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Dynamical tangles in third-order oscillator with single jump function.

Jiří Petržela¹, Tomas Gotthans¹, Milan Guzan²

¹Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic.

Thescientificworldjournal

|December 30, 2014

Summary

This study explores chaotic dynamics in a nonlinear oscillator using advanced mathematical theorems and numerical analysis. It visualizes complex behaviors and clarifies how phenomena evolve over time.

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Area of Science:

Nonlinear Dynamics
Chaos Theory
Differential Equations

Background:

Nonlinear oscillators are fundamental in physics and engineering.
Understanding their complex dynamical behavior is crucial.
Previous work focused on simpler Newtonian dynamics.

Purpose of the Study:

To conduct a detailed study of the dynamical behavior of a nonlinear oscillator.
To analyze solutions, invariant manifolds, and basins of attraction.
To prove the existence of chaotic motion and link to prior research.

Main Methods:

Analysis of a third-order differential equation with scalar jump nonlinearity.
Utilizing the primitive geometry of the vector field for numerical analysis.
Application of Shilnikov theorems to demonstrate chaos.

Main Results:

Exhaustive numerical analysis of possible solutions.
Visualizations of invariant manifolds and basins of attraction.
Proof of chaotic motion through Shilnikov theorems.

Conclusions:

The study provides a comprehensive understanding of the oscillator's chaotic dynamics.
It links current findings with previous research on Newtonian dynamics.
Offers insights into the temporal evolution of complex phenomena.