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Investigation of Early Warning Indexes in a Three-Dimensional Chaotic System with Zero Eigenvalues
Lianyu Chen1, Fahimeh Nazarimehr2, Sajad Jafari2,3
1School of Electrical and Information Engineering, Jiangsu University of Technology, Changzhou 213001, China.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
This study introduces a rare 3D chaotic system with zero eigenvalues, revealing its unstable equilibrium and period-doubling route to chaos. Entropy analysis predicts bifurcation points, offering early warning signals for chaotic dynamics.
Area of Science:
- Nonlinear Dynamics and Chaos Theory
- Mathematical Physics
Background:
- Chaotic systems are fundamental in understanding complex phenomena across various scientific disciplines.
- Investigating systems with unique properties, such as zero eigenvalues, is crucial for advancing chaos theory.
- Predicting bifurcations and transitions to chaos remains a significant challenge in dynamical systems analysis.
Purpose of the Study:
- To propose and analyze a novel three-dimensional chaotic system characterized by all eigenvalues being zero.
- To explore the dynamical properties, including stability and bifurcations, of the proposed chaotic system.
- To investigate the efficacy of entropy measures as early warning signals for predicting bifurcation points.
Main Methods:
- Development of a rare three-dimensional chaotic system with zero eigenvalues.
- Numerical analysis to determine the stability of the system's equilibrium point at the origin.
- Bifurcation analysis employing Shannon entropy and Kolmogorov-Sinai entropy to identify and predict bifurcation points, with comparisons to Lyapunov exponents.
Main Results:
- A novel 3D chaotic system with all zero eigenvalues was successfully proposed.
- The system exhibits an unstable equilibrium point at the origin.
- Bifurcation analysis revealed a period-doubling route to chaos, and entropy measures effectively predicted bifurcation points, correlating with Lyapunov exponent results.
Conclusions:
- The proposed chaotic system offers a unique platform for studying complex dynamics.
- Entropy evaluation serves as a reliable method for predicting bifurcation points, providing valuable early warning signals.
- This research contributes to a deeper understanding of chaotic system behavior and bifurcation prediction techniques.
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