Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 777–802 | Cite as

Global Asymptotic Stability for Oscillators with Superlinear Damping

  • Jitsuro Sugie
  • Tsunehiko Shimadu
  • Takashi Yamasaki
Article

Abstract

A necessary and sufficient condition is established for the equilibrium of the damped superlinear oscillator
$$x^{\prime\prime} + a(t)\phi_q(x^{\prime}) + \omega^2x = 0$$

to be globally asymptotically stable. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation

$$u^{\prime} + \omega^{q-2}a(t)\phi_q(u) + 1 = 0$$
is divergent or convergent. Since this nonlinear differential equation cannot be solved in general, it can be said that the presented result is expressed by an implicit condition. Explicit sufficient conditions and explicit necessary conditions are also given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, it is proved that a certain growth condition of a(t) guarantees the global asymptotic stability for the equilibrium of the damped superlinear oscillator.

Keywords

Damped oscillator Superlinear differential equations Global asymptotic stability Newtonian damping Growth condition 

Mathematics Subject Classification

34D05 34D23 34D45 37B25 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jitsuro Sugie
    • 1
  • Tsunehiko Shimadu
    • 1
  • Takashi Yamasaki
    • 1
  1. 1.Department of Mathematics and Computer ScienceShimane UniversityMatsueJapan

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