On the periodic solutions of a perturbed double pendulum

Journal of Applied Mathematics and Mechanics, 1989

Nonlinear Analysis: Theory, Methods & Applications, 2017

Journal of Dynamics and Differential Equations, 2010

In this paper we study the existence and multiplicity of periodic solutions of pendulum-like perturbations of bounded or singular φ-Laplacians. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.

Journal of Differential Equations, 1987

In both works some symmetry hypotheses on the forcing terms were considered. This paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms. Note that as a model example it is possible to consider the motion of N coupled pendulums (see the already mentioned paper by J. A. Marlin) or the oscillations of an N-coupled point Josephson junction with external time-dependent disturbances studied in the autonomous case by M.

Topological methods in nonlinear analysis

We establish multiplicity results of periodic solutions for relativistic pendulum type systems of ordinary differential equations. We provide a different approach to the problems and answer some questions raised in \cite{6}, \cite{7} by Brezis and Mawhin recently.

Journal of Mathematical Analysis and Applications, 2000

Chaos, Solitons & Fractals, 2003

The simple pendulum is a paradigm in the study of oscillations and other phenomena in physics and nonlinear dynamics. This explains why it has deserved much attention, from many viewpoints, for a long time. Here, we attempt to describe what we call a generalized perturbed pendulum, which comprises, in a single model, many known situations related to pendula, including different forcing and nonlinear damping terms. Melnikov analysis is applied to this model, with the result of general formulae for the appearance of chaotic motions that incorporate several particular cases. In this sense, we give a unified view of the pendulum.

Acta Mathematica Sinica, 1987

Abotrac*t. Existence of 2n-periodic solutions to the equation ~ + g(x)=p(t) is proved, under sharp nonresonance conditions on the interaction of g(x)/x with two consecutive eigenvalues m 2 and (m + 1)2: touch with the eigenvalues is allowed.

HAL (Le Centre pour la Communication Scientifique Directe), 2020

We study a singularly perturbed second-order differential equation describing a slowly and periodically varying hamiltonian system. Typical dynamics governed by this type of system are, for example, equations of forced pendulum, of Duffing or of the "shallow water sloshing" problem. Using symmetries of this equation and singular perturbation tools, we describe dynamics, by splitting the phase space in regions where the motion is oscillatory and others where it is unbounded, and study dynamics in each kind of regions. Finally we establish the existence periodic solutions and give the structure of these solutions in term of response curves. In particular, our results extend and complete the ones stated in and answer to some open questions within. We also give new results about multiplicity of periodic solutions of forced pendulum equation. To illustrate our results, we conclude this work by a numerical study of these classical examples.

Journal of Differential Equations, 1996

derivative are both bounded and present an oscillating behavior. In this framework the most deeply studied problem in the literature is the case in which G is a periodic function. A natural and largely investigated problem of this type is given by the forced pendulum equation article no. 0160

São Paulo Journal of Mathematical Sciences, 2015

We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non-linear planar double pendulum perturbed by smooth or non-smooth functions.

Journal of Differential Equations, 1989

1999

Qualitative analysis of a pendulum with a periodically varying length is conducted. It is proved that there are two periodic solutions having a prescribed amplitude A( and a period ¹ which is an even multiple k of the excitation period. Stability analysis is carried out for the principal parametric oscillations (k"2). In this connection it is shown that such a pendulum cannot serve as a mathematical model of swing as it is generally considered.

2009

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Two types of transitions to chaos of the pendulum depending on problem parameters are investigated numerically.

Journal of Mathematical Analysis and Applications, 1987