Scaling behavior of nonhyperbolic coupled map lattices

Area of Science:

• Dynamical systems
• Statistical physics
• Nonlinear dynamics

Background:

• Coupled map lattices (CMLs) model complex systems using discretized equations.
• Nonhyperbolic maps are crucial for understanding strong chaos in physical systems.
• Tchebyscheff maps serve as a prototype for studying CMLs.

Purpose of the Study:

• Investigate the scaling behavior of observables in diffusively coupled Tchebyscheff maps.
• Analyze the influence of coupling constants on chaotic dynamics.
• Explore the presence and nature of oscillations in system properties.

Main Methods:

• Analytical derivation of scaling laws in the low-coupling limit (a -> 0).
• Development of a first-order perturbation theory for invariant densities.
• Numerical simulations to validate theoretical predictions.

Main Results:

• Proven that observable expectations scale with the square root of the coupling constant (sqrt(a)).
• Demonstrated log-periodic oscillations with period ln(N^2) in observable scaling.
• Analytically calculated invariant one-point densities showing phase-space oscillations.

Conclusions:

• Nonhyperbolic CMLs exhibit distinct scaling behavior compared to hyperbolic systems.
• Log-periodic oscillations are a key feature of these chaotic systems.
• The developed theory accurately predicts numerical observations in coupled map lattices.