Random Melnikov Method and Induced Chaos in Bistable Vibro-Impact Oscillators with Bounded Noise

Abstract

In this paper, the Melnikov method for an abstract stochastic nonsmooth hybrid system is derived in detail and employed to study the homoclinic bifurcations and induced chaos in a bistable vibro-impact SD oscillator with bounded noise first proposed by Li et al. 2021, in which the geometric nonlinearity is used to produce the irrational restoring force and an impact map is introduced to describe energy loss during collisions. The initial Poincaré section is selected on the right rigid constraint so that rigorous perturbation analysis can be carried out to avoid the extension of trajectories near the switching manifold. The stochastic Melnikov function with geometrical intuition is obtained to detect the threshold of parameters for homocinic tangency of the perturbed stochastic stable and unstable manifolds by calculating the energy difference of their initial points on the Poincaré section. Finally, the analytical Melnikov analysis combined with numerical simulations is carried out to validate the developed Melnikov method for analyzing the global dynamics of the bistable vibro-impact SD oscillator subject to bounded noise.

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

“…Using Eqs. ( 21)-( 22) and the similar derivation presented in the literature [45] obtains the variance σ 2 M r as shown in this theorem. By the Melnikov criterion given in Theorem 1, the inequality (37) can be obtained.…”

“…We note that the aforementioned derivation details can be found in the literature [45] and the corresponding Melnikov function of the left symmetric homoclinic orbit can similarly obtained. Let M d (t 0 ) = −A 1 + A 2 − A 3 and M r (t 0 ) = A 4 , Eq.…”

“…dτ, where the detailed calculation process is omitted and the similar derivation can be found in the Appendix of the literature [45] through overcoming the integral problem of the homoclinic orbit with implicit expression. Theorem 3.…”

“…Zhou and Jin [44] used the mean square criterion and phase space flux function theory to study the effect of noise intensity on chaotic dynamics of bistable coupled SD oscillators under Gaussian colored noise and harmonic excitation. Li et al [45] derived in detail the random Melnikov method for abstract stochastic non-smooth hybrid systems under bounded noise and applied it to study the global stochastic dynamics of the bistable vibro-impact SD oscillator.…”