Analysis of bifurcations for non-linear stochastic non-smooth vibro-impact system via top Lyapunov exponent

Bifurcations are discussed by the criterion of top Lyapunov exponent. Based on the local map and Kaminski’s algorithms, a general formulation of the top Lyapunov exponents is proposed for non-linear vibro-impact oscillators with Gaussian white noise perturbation. The analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic Duffing vibro-impact oscillator. Both results are consistent.  

非对称碰撞约束下形状记忆合金梁的随机响应与分岔

形状记忆合金(SMA)是二十一世纪具有形状记忆效应的新型智能材料.针对具有非对称约束的SMA梁,本文构造了碰撞振动系统.在无碰撞和有碰撞两种情况下,利用随机平均法给出了近似解析结果.数值模拟作为验证解析结果的工具.结果表明,系统能量的概率响应曲线具有非光滑特性.当约束位置发生变化时,系统会出现随机P分岔和D分岔.  

Chaos prediction and control of Goodwin’s nonlinear accelerator model

Chaos prediction and its control of the Goodwin model under the deterministic or stochastic excitation are studied theoretically and numerically. Applying the Melnikov technique, the threshold conditions for the occurrence of chaos are obtained theoretically. The stable and unstable manifolds of saddle are computed to verify the effectiveness of the analytical prediction in the deterministic case. Also, the safe basins are introduced to show how the externally stochastic perturbation affects the safety of the economic system as the noise amplitude increases. Finally, the analytical criterion of controlling chaos is derived via the delayed feedback control method. Numerical investigations including the top Lyapunov exponent, Poincare section, and phase portraits are carried out to demonstrate the validity and effectiveness of the theoretical results.