Final-state sensitivity characterized by phase transition in singularity spectra of intermingled basins

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Statistical Physics

Background:

• Intermingled basins of attraction exhibit complex final-state sensitivity.
• Understanding the scaling properties of these basins is crucial for predicting system behavior.
• Previous analyses often simplified basin structures, neglecting intricate details.

Purpose of the Study:

• To investigate final-state sensitivity in systems with intermingled basins of attraction.
• To introduce and utilize a new dimension, the conditional external dimension, to characterize basin uncertainty.
• To perform a multifractal analysis of the basin structure for an analytically tractable model.

Main Methods:

• A scaling assumption was applied to analyze the basin structure.
• A multifractal analysis was performed on a non-skew product, analytically tractable model.
• The conditional external dimension was calculated as a key parameter.

Main Results:

• The scaling assumption was validated for the model system.
• The conditional external dimension was determined by transient motions not converging to attractors.
• These transient motions were shown to induce a phase transition in the singularity spectrum.

Conclusions:

• The conditional external dimension provides a quantitative measure of uncertainty in intermingled basins.
• Transient dynamics play a critical role in shaping the multifractal properties of basin boundaries.
• The findings reveal a connection between transient motion, phase transitions, and the singularity spectrum.