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 t...

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Bibliographic Details
Published in:Nonlinear dynamics 2024-06, Vol.112 (12), p.10147-10155
Main Authors: Nieto, Alexandre R., Capeáns, Rubén, Sanjuán, Miguel A. F.
Format: Article
Language:English
Subjects:
Automotive Engineering
Charged particles
Classical Mechanics
Control
Dynamical Systems
Engineering
Islands
Mechanical Engineering
Orbits
Parameters
Stability
Vibration
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Summary: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 .
ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-024-09566-z