Nonlinear dynamics theory of stochastic layers in Hamiltonian systems

[+] Author and Article Information
Albert CJ Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, Edwardsville IL 62026-1805aluo@siue.edu

Appl. Mech. Rev 57(3), 161-172 (Jun 10, 2004) (12 pages) doi:10.1115/1.1683699 History: Online June 10, 2004
Copyright © 2004 by ASME
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Resonant conditions in the neighborbood of separatrix for the twin well-Duffing oscillator (α12=1.0)
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Outer and inner stochastic layers for an undamped twin-well Duffing oscillator
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The phase portrait of the unperturbed system of Eq. (1) near a hyperbolic point p.q0(t) is a separatrix going through the hyperbolic point p and splitting the phase into three parts near the hyperbolic point, and the corresponding orbits qα(t), qβ(t) and qγ(t) are termed the α-, β- and γ-orbits.
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The ε-neighborhood of orbit q0(t). The bold solid curves represent the separatrix q0(t) and its ε-neighborhood boundaries qσε(t) determined by ‖qσε(t)−q0(t)‖=ε where σ={α,β,γ}. The solid curves depict all orbits qσ(t) in the ε-neighborhood. The energies on the boundary orbits are given through Eσε=H0(qσε(t)).
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A stochastic layer of Eq. (14) formed by the Poincaré mapping set of q(t) in the ε-neighborhood of q0(t) when t≥0 to ∞. The separatrix separates the stochastic layer into three sub-stochastic layers (ie, α-layer and β-layer and γ-layer).
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Energy spectrum for the slowly forced Duffing oscillator at Q0=0.01(α12=1.0)
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Poincare mapping section of a slowly forced Duffing oscillator for Ω=0.1 and Q0=0.01(α12=1.0)
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Resonant conditions of the perturbed system in the stochastic layer
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Stochastic layer width (w) defined for a parametrically excited pendulum with parameters (α=1.0,Q0=0.05,Ω=2.5)
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Maximum and minimum energy spectra for stochastic layers in a parametrically excited pendulum with parameters (α=1.0,Q0=0.05)
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The stochastic layer width of a parametrically excited pendulum with parameters (α=1.0,Q0=0.05) at x=±2mπ(m=0,1,2,⋯)




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