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Atypical bifurcation for periodic solutions of $φ$-Laplacian systems

Pierluigi Benevieri, Guglielmo Feltrin

TL;DR

The paper develops an atypical (cobifurcation) bifurcation theory for $T$-periodic solutions of parameter-dependent $\phi$-Laplacian systems $ (\phi(x'))'=F(\lambda,t,x,x')$ by leveraging Mawhin’s coincidence degree. It constructs a functional first-order formulation, proves a general bifurcation theorem (Theorem GenBifthm) via a finite-dimensional reduction and degree-theoretic arguments, and shows that nontrivial solution branches emanating from trivial solutions either are unbounded or meet domain boundaries. The authors then apply the framework to Liénard-type equations, obtaining branches of periodic solutions that are unbounded in the $\lambda$-direction, unbounded in the solution component with $\lambda$ bounded, or bounded in both, under concrete structural hypotheses. The results broaden classical continuation methods by explicitly tracking parameter-dependent branches and provide topological criteria (nonzero Brouwer degree) to guarantee connected sets of nontrivial periodic solutions. Overall, the work advances a topological, degree-theoretic approach to atypical bifurcation for nonlinear differential operators with nonlinearity in the highest derivative, with concrete implications for nonlinear oscillations and pattern-formation models.

Abstract

In this paper, we study the $T$-periodic solutions of the parameter-dependent $φ$-Laplacian equation \begin{equation*} (φ(x'))'=F(λ,t,x,x'). \end{equation*} Based on the topological degree theory, we present some atypical bifurcation results in the sense of Prodi-Ambrosetti, i.e., bifurcation of $T$-periodic solutions from $λ=0$. Finally, we propose some applications to Liénard-type equations.

Atypical bifurcation for periodic solutions of $φ$-Laplacian systems

TL;DR

The paper develops an atypical (cobifurcation) bifurcation theory for -periodic solutions of parameter-dependent -Laplacian systems by leveraging Mawhin’s coincidence degree. It constructs a functional first-order formulation, proves a general bifurcation theorem (Theorem GenBifthm) via a finite-dimensional reduction and degree-theoretic arguments, and shows that nontrivial solution branches emanating from trivial solutions either are unbounded or meet domain boundaries. The authors then apply the framework to Liénard-type equations, obtaining branches of periodic solutions that are unbounded in the -direction, unbounded in the solution component with bounded, or bounded in both, under concrete structural hypotheses. The results broaden classical continuation methods by explicitly tracking parameter-dependent branches and provide topological criteria (nonzero Brouwer degree) to guarantee connected sets of nontrivial periodic solutions. Overall, the work advances a topological, degree-theoretic approach to atypical bifurcation for nonlinear differential operators with nonlinearity in the highest derivative, with concrete implications for nonlinear oscillations and pattern-formation models.

Abstract

In this paper, we study the -periodic solutions of the parameter-dependent -Laplacian equation \begin{equation*} (φ(x'))'=F(λ,t,x,x'). \end{equation*} Based on the topological degree theory, we present some atypical bifurcation results in the sense of Prodi-Ambrosetti, i.e., bifurcation of -periodic solutions from . Finally, we propose some applications to Liénard-type equations.
Paper Structure (11 sections, 11 theorems, 90 equations, 1 figure)

This paper contains 11 sections, 11 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Preliminaries on degree theory
  3. The coincidence degree
  4. A composition property for the Brouwer degree
  5. A functional formulation of problem \ref{['mainpb']}
  6. Proof of Theorem \ref{['genbifthm']} and consequences
  7. Some applications
  8. A branch of periodic solutions unbounded in the $\lambda$-component
  9. An unbounded branch of periodic solutions, bounded in the $\lambda$-component
  10. A bounded branch of periodic solutions
  11. A topological remark

Key Result

Theorem 1.1

Assume that hp-H1, hp-H2 and hp-H3 hold true. Let $\Omega$ be an open subset of $\mathbb{R} \times \mathcal{C}^1_T$ and suppose that the Brouwer degree where $\widetilde{\Omega}_0\coloneqq\{x\in \mathbb{R}^n \colon (0,x)\in \Omega\}$ (with the natural identification between $\mathbb{R}^n$ and the subspace of constant maps in $\mathcal{C}^1_T$). Then, there exists a connected set $\Gamma \subsete

Figures (1)

  • Figure 1: Qualitative representation of the branches of nontrivial solutions in the three examples considered in Section \ref{['section-5']}.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof : Proof of Theorem \ref{['genbifthm']}
  • Remark 4.1
  • Remark 4.2
  • ...and 18 more