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Forced oscillations for generalized $Φ$-Laplacian equations with Carathéodory perturbations

Alessandro Calamai, Maria Patrizia Pera, Marco Spadini

TL;DR

The paper studies $T$-periodic forced solutions of two parametric ODEs with a generalized $\Φ$-Laplacian in the Carathéodory setting, using topological Brouwer degree to derive global bifurcation results. By introducing an auxiliary variable and recasting the problem on $\mathbb{R}^{2n}$, the authors obtain unbounded branches of nontrivial forced pairs emanating from stationary states under explicit degree conditions on averaged perturbations. The approach builds on recent CaSp24 results for systems on manifolds but expresses the conclusions purely in terms of Brouwer degree in Euclidean spaces, making the results broadly accessible. Visual representations via finite-dimensional projections illustrate the possible branch geometries and multiplicity patterns of $T$-periodic solutions as the parameter $\lambda$ varies.

Abstract

Using topological methods, we study the structure of the set of forced oscillations of a class of parametric, implicit ordinary differential equations with a generalized $Φ$-Laplacian type term. We work in the Carathéodory setting. Under suitable assumptions, involving merely the Brouwer degree in Euclidean spaces, we obtain global bifurcation results. In some illustrative examples we provide a visual representation of the bifurcating set.

Forced oscillations for generalized $Φ$-Laplacian equations with Carathéodory perturbations

TL;DR

The paper studies -periodic forced solutions of two parametric ODEs with a generalized -Laplacian in the Carathéodory setting, using topological Brouwer degree to derive global bifurcation results. By introducing an auxiliary variable and recasting the problem on , the authors obtain unbounded branches of nontrivial forced pairs emanating from stationary states under explicit degree conditions on averaged perturbations. The approach builds on recent CaSp24 results for systems on manifolds but expresses the conclusions purely in terms of Brouwer degree in Euclidean spaces, making the results broadly accessible. Visual representations via finite-dimensional projections illustrate the possible branch geometries and multiplicity patterns of -periodic solutions as the parameter varies.

Abstract

Using topological methods, we study the structure of the set of forced oscillations of a class of parametric, implicit ordinary differential equations with a generalized -Laplacian type term. We work in the Carathéodory setting. Under suitable assumptions, involving merely the Brouwer degree in Euclidean spaces, we obtain global bifurcation results. In some illustrative examples we provide a visual representation of the bifurcating set.

Paper Structure

This paper contains 9 sections, 10 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A quick refresher on the Brouwer degree in Euclidean spaces
  3. Setting of the problem and statement of the main results
  4. Preliminary results
  5. Rearrangement of equations \ref{['eq:0-pertb-intro']} and \ref{['eq:g-pertb-intro']}
  6. Proof of the main results
  7. Proof of Theorem \ref{['th:main1']}
  8. Proof of Theorem \ref{['th:main2']}.
  9. Visual representation of branches

Key Result

Theorem 3.5

Let $\Omega$ be an open subset of $[0,\infty)\times C^1_T(U)$. Define the vector field $w\colon U\to\mathbb{R}^n$ as follows: and assume that $\deg(w,\Omega_U)$ is well-defined and nonzero. Then there exists a connected set $\Gamma$ of nontrivial $T$-forced pairs in $\Omega$ of eq:0-pertb-intro whose closure in $[0,\infty)\times C^1_T(U)$ intersects the set $\{ (0,\overline{p})\in [0,\infty) \tim

Figures (5)

  • Figure 1: Representation of a portion of $\Gamma$ of Example \ref{['ex:1']}
  • Figure 2: Representation of $\Gamma_0$ and $\Gamma_1$ of Example \ref{['ex:2']}
  • Figure 3: $C^1$-norm and orbit diameter of points in $\Gamma_{-1}$ and $\Gamma_1$ of Example \ref{['ex:3']}
  • Figure 4: A portion of the set of starting points $(\lambda,p,v)\in\Sigma_{-1}\cup\Sigma_{+1}$ of equation \ref{['eq:example3']} in Example \ref{['ex:3']}.
  • Figure 5: A portion of the set of starting points $(\lambda,p,v)\in\Sigma\cap([-1.5,1.5]\times[-0.2,0.15]\times[0,1.1])$ of equation \ref{['eq:example4']} in Remark \ref{['re:finale']}.

Theorems & Definitions (28)

  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Remark 3.7
  • Definition 4.1
  • Theorem 4.2: CaSp24
  • Definition 4.3
  • ...and 18 more