Delayed self-feedback echo state network for long-term dynamics of hyperchaotic systems.

Area of Science:

• Nonlinear Science
• Complex Systems Analysis
• Data-Driven Modeling

Background:

• Analyzing long-term hyperchaotic system behavior presents significant challenges.
• Conventional Echo State Networks (ESNs) struggle with capturing complex dynamics and require optimal parameter tuning.

Purpose of the Study:

• To propose a novel data-driven model, the delayed self-feedback echo state network (self-ESN), for enhanced long-term prediction of hyperchaotic systems.
• To improve the memory performance and prediction accuracy of ESNs for complex dynamical systems.

Main Methods:

• Developed the self-ESN by incorporating a delayed self-feedback term into the reservoir's dynamic equation.
• Introduced and analyzed the local echo state property (ESP) to guide the selection of feedback delay and gain.
• Conducted numerical experiments on various hyperchaotic systems, including 4D systems, networks, and infinite-dimensional delayed systems.

Main Results:

• The self-ESN demonstrated improved memory performance by connecting current and previous reservoir states.
• Theoretical analysis provided guidance for selecting feedback gain and delay, enhancing prediction accuracy.
• Numerical experiments validated the self-ESN's capability in reconstructing bifurcation diagrams, predicting chaotic synchronization, and analyzing spatiotemporal patterns.

Conclusions:

• The proposed self-ESN effectively captures the dynamic characteristics of hyperchaotic systems.
• The model offers a robust strategy for long-term prediction and analysis of complex dynamical systems.
• Self-ESN overcomes limitations of conventional ESNs in parameter optimization and memory performance.