Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps

TL;DR

This paper investigates the breakdown of rotational invariant tori in 2D and 4D standard maps, using three methods—analyticity analysis, Newton embedding, and Greene extension—to understand stability thresholds in conservative and dissipative systems.

Contribution

It introduces and compares three methods for analyzing torus breakdown in high-dimensional standard maps, including new applications to 4D systems with various symplectic properties.

Findings

Padé and Newton methods yield reliable estimates of breakdown thresholds.

The Greene extension method is computationally challenging and less conclusive.

Different map types exhibit distinct breakdown behaviors and thresholds.

Abstract

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Pad\'e approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus. We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative ( more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The 4D maps are obtained coupling $(i)$ two symplectic standard maps, or $(ii)$ two conformally symplectic standard maps, or $(iii)$ a symplectic and a conformally symplectic standard map. Concerning the results, Pad\'e and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.