Systematic search for islets of stability in the standard map for large parameter values

In the seminal paper, Chirikov (Phys Rep 52:263–379, 1979) showed that the standard map does not exhibit a boundary to chaos, but rather that there are small islands ("islets") of stability for arbitrarily large values of the nonlinear perturbation. In this context, he established that the area of the islets in the phase space and the range of parameter values where they exist should decay following power laws with exponents -2 and -1, respectively. In this paper, we carry out a systematic numerical search for islets of stability and we show that the power laws predicted by Chirikov hold. Furthermore, we use high-resolution 3D islets to reveal that the islets' volume decays following a similar power law with exponent -3.

Publication:

Nonlinear Dynamics

Pub Date:

June 2024

DOI:

10.1007/s11071-024-09566-z

10.48550/arXiv.2404.12027

arXiv:

arXiv:2404.12027

Bibcode:

2024NonDy.11210147N

Keywords:

Hamiltonian systems;
Area-preserving maps;
KAM tori;
Numerical simulations;
Nonlinear Sciences - Chaotic Dynamics

E-Print:

7 pages

Systematic search for islets of stability in the standard map for large parameter values

Abstract