The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps

Abstract

We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein's greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities.

Figures (19)

• Figure 1: Cobweb diagrams showing the attractor of the skew tent map \ref{['eq:skewtent_map']} with two different combinations of the parameter values. In (a) the attractor is an interval; in (b) the attractor is the union of four disjoint intervals. The maps are also instances of the BCNF \ref{['eq:BCNF2']} with $\xi = (1.3,0,-2,0)$ in panel (a) and $\xi = (1.1,0,-1.1,0)$ in panel (b).
• Figure 2: A two-parameter bifurcation diagram of the skew tent map family \ref{['eq:skewtent_map']}. In $\mathcal{R}_0$ the attractor is an interval. In each $\mathcal{R}_n$ with $n \ge 1$ the attractor is comprised of $2^n$ disjoint intervals. Below $\phi_0(\tau_L,\tau_R) = 0$ the map has no attractor; above $\tau_R = -1$ it has a stable fixed point in $x>0$; in the top-left region it has a stable $LR$-cycle (period-two solution). The triangles indicate the parameter values used in Fig. \ref{['fig:cobwebs_skewtent']}.
• Figure 3: Phase portraits of the BCNF \ref{['eq:BCNF2']} for four different parameter combinations: (a) $\xi = (2.1, 0.06, -1.7, 0.18)$; (b) $\xi = (2.1, 0.4, -1.7, -0.55)$; (c) $\xi = (2.2, -0.3, -1.7, 0.1)$; (d) $\xi = (1.8, -0.75, -1.6, -0.4)$. Each plot shows the linear segments of the stable and unstable manifolds emanating from the fixed points $X$ and $Y$, as well as an adjoining segment of the stable manifold of $Y$. We also indicate some of their intersections with $y=0$: $T$, $D$, and $C$; formulas for these are given by \ref{['eq:T']}, \ref{['eq:D']}, and \ref{['eq:C']}. To illustrate the chaotic attractor the black dots show 2000 iterates of a typical forward orbit after transient dynamics has decayed.
• Figure 4: Phase portraits of the BCNF \ref{['eq:BCNF2']}. Panel (a) uses the example parameter point $\xi^{(1)}_{\rm ex}$ given by \ref{['eq:xi1ex']} where the attracor $\tilde{\Lambda}$ has two connected components, one of which lies entirely in $\Pi_\xi$ (shaded). Panel (b) uses the parameter point $g ( \xi^{(1)}_{\rm ex} )$ where the attractor $\Lambda$ has one connected component in $h_\xi(\Pi_\xi)$ (shaded).
• Figure 5: Phase portraits of \ref{['eq:BCNF2']} with $\delta_L = 0.1$, $\delta_R=0.1$, $\tau_R=-2$ and three different values of $\tau_L$. These parameter values correspond to the black triangles in Fig. \ref{['fig:reg1']}-a. Panel (b) uses $\tau_L$ such that $\phi^+(\xi) = 0$ to ten decimal places. In (a) ${\rm cl}(W^u(X))$ is a chaotic attractor; in (b) $W^s(Y)$ and $W^u(Y)$ form a homoclinic corner by intersecting at $D=C$; in (c) there is no attractor.

Theorems & Definitions (22)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Proposition 4.4