Manifolds on the verge of a hyperbolicity breakdown

Abstract

We study numerically the disappearance of normally hyperbolic invariant tori in quasiperiodic systems and identify a scenario for their breakdown. In this scenario, the breakdown happens because two invariant directions of the transversal dynamics come close to each other, losing their regularity. On the other hand, the Lyapunov multipliers associated with the invariant directions remain more or less constant. We identify notable quantitative regularities in this scenario, namely that the minimum angle between the two invariant directions and the Lyapunov multipliers have power law dependence with the parameters. The exponents of the power laws seem to be universal. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2150947͔Quasiperiodically forced systems occur in many situations in physics, mathematics, engineering, etc. In many cases, the external quasiperiodic perturbations induce quasiperiodic motions, which correspond to invariant tori. It is important to understand when these invariant tori persist under perturbations, and to identify the mechanisms of their breakdown. It has been known for a long time that the persistence of a torus is related to the exponential growth of the linearization along certain directions (normal hyperbolicity), and that normal hyperbolicity may be lost because of bifurcations such as saddle-node and period doubling, among others. The common feature of these standard bifurcations is that some Lyapunov multipliers approach 1, while the invariant directions remain smooth. In this paper we propose a new mechanism, in which two invariant directions of the linearized dynamics come close to each other, losing their regularity, and the corresponding Lyapunov multipliers remain more or less constant, away from each other and away from 1. Hence, the phenomenon is described by two observables: the minimum angle between the invariant directions and the Lyapunov multipliers. We also identify notable universal power laws of these observables.

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

“…The difficulty lies in proving that M again has a continuous invariant splitting (1.8) with bounded projections. This is one possible reason for breakdown of normal hyperbolicity [HL06]. ♦ These two theorems reduce to the corollaries formulated below, when M is compact or when Q = R m+n with standard Euclidean metric and M = R m × {0}.…”

On the Computation of Reducible Invariant Tori on a Parallel Computer

“…When ε = 0, the point x = y = 0 is invariant by the flow of (24), and this implies that the map P satisfies P (0) = 0 (note that if we consider the variable θ, this set is a d-dimensional invariant torus for (27)). To apply the previous algorithm, we will use z 0 (θ) ≡ 0 as an initial approximation to the torus and the Jacobian of P at z = 0 (for ε = 0) as an approximation to the reduced matrix (note that, for ε = 0, the Jacobian of P does not depend on θ) and the identity matrix as an approximation to the Floquet transformation.…”

“…We will focus on invariant tori of dimension d, that is, tori that can be parametrized as z 0 : T d → R 2 and that satisfy the equation z 0 (θ + ω) = P (z 0 (θ), θ). They are the simplest invariant objects of a system like (27). As mentioned before, the linearization around such a torus is given by (3), where…”

Strange nonchaotic attractors in Harper maps

“…Our choice aims to emphasize that the results presented here are independent of the arithmetical properties of the frequency. 19,20 B. A topological argument for the existence of SNA When ⌽ u is a continuous attracting invariant curve, and thus a is in a gap of the spectrum, there is a topological index which counts its winding around P, which must coincide with that of ⌽ s .…”

“…The connection between ͑a͒ and ͑c͒ in similar models has been used to study the linearized dynamics around invariant tori in quasiperiodic systems. 19 Specifically, the formation of SNA in this linearized dynamics is suggested to be a mechanism of breakdown of invariant tori. 20 Even if we deal with one-dimensional quasiperiodically forced systems, many of the techniques that we use in this paper also apply to higher dimensional or less regular settings, including other types of dynamics on the external forcing.…”