Stochastic chaos induced by diffusion processes with identical spectral density but different probability density functions. (English) Zbl 1378.34081
Summary: Stochastic chaos induced by diffusion processes, with identical spectral density but different probability density functions (PDFs), is investigated in selected lightly damped Hamiltonian systems. The threshold amplitude of diffusion processes for the onset of chaos is derived by using the stochastic Melnikov method together with a mean-square criterion. Two quasi-Hamiltonian systems, namely, a damped single pendulum and damped Duffing oscillator perturbed by stochastic excitations, are used as illustrative examples. Four different cases of stochastic processes are taking as the driving excitations. It is shown that in such two systems the spectral density of diffusion processes completely determines the threshold amplitude for chaos, regardless of the shape of their PDFs, Gaussian or otherwise. Furthermore, the mean top Lyapunov exponent is employed to verify analytical results. The results obtained by numerical simulations are in accordance with the analytical results. This demonstrates that the stochastic Melnikov method is effective in predicting the onset of chaos in the quasi-Hamiltonian systems.{
©2016 American Institute of Physics}
©2016 American Institute of Physics}
MSC:
| 34F05 | Ordinary differential equations and systems with randomness |
| 60J60 | Diffusion processes |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 62M15 | Inference from stochastic processes and spectral analysis |
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