Scaling properties of discontinuous maps

Area of Science:

• Dynamical systems
• Nonlinear dynamics
• Statistical physics

Background:

• Discontinuous maps exhibit complex scaling properties influenced by control parameters.
• Understanding diffusion regimes (slow and quasilinear) is crucial for analyzing dynamical systems.
• Chirikov's standard map serves as a benchmark for studying chaotic dynamics and diffusion.

Purpose of the Study:

• To investigate the scaling properties of the squared action variable I(2) in discontinuous maps.
• To compare the scaling laws of discontinuous maps with Chirikov's standard map across different nonlinearity regimes.
• To analyze the impact of the absence of Kolmogorov-Arnold-Moser tori on diffusion scaling.

Main Methods:

• Analysis of the average value of the squared action variable I(2).
• Focus on two dynamical regimes: slow diffusion (K < K(c)) and quasilinear diffusion (K > K(c)).
• Comparison of scaling laws in the limits of weak (K ≪ K(c)) and strong (K ≫ K(c)) nonlinearity.

Main Results:

• Discontinuous maps exhibit scaling laws similar to Chirikov's standard map in weak and strong nonlinearity regimes.
• The squared action variable I(2) scales as nK(β) for n ≫ 1 in both diffusion regimes.
• The exponent β is approximately 5/2 for K ≪ K(c) (slow diffusion) and 2 for K ≫ K(c) (quasilinear diffusion).

Conclusions:

• The absence of Kolmogorov-Arnold-Moser tori influences the diffusion scaling in discontinuous maps.
• Discontinuous maps share fundamental scaling behaviors with standard maps despite structural differences.
• The identified scaling exponents provide insights into the diffusive behavior of these dynamical systems.