Complex dynamics in physical pendulum equation with suspension axis vibrations

Abstract

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov's method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov's method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω + εν, n = 1 − 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 − 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f 0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from periodone orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven't find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

“…The case (A) (11). The bifurcation diagram in the (δ, x) plane for α = 0.1, f 0 = 3.5, f 1 = 0.5, γ = 0.01, ω = 0.55, Ω = 3.3 is shown in Fig.…”

Chaos control in a special pendulum system for ultra-subharmonic resonance

“…where m, n are positive integers, the value of Φ + is denoted Φ + min , thus the suitable initial phase Ψ + min = Φ + min + θΩ/ω. Although condition ( 21) is a necessary one for the inequality (19) to hold for all t 1 , it gives good situations in which M + 2 (τ 0 ) is as near as possible to be tangency condition for B 2 = B min . Therefore, one has a good chance to suppress chaos.…”

“…For α = 0, γ = 0 and Ψ = 0, D'Humieres et al [16] investigated the chaotic behaviors of system (1) through the experimental method, they found some interesting behaviors in chaotic region, for example, the intermittent behavior, symmetry breaking of periodic orbits and period-3 bifurcations. when Ψ = 0, Fu et al [19] investigated the bifurcation and chaos of system (1) by using Melnikov methods and second-order averaging method. Chen et al [11] investigated the chaos control of system (1) for primary and subharmonic resonance.…”

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

“…Alternatively, an equivalent interval can be used, commonly one between −180 • and +180 • . In this case, although the graphical representation of the system behavior is improved, the new normalized phase portrait [Fu et al, 2010;Pavlovskaia et al, 2012] usually shows discontinuities corresponding to the apparent jump between the two extremes of the range that, in fact, are the same angular position.…”