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Periodic second-order systems and coupled forced Van der Pol oscillators

Feliz Minhós, Sara Perestrelo

Abstract

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.

Periodic second-order systems and coupled forced Van der Pol oscillators

Abstract

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.
Paper Structure (5 sections, 2 theorems, 70 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Main Result
  4. Coupled forced Van der Pol oscillators
  5. Conclusions

Key Result

Lemma 2

Suppose that the continuous functions $f,g:[0,T]\times \mathbb{R}^4 \rightarrow \mathbb{R}$ satisfy a Nagumo-type condition relative to the intervals $[\gamma_1(t), \Gamma_1(t)]$ and $[\gamma_2(t), \Gamma_2(t)]$, for all $t \in [0,T]$. Then, for every solution $(z(t),w(t)) \in X^2$ of (eq: OP), (eq: there are $N_1, N_2 > 0$ such that

Figures (3)

  • Figure 1: Orderless $\alpha_i, \beta_i$ functions, with $i=1,2$.
  • Figure 2: Shifted functions, $\alpha_i^0, \beta_i^0, i=1,2$, localizing the solution pair $(z^*(t), w^*(t))$.
  • Figure 3: Shifted functions, $\alpha_i^0, \beta_i^0, i=1,2$, localizing the solution pair $(z^*(t), w^*(t))$.

Theorems & Definitions (7)

  • Definition 1
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 4
  • proof
  • Example 5