Global Asymptotic Stability for Oscillators with Superlinear Damping

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.

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Correspondence to Jitsuro Sugie.

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Sugie, J., Shimadu, T. & Yamasaki, T. Global Asymptotic Stability for Oscillators with Superlinear Damping. J Dyn Diff Equat 24, 777–802 (2012). https://doi.org/10.1007/s10884-012-9256-3

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Keywords

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

Mathematics Subject Classification

  • 34D05
  • 34D23
  • 34D45
  • 37B25