Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data

Area of Science:

• Nonlinear Dynamics and Chaos Theory
• Time Series Analysis
• Statistical Physics

Background:

• Chaotic time series analysis often involves filtering to remove noise or extract features.
• The impact of filtering on the underlying attractor dimension of chaotic systems is not fully understood.
• Traditional methods may not adequately capture the complex changes induced by severe filtering.

Purpose of the Study:

• To investigate the effect of infinite-extent time filters on the attractor dimension of chaotic time series.
• To identify the origin of attractor dimension increase due to severe filtering.
• To establish a quantitative relationship between filtering-induced discontinuities and resolution.

Main Methods:

• Application of novel statistics for continuity and differentiability to analyze filtered chaotic time series.
• Mathematical analysis of the relationship between discontinuities and attractor dimension.
• Direct visualization of the filtered Henon attractor to observe changes.

Main Results:

• Severe filtering introduces discontinuities or non-differentiable functions between unfiltered and filtered attractors.
• These discontinuities are identified as the primary cause for the increase in attractor dimension.
• A scaling relation is observed for the density of discontinuities as a function of resolution.

Conclusions:

• Novel continuity and differentiability statistics effectively reveal the impact of severe filtering on chaotic attractors.
• The increase in attractor dimension is directly linked to the emergence of discontinuities caused by filtering.
• The findings provide a new perspective on understanding and quantifying the effects of filtering in chaotic systems.