Riddled-like basins of transient chaos

Area of Science:

• Nonlinear dynamics
• Chaos theory
• Chemical kinetics

Background:

• Transient chaos occurs in dynamical systems near a crisis.
• Riddled basins are a complex dynamical phenomenon where initial conditions are intricately mixed.
• Finite accuracy in measurements and computations is a practical limitation.

Purpose of the Study:

• To investigate the presence of riddled-like behavior in the initial conditions of transient chaos.
• To determine if this behavior persists under finite accuracy constraints.
• To compare this phenomenon with established riddled basin dynamics.

Main Methods:

• Numerical simulations using the logistic map.
• Modeling a quantitatively described chemical reaction.
• Analysis of initial condition sets under varying precision levels.

Main Results:

• Transient chaos near a crisis point demonstrates riddled-like behavior for any finite accuracy.
• This apparent riddling is computationally and experimentally indistinguishable from true riddled basins.
• The findings hold for both abstract mathematical models and realistic chemical systems.

Conclusions:

• Finite accuracy in computational or experimental settings leads to observable riddled-like behavior in transient chaos.
• This behavior mimics riddled basins, posing challenges for precise system analysis.
• The study highlights the practical implications of precision limitations in chaos studies.