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Stochastic chaos induced by diffusion processes with identical spectral density but different probability density
1Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.
Chaos (Woodbury, N.Y.)
|January 2, 2017
Summary
Stochastic chaos onset in damped systems is determined by diffusion process spectral density, not probability density functions (PDFs). The stochastic Melnikov method accurately predicts this chaos threshold.
Area of Science:
- Nonlinear dynamics
- Stochastic processes
- Chaos theory
Background:
- Hamiltonian systems can exhibit stochastic chaos.
- Diffusion processes can induce chaos with varying probability density functions (PDFs).
Purpose of the Study:
- Investigate stochastic chaos in damped Hamiltonian systems.
- Determine the threshold amplitude for chaos induced by diffusion processes.
- Analyze the influence of spectral density and PDFs on chaos onset.
Main Methods:
- Stochastic Melnikov method
- Mean-square criterion
- Analysis of damped single pendulum and Duffing oscillator models
- Numerical simulations and Lyapunov exponent calculations
Main Results:
- Spectral density of diffusion processes dictates the chaos threshold amplitude.
- The shape of probability density functions (PDFs) does not affect the chaos threshold.
- Analytical predictions align with numerical simulation results.
Conclusions:
- The stochastic Melnikov method is effective for predicting chaos onset in quasi-Hamiltonian systems.
- Spectral density is the key factor in diffusion-induced chaos thresholds.
- Understanding these dynamics is crucial for analyzing complex systems.
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