Estimation of dynamical invariants without embedding by recurrence plots.

Area of Science:

• Nonlinear dynamics
• Chaos theory
• Time series analysis

Background:

• Dynamical invariants, such as Renyi entropy and correlation dimension, are crucial for characterizing complex systems.
• Traditional nonlinear data analysis methods often require specific embedding parameters, limiting their applicability.
• Recurrence plots (RPs) offer a visual and quantitative tool for analyzing dynamical systems.

Purpose of the Study:

• To demonstrate that recurrence plots (RPs) can reliably estimate the second-order Renyi entropy and correlation dimension.
• To highlight the advantage of RPs in estimating these invariants irrespective of embedding dimension and delay.
• To assess the robustness and computational efficiency of RP-based estimation of dynamical invariants.

Main Methods:

• Estimation of dynamical invariants (Renyi entropy, correlation dimension) directly from recurrence plots (RPs).
• Analysis using arbitrary embedding dimensions and delays, and also without embedding.
• Validation of the method using well-known chaotic systems (Rossler, funnel attractor, Mackey-Glass) and fluid dynamics experimental data.

Main Results:

• Recurrence plots allow for robust and numerically inexpensive estimation of the second-order Renyi entropy and correlation dimension.
• These dynamical invariants can be accurately estimated from RPs even when no embedding is used.
• The method successfully identified low-dimensional chaos in fluid dynamical experimental data, confirming prior findings.

Conclusions:

• Recurrence plots provide a powerful and versatile tool for estimating key dynamical invariants in nonlinear systems.
• The independence from embedding parameters makes RPs advantageous over other nonlinear data analysis techniques.
• The approach is efficient and robust, suitable for both theoretical models and experimental data analysis, including fluid dynamics.