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Excitation-Induced Stability and Phase Transition: A Review

Wayne State University, Department of Mechanical Engineering, Detroit, MI 48202, USA, ibrahim{at}eng.wayne.edu

Dynamical systems may experience undesirable behavior or instability, which can be eliminated using feedback control means. However, in the absence of feedback control, the stability of some systems may be increased by imposing parametric excitation. In other cases, the exit time of the system response from the stable to the unstable domain may be prolonged by imposing external noise, a phenomenon termed noise-enhanced stability (NES). This article presents an assessment of the mechanisms of stabilization via multiplicative noise and noise-enhanced stability. The first part deals with stabilization via deterministic parametric excitation of gravity-defying systems such as the inverted simple and spherical pendulums, aeroelastic structures, human walking, and quantum nonlinear couplers. The second part introduces the concept of noise-induced transition (NIT) in one-dimensional nonlinear systems and ship roll motion. Stabilization of originally unstable systems via multiplicative noise is treated in the third part. The fourth part addresses the influence of additive noise in delaying the exit time of system response to an unstable domain. This topic is related to the phenomenon of stochastic resonance (SR) and NES of systems with one or more metastable states and fluctuating potential. Finally, this review article also introduces some applications in other fields such as the Ising model, ecosystems, and tumor-immune system models.

Key Words: Stabilization • parametric excitation • additive noise • noise-induced transition • noise-induced stability • noise-enhanced stability • stochastic resonance • inverted pendulum • Langevin equation • tumor-immune systems • Ising model • ecosystems • flutter of aeroelastic structures • ship capsizing

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