a topological attractor with a positive Lyapunov exponent.

Figures (8)

  • Figure 1: A sketch of the skew tent map \ref{['eq:skewTentMap']} with slopes $\alpha = 1.9$ and $-\beta = -1.7$.
  • Figure 2: The shaded region indicates values of $\frac{1}{\alpha}$, $\frac{1}{\beta}$, and $r(n-1)$ satisfying \ref{['eq:rA']}, \ref{['eq:rB']}, and \ref{['eq:rC']}.
  • Figure 3: A phase portrait of the three-dimensional BCNF, \ref{['eq:bcnf']} with \ref{['eq:bcnfALARb']}, with parameter values \ref{['eq:3dexample']}. The attractor is shown in black. The points $X$ and $Y$ are the saddle fixed points \ref{['eq:Y']} and \ref{['eq:X']}.
  • Figure 4: A phase portrait of a ten-dimensional instance of the BCNF \ref{['eq:bcnf']} with \ref{['eq:bcnfALARb']}. The parameter values are such that the eigenvalues of $A_L$ and $A_R$ are given by \ref{['eq:10dexample']} with $r = 0.0039$.
  • Figure 5: A two-parameter bifurcation diagram of the two-dimensional BCNF with $\alpha = \beta = 1.3$. The central square is where the conditions of Theorem \ref{['th:main']} are satisfied, the region between the black curves are where the results of earlier work GhMc23 verifies robust chaos, while the green region is where numerical simulations suggest a chaotic attractor exists.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['pr:charPolyCoeffs']}
  • Corollary 4.2