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Published on: May 30, 2014
Chaotic motion of a weakly nonlinear, modulated oscillator
1Institute of Geophysics and Planetary Physics, University of California, La Jolla, CA 92093.
Summary
This study investigates a weakly nonlinear oscillator under amplitude-modulated force. The research reveals conditions under which the oscillator
Area of Science:
- Nonlinear Dynamics
- Mechanical Vibrations
- Chaos Theory
Background:
- Nonlinear oscillators are fundamental in physics and engineering.
- Amplitude-modulated forces introduce complex dynamics.
- Resonant phenomena in nonlinear systems can lead to chaotic behavior.
Purpose of the Study:
- To analyze the dynamics of a weakly nonlinear oscillator driven by an amplitude-modulated force near resonance.
- To derive and study the equations governing the slow modulation of the oscillator's response envelope.
- To identify the conditions for chaotic behavior in the system's envelope.
Main Methods:
- Derivation of a reduced set of three first-order ordinary differential equations for the response envelope.
- Analysis of the nonlinear system in the resonant neighborhood where frequency detuning, modulation frequency, and damping are small relative to the forcing amplitude.
- Investigation of the parameter space to determine the onset of chaotic dynamics in the envelope.
Main Results:
- The oscillator's response is a slowly modulated sine wave.
- The envelope dynamics are governed by three autonomous ordinary differential equations.
- The envelope exhibits periodic behavior for large damping relative to the forcing amplitude.
- Chaotic behavior in the envelope emerges for specific ranges of detuning and modulation frequencies when damping is sufficiently small.
Conclusions:
- The derived envelope equations accurately describe the system's dynamics in the resonant regime.
- The transition from periodic to chaotic behavior in the envelope is dependent on the interplay between damping, detuning, and modulation frequency.
- This work provides insights into the complex dynamics of driven nonlinear oscillators, with implications for understanding chaotic phenomena.
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