A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

Abstract

Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

Figures (3)

• Figure 1: Stable (blue) and unstable (bronze) manifold of the fixed point (black filled circle) of the map $\bm{T}$ given in (\ref{['b']}) and (\ref{['c']}) for $\mu=0.7$ and $\alpha=0.3$. Manifolds of finite length have been computed numerically by forward, respectively backward, iteration (and an additional adaptive bisection scheme) of initial conditions close to the fixed point and located on the stable, respectively unstable, direction.
• Figure 2: Local expansion rates on a $100 \times 100$ square grid of $\bm{\theta}$-values, computed from backward orbits of length 200. (a) Local expansion rate $\lambda_u(\bm{\theta})$. (b) First order difference quotient defined in (\ref{['d']}), for $h=10^{-4}$. (c) Second order difference quotient defined in (\ref{['e']}), for $h=10^{-4}$.
• Figure 3: (a) Log-log plot of the difference between $\Delta \lambda_u$ and a numerically "exact" estimate for the partial derivative (obtained via the difference quotient at $h=10^{-16}$) in dependence on the offset $h$. Data are shown for different values of $\bm{\theta}$ on a $40\times 40$ square grid (blue), and for five selected $\bm{\theta}$-values (highlighted, bronze). (b) Semi-log plot of the dependence of the second order difference quotient $\Delta_2 \lambda_u$ on the offset $h$. The second order difference quotient is scaled by $|\ln h|$. Data are shown for different values of $\bm{\theta}$ on a $40\times 40$ regular grid (blue), and for five selected values (highlighted, bronze).