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On the boundedness of solutions of a forced discontinuous oscillator

Tere M-Seara, Luan V. M. F. Silva, Jordi Villanueva

Abstract

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.

On the boundedness of solutions of a forced discontinuous oscillator

Abstract

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.
Paper Structure (6 sections, 5 theorems, 152 equations, 3 figures)

This paper contains 6 sections, 5 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. A parameterization result for invariant curves of a twist map of the annulus: Proof of Theorem \ref{['TB']}
  4. The Impact Map: Proof of Proposition \ref{['P']}
  5. The Localized Impact Map: Proof of Proposition \ref{['P1']}
  6. Exact sympletic structure: Proof of Proposition \ref{['ExactSympleticGeneral']}

Key Result

Proposition 1

Let $H(x,y,t)$ be a non-autonomous Hamiltonian, with respect to the $2$-form $\mathrm{d} x\wedge \mathrm{d} y$, and with $2\pi$-periodic dependence in $t$. By adding $E$ as a conjugate variable of $t$, we introduce the autonomous Hamiltonian ${\mathcal{H}}(x,t,y,E)=H(x,y,t)+E$, with respect to the $ with $t^{\tau}(x_0,t_0,y_0,E_0)=t_0 +\tau$, where $\tau$ represents the new time, and $(x_0,t_0,y_0

Figures (3)

  • Figure 1: Phase portrait of \ref{['eq1']} for the unperturbed case ($\varepsilon=0$).
  • Figure 2: Impact map $\mathcal{P}_{\varepsilon}= \mathcal{P}_{\varepsilon}^{-}\circ\mathcal{P}_{\varepsilon}^{+}$.
  • Figure 3: Intersections between the invariant tori in the unperturbed scenario and the subset $\Sigma^{+}$. In the curve $\mathbb{T}\times\{0\}\times\{y_0\}$, we can see the dynamics generated by the impact map $\mathcal{P}_0$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3
  • Proposition 3
  • proof
  • proof : Proof of Theorem \ref{['TB']}
  • Lemma 4
  • proof
  • ...and 2 more