The coexistence of quasi-periodic and blow-up solutions for a class of impact oscillators without the twist condition

Yanmei Sun; Yanmei Sun

doi:10.3934/math.2025467

AIMS Mathematics

2025, Volume 10, Issue 5: 10263-10282. doi: 10.3934/math.2025467

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Research article

The coexistence of quasi-periodic and blow-up solutions for a class of impact oscillators without the twist condition

Yanmei Sun ^,

School of Mathematics and Statistics, Weifang University, Weifang 261061, China

Received: 10 February 2025 Revised: 13 April 2025 Accepted: 24 April 2025 Published: 06 May 2025
MSC : 34C11, 34C15, 37J40, 70K43

The purpose of this paper was the study of the dynamic behavior of impact oscillators:
$\begin{align} \begin{split} \left\{ \begin{array}{l} x''+a(x)\, x^{2n+1}+ \sum\limits_{l = 0}^{m}p_l(t)\, x^{l} = 0, \, \text{for}\ x(t)>0, \, \\ x(t)\geq 0, \\ x'(t^{+}_{0}) = -x'(t^{-}_{0}), \, \text{if}\ x(t_{0}) = 0, \end{array} \right. \nonumber \end{split} \end{align}$
where the positive function $ a(x) $ is a smooth $ T $-periodic oscillator violating the monotone twist condition. We have proved that the above equation has an infinite number of bounded solutions as well as a solution that escapes to infinity in a finite amount of time.
- KAM theorem,
- impact oscillators,
- boundedness of solutions,
- quasi-periodic solutions,
- unbounded solutions
Citation: Yanmei Sun. The coexistence of quasi-periodic and blow-up solutions for a class of impact oscillators without the twist condition[J]. AIMS Mathematics, 2025, 10(5): 10263-10282. doi: 10.3934/math.2025467

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Abstract

The purpose of this paper was the study of the dynamic behavior of impact oscillators:

$\begin{align} \begin{split} \left\{ \begin{array}{l} x''+a(x)\, x^{2n+1}+ \sum\limits_{l = 0}^{m}p_l(t)\, x^{l} = 0, \, \text{for}\ x(t)>0, \, \\ x(t)\geq 0, \\ x'(t^{+}_{0}) = -x'(t^{-}_{0}), \, \text{if}\ x(t_{0}) = 0, \end{array} \right. \nonumber \end{split} \end{align}$

where the positive function $ a(x) $ is a smooth $ T $-periodic oscillator violating the monotone twist condition. We have proved that the above equation has an infinite number of bounded solutions as well as a solution that escapes to infinity in a finite amount of time.

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