Global attraction to the origin in a parametrically driven nonlinear oscillator

Bartuccelli, Michele V.; Deane, Jonathan H. B.; Gentile, Guido; Gourley, Stephen A.

doi:10.1016/s0096-3003(03)00317-5

Cited by 10 publications

(11 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The numerical computations reported in Section 5 show that in fact the origin attracts almost every trajectory even for smaller values of the damping coefficient: taking γ > γ 1 or ζ > ζ 1 , with γ 1 ≈ 0.060 and ζ 1 ≈ 0.090, turns out to be enough. As expected, condition (14) on ζ for the pendulum with varying length has some similarity to that in (11) for the pendulum with oscillating support. However for the system of the pendulum with varying length we find that damping coefficient must be taken larger, so as to exceed the forcing also created by the term g(t).…”

Section: Global Attraction To the Originsupporting

confidence: 71%

See 1 more Smart Citation

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

Wright

Bartuccelli

Gentile

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

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.

show abstract

Section: Global Attraction To the Originsupporting

confidence: 71%

“…To compute the conditions for which the origin attracts the full phase space, up to a zero-measure set, we use the approach outlined in [11]; see also [10,64]. First we consider the system (2) and define f (t) as…”

Section: Global Attraction To the Originmentioning

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

View full text Add to dashboard Cite

show abstract

“…In the presence of dissipation, it is unlikely that solutions other than the attractive ones found with the method we have used, would be relevant for the dynamics -cf. for instance the problems investigated in [4,2,3,1,5]. In general the situation can be delicate; for instance when one investigates quasi-periodic solutions corresponding to lower-dimensional tori of quasi-integrable systems, where uniqueness becomes a subtle problem -cf.…”

Section: Conclusion and Final Commentsmentioning

confidence: 99%

Bifurcation Curves of Subharmonic Solutions and Melnikov Theory Under Degeneracies

2007

View full text Add to dashboard Cite

We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves from the nonexistence to the existence of subharmonic solutions. In particular our results apply also when one has degeneracy to first order -i.e. when the subharmonic Melnikov function vanishes identically. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalisations to higher orders of the subharmonic Melnikov function are also identically zero. In general the bifurcation curves are not analytic, and even when they are smooth they can form cusps at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The technique we use is completely different from that of Chow and Hale, and it is essentially based on rigorous perturbation theory.

show abstract

“…Again formula (8) clearly shows that also this result is effectively a time-reparametrization of the time-independent case.…”

mentioning

confidence: 69%

“…Partial results for such systems may be found in the literature, for instance, in [12], [3], [11], [13], [9], [10] and [2]. The case of functions f whose derivative can change sign is much more difficult, but it is beyond the intentions of this paper; recent results in this directions have been obtained in [6], [7] and [8]. For a list of related problems of physical relevance we refer, for instance, to [1].…”

Section: Introductionmentioning

confidence: 99%