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The Sprott B system
Ferdinand Verhulst1, Taoufik Bakri2
1Mathematisch Instituut, P.O. Box 80.010, 508TA Utrecht, The Netherlands.
Chaos (Woodbury, N.Y.)
|October 7, 2024
Summary
The Sprott B system, a generalization of Sprott A, exhibits explicit periodic solutions and complex dynamics including dissipative KAM tori and strange attractors. Its parameter-dependent behavior reveals instability intervals and violent vibrations.
Area of Science:
- Nonlinear Dynamics
- Dynamical Systems Theory
- Chaos Theory
Background:
- The Sprott A system is a well-known one-parameter system in nonlinear dynamics.
- Thermostatic systems offer a framework for studying complex behaviors.
- Understanding parameter-dependent dynamics is crucial for predicting system evolution.
Purpose of the Study:
- To analyze the generalized Sprott B thermostatic system.
- To investigate the explicit periodic solutions and complex dynamics of Sprott B.
- To explore the influence of parameter variations on system stability and behavior.
Main Methods:
- Mathematical analysis of the Sprott B system equations.
- Identification of periodic solutions and their stability properties.
- Investigation of bifurcations and transitions to chaos as a parameter varies.
Main Results:
- Sprott B possesses an explicit periodic solution for all positive parameter values 'a'.
- Dissipative KAM tori and canards are observed, similar to Sprott A.
- Infinite instability intervals reveal stable/unstable solutions, tori, and strange attractors.
- Non-hyperbolic slow manifolds lead to violent vibrations for large 'a'.
Conclusions:
- The Sprott B system displays rich and complex dynamics, including chaos.
- Parameter variations lead to intricate bifurcations and emergent behaviors.
- The study provides insights into generalized thermostatic systems and their potential for complex phenomena.
