# Dynamics of a Forced Oscillator having an Obstacle

Chapter

## Abstract

Consider the scalar differential equation where The existence of 2π-periodic solutions has been analyzed by many authors using different variational and topological methods. For the linear case it (

$$\ddot x + g(x) = f(t)$$

(1.1)

*f*is 2π-periodic, say \(f \in C(\mathbb{T}{\text{)}} {\text{with}} \mathbb{T}{\text{ = }}\mathbb{R}{\text{/2}}\pi \mathbb{Z}\) and*g*satisfies$$\mathop {\lim }\limits_{x \to \pm \infty } g(x) = \pm \infty , \mathop {\lim \sup }\limits_{|x| \to \infty } \frac{{g(x)}}{x} < \infty .$$

(1.2)

*g*(*x*) =*w*^{2}*x*) is well known that the existence of a periodic solution is equivalent to the boundedness of all solutions, and one can ask whether such an equivalence still holds in nonlinear cases. In this paper we report on several results which give partial answers to this question. First we shall assume that*g*satisfies the assumptions of Lazer and Leach in [13] and we shall show that the condition for existence of a periodic solution obtained in that paper guarantees, in many cases, the boundedness of all solutions. For this class of nonlinearities the situation resembles the linear theory. Later we shall consider the asymmetric nonlinearities that were first discussed by Fucik [11] and Dancer [6, 5]. The situation now is more delicate because unbounded and periodic solutions can coexist. After this brief review of published results we shall analyze in detail the problem of boundedness for a forced linear oscillator which bounces elastically against a wall.## Keywords

Periodic Solution Periodic Problem Invariant Curf Unbounded Solution Twist Mapping
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J.M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,
*Nonlinearity*9 (1996), 1099–1111.MathSciNetMATHCrossRefGoogle Scholar - [2]J.M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator,
*J. Differential Equations*143 (1998), 201–220.MathSciNetMATHCrossRefGoogle Scholar - [3]P. Boyland, Dual billiards, twist maps and impact oscillators,
*Nonlinearity*9 (1996), 1411–1438.MathSciNetMATHCrossRefGoogle Scholar - [4]W. Dambrosio, A note on the existence of unbounded solutions to a perturbed asymmetric oscillator, to appear.Google Scholar
- [5]E.N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations,
*Bull. Austral. Math. Soc.***15**(1976), 321–328.MathSciNetMATHCrossRefGoogle Scholar - [6]E.N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations,
*Proc. Roy.**Soc. Edinburgh Sect. A***76**(1977), 283–300.MathSciNetMATHGoogle Scholar - [7]R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem,
*Ann. Scuola Norm. Sup. Pisa Cl. Sci.***14**(1987), 79–95.MathSciNetMATHGoogle Scholar - [8]C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,
*J. Differential Equations***147**(1998), 58–78.MathSciNetMATHCrossRefGoogle Scholar - [9]C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance,
*Nonlinearity***13**(2000), 493–505.MathSciNetMATHCrossRefGoogle Scholar - [10]D.G. de Figueiredo and B. Ruf, On a superlinear Sturm—Liouville equation and a related bouncing problem,
*J. Reine Angew. Math.***421**(1991), 1–22.MathSciNetMATHGoogle Scholar - [11]S. Fuóik, Solvability of nonlinear equations and boundary value problems,
*Mathematics and its Applications*,**4**, D. Reidel Publishing Co., Dordrecht-Boston, 1980.Google Scholar - [12]M. Kunze, Remarks on boundedness of semilinear oscillators, in Nonlinear analysis and its applications to differential equations (Lisbon, 1998), L. Sanchez, ed., 311–319,
*Prog. Nonlinear Differential Equations Appl.*,**43**, Birkhäuser, Boston, 2001.Google Scholar - [13]A.C. Lazer and D.E. Leach, Bounded perturbations of forced harmonic oscillators at resonance,
*Ann. Mat. Pura Appl.***82**(1969), 49–68.MathSciNetMATHCrossRefGoogle Scholar - [14]A.C. Lazer and J.P. McKenna, A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearity,
*Proc. Amer. Math. Soc.***106**(1989), 119–125.MathSciNetMATHCrossRefGoogle Scholar - [15]A.C. Lazer and J.P. McKenna, Periodic bouncing for a forced linear spring with obstacle,
*Differential Integral Equations***5**(1992), 165–172.MathSciNetMATHGoogle Scholar - [16]M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,
*Comm. Math. Phys.***143(1991)**,43–83.MathSciNetMATHCrossRefGoogle Scholar - [17]M. Levi and J. You, Oscillatory escape in a Duffing equation with polynomial potential,
*J. Differential Equations***140**(1997), 415–426.MathSciNetMATHCrossRefGoogle Scholar - [18]B. Liu, Boundedness in nonlinear oscillations at resonance,
*J. Differential Equations***153**(1999), 142–174.MathSciNetMATHCrossRefGoogle Scholar - [19]B. Liu, Boundedness in asymmetric oscillations,
*J. Math. Anal. Appl.***231**(1999), 355–373.MathSciNetMATHCrossRefGoogle Scholar - [20]G.R. Morris, A case of boundedness in Littlewood’s problem on oscillatory differential equations, Bull. Austral. Math. Soc. 14 (1976), 71–93.MathSciNetMATHCrossRefGoogle Scholar
- [21]R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. 53 (1996), 325–342.MathSciNetMATHCrossRefGoogle Scholar
- [22]R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999), 381–413.MathSciNetMATHCrossRefGoogle Scholar
- [23]R. Ortega, Invariant curves of mappings with averaged small twist, Advanced Nonlinear Studies 1 (2001), 14–39.MathSciNetMATHGoogle Scholar
- [24]G. Seifert, Resonance in undamped second-order nonlinear equations with periodic forcing, Quart. Appl. Math. 48 (1990), 527–530.MathSciNetMATHGoogle Scholar
- [25]C.L. Siegel, and J.K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, 187, Springer-Verlag, New York-Heidelberg, 1971.Google Scholar
- [26]X. Yuan, Quasiperiodic motion and boundedness for equations with jumping nonlinearity, to appear.Google Scholar
- [27]V. Zharnitsky, Instability in Fermi-Ulam “ping-pong” problem, Non-linearity 11 (1998), 1481–1487.MathSciNetMATHGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2002