Subharmonic and homoclinic bifurcations in a parametrically forced pendulum

Koch, B. P.; Leven, R.

doi:10.1016/0167-2789(85)90082-x

Cited by 119 publications

(51 citation statements)

References 28 publications

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“…The behaviour observed here is similar to that of the basin erosion phenomenon described in earlier work for external excitation only [17]; it can be seen that there is little change in the size or position of the safe basin up to G = 0.10. As the magnitude of the forcing is increased there is then a homoclinic tangling of the stable and unstable manifolds of the hill-top saddle, resulting in a fractal basin boundary [19][20][21], with thin finger-like projections penetrating the safe basin. Indeed at G = 0.15, a value of forcing just above the homoclinic tangency (obtained numerically), the fractal structure is hardly visible on the scale considered in Figure 2; in Figure 3, a blow-up of the basin and manifold structure, it can clearly seen that the stable manifold (the basin boundary), W s (Dh), and the unstable manifold WU(Dh) are homoclinically tangled.…”

Section: Erosion Of the Safe Basinmentioning

confidence: 99%

Global transient dynamics of nonlinear parametrically excited systems

Soliman

1994

Nonlinear Dyn

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In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system.We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.

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Section: Erosion Of the Safe Basinmentioning

confidence: 99%

Global transient dynamics of nonlinear parametrically excited systems

Soliman

1994

Nonlinear Dyn

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“…This fixes the resonance p : q. To first order, in both cases, we need p = 1 and q to be even for B 1 not to vanish, see also [42]. The corresponding values of the coefficients B 1 = B 1 (p/q) for the system with α = 0.5 are given in Tables 1 and 2; the values k 1 and k 2 are the elliptic moduli corresponding to each solution.…”

Section: Thresholds Values Of the Subharmonic Solutionsmentioning

confidence: 99%

Comparisons between the pendulum with varying length and the pendulum with oscillating support

Wright

Bartuccelli

Gentile

2017

Journal of Mathematical Analysis and Applications

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We consider two forced dissipative pendulum systems, the pendulum with vertically oscillating support and the pendulum with periodically varying length, with a view to draw comparisons between their behaviour. We study the two systems for values of the parameters for which the dynamics are non-chaotic. We focus our investigation on the persisting attractive periodic orbits and their basins of attraction, utilising both analytical and numerical techniques. Although in some respect the two systems have similar behaviour, we find that even within the perturbation regime they may exhibit different dynamics. In particular, for the same value of the amplitude of the forcing, the pendulum with varying length turns out to be perturbed to a greater extent. Furthermore the periodic attractors persist under larger values of the damping coefficient in the pendulum with varying length. Finally, unlike the pendulum with oscillating support, the pendulum with varying length cannot be stabilised around the upward position for any values of the parameters.

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“…The bifurcation in an inverted pendulum with the high frequency excitation was proved by using analytical and experimental investigations in [9]. Subharmonic and homoclinic bifurcations in a parametrically forced pendulum was studied in [10]. Recently, a rotating pendulum linked by an oblique spring with fixed end has been proposed and investigated in [11].…”

Section: Introductionmentioning

confidence: 99%

Multiple Bifurcations of a Cylindrical Dynamical System

Han¹,

Cao²

2016

Journal of Theoretical and Applied Mechanics

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Abstract. This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the homoclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and numerical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

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Subharmonic and homoclinic bifurcations in a parametrically forced pendulum

Cited by 119 publications

References 28 publications

Global transient dynamics of nonlinear parametrically excited systems

Global transient dynamics of nonlinear parametrically excited systems

Comparisons between the pendulum with varying length and the pendulum with oscillating support

Multiple Bifurcations of a Cylindrical Dynamical System

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