Dynamics of a Forced Oscillator having an Obstacle

  • R. Ortega
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 49)

Abstract

Consider the scalar differential equation
$$\ddot x + g(x) = f(t)$$
(1.1)
where 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)
The existence of 2π-periodic solutions has been analyzed by many authors using different variational and topological methods. For the linear case it (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

Peri Sine 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J.M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity 9 (1996), 1099–1111.MathSciNetMATHCrossRefGoogle Scholar
  2. [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. [3]
    P. Boyland, Dual billiards, twist maps and impact oscillators, Nonlinearity 9 (1996), 1411–1438.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W. Dambrosio, A note on the existence of unbounded solutions to a perturbed asymmetric oscillator, to appear.Google Scholar
  5. [5]
    E.N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc. 15 (1976), 321–328.MathSciNetMATHCrossRefGoogle Scholar
  6. [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. [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. [8]
    C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations 147 (1998), 58–78.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity 13 (2000), 493–505.MathSciNetMATHCrossRefGoogle Scholar
  10. [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. [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. [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. [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. [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. [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. [16]
    M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. 143(1991),43–83.MathSciNetMATHCrossRefGoogle Scholar
  17. [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. [18]
    B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations 153 (1999), 142–174.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl. 231 (1999), 355–373.MathSciNetMATHCrossRefGoogle Scholar
  20. [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. [21]
    R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. 53 (1996), 325–342.MathSciNetMATHCrossRefGoogle Scholar
  22. [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. [23]
    R. Ortega, Invariant curves of mappings with averaged small twist, Advanced Nonlinear Studies 1 (2001), 14–39.MathSciNetMATHGoogle Scholar
  24. [24]
    G. Seifert, Resonance in undamped second-order nonlinear equations with periodic forcing, Quart. Appl. Math. 48 (1990), 527–530.MathSciNetMATHGoogle Scholar
  25. [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. [26]
    X. Yuan, Quasiperiodic motion and boundedness for equations with jumping nonlinearity, to appear.Google Scholar
  27. [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

Authors and Affiliations

  • R. Ortega
    • 1
  1. 1.Departamento de Matemática Aplicada Facultad de CienciasUniversidad de GranadaGranadaSpain

Personalised recommendations