Universal criterion for the breakup of invariant tori in dissipative systems

TL;DR

This paper investigates the transition from quasiperiodicity to chaos in dissipative systems, proposing a universal criterion based on the slope of the critical circle map, supported by numerical evidence and applicable to various models.

Contribution

It introduces a universal criterion linking the critical circle map slope to the Jacobian determinant, with a decimation scheme applicable to multiple quasiperiodic systems.

Findings

The critical slope is proportional to the Jacobian determinant.

The universal constant emerges in the zero-Jacobian limit.

Numerical results support the universality in dissipative maps.

Abstract

The transition from quasiperiodicity to chaos is studied in a two-dimensional dissipative map with the inverse golden mean rotation number. On the basis of a decimation scheme, it is argued that the (minimal) slope of the critical iterated circle map is proportional to the effective Jacobian determinant. Approaching the zero-Jacobian-determinant limit, the factor of proportion becomes a universal constant. Numerical investigation on the dissipative standard map suggests that this universal number could become observable in experiments. The decimation technique introduced in this paper is readily applicable also to the discrete quasiperiodic Schrodinger equation.