Asymptotic decay of solutions to an essentially nonlinear system with two-frequency resonances

Abstract

This paper is focused on the analysis of the asymptotic behavior of solutions for a nonlinear system with pairs of purely imaginary roots under the presence of two-frequency resonances of the fourth order. The main purpose of the paper is to estimate the norm of solutions, provided that the trivial solution is asymptotically stable regardless of forms higher than the third order. For this study, asymptotic stability conditions are presented and a Lyapunov function is constructed. These results are applied for estimating the decay rate of oscillations of a spring-pendulum system with partial dissipation.

Keywords:

• Asymptotic estimate
• critical case
• Lyapunov function
• resonance
• stability

AMS Subject Classifications:

• 34D05
• 34D20
• 70E55

Acknowledgements

The author is grateful to Professor Alexander Zuyev for his constant attention to this work and for valuable and constructive suggestions.

Notes

No potential conflict of interest was reported by the author.