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Unified analysis of complex nonlinear motions via densities. (English) Zbl 1003.70018
The authors propose a method of analysis of deterministic and randomly perturbed chaotic responses of nonlinear systems using probability densities which are solutions of Fokker-Planck equation. Transient and steady state probability densities are computed by numerically solving this equation, and their asymptotic behavior is examined according to Foguel “alternative theorem”. The authors investigate a model second-order nonlinear differential equation with a periodic external exciting force and an additive zero-mean (delta-correlated white noise). Homoclinic and heteroclinic orbits in unperturbed rolling motion are obtained, and deterministic chaos is discussed. It is pointed out that possible deterministic chaotic responses appear when stable and unstable manifolds transversely intersect each other. In the presence of random perturbations, depending on the stability of probability density, two distinct asymptotic properties are observed: invariant and sweeping. Here, homoclinic and heteroclinic dynamics are examined individually in detail. Numerical results exhibit periodicity in the evolution of densities, which implies the existence of an invariant measure, time-averaging density. Finally, the authors discuss some engineering application of the proposed approach.

MSC:
70L05 Random vibrations in mechanics of particles and systems
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
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