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On the dynamics of non-autonomous systems in a~neighborhood of a~homoclinic trajectory

Alessandro Calamai, Matteo Franca, Michal Pospisil

TL;DR

This work analyzes the local dynamics near a homoclinic trajectory of a 2D piecewise‑smooth nonautonomous system $\dot{x}=f(x)+\varepsilon g(t,x,\varepsilon)$, establishing a time‑dependent Poincaré map to track loops close to $\boldsymbol{\Gamma}$. Through a four‑leg loop decomposition and a combination of exponential dichotomy and fixed‑point methods, the authors derive precise asymptotics for fly time and space displacement, with forward/backward estimates tied to the system’s hyperbolic eigenvalues $\lambda_s^{\pm},\lambda_u^{\pm}$ and logarithmic corrections in the loop distance $d$. They formulate two main results (Theorems key and keymissed) under no‑sliding assumptions, linking persistence of the homoclinic orbit to nondegenerate zeros of the Melnikov function $\mathcal{M}$ and outlining when Melnikov chaos may occur in discontinuous settings. The paper further discusses future directions, including subharmonic Melnikov theory, safe regions near $\boldsymbol{\Gamma}$ to avoid chaos, and extensions to settings where sliding obstructions are absent.

Abstract

This article is devoted to the study of a $2$-dimensional piecewise smooth (but possibly) discontinuous dynamical system, subject to a non-autonomous perturbation; we assume that the unperturbed system admits a homoclinic trajectory $\vecγ(t)$. Our aim is to analyze the dynamics in a neighborhood of $\vecγ(t)$ as the perturbation is turned on, by defining a Poincaré map and evaluating fly time and space displacement of trajectories performing a loop close to $\vecγ(t)$. Besides their intrinsic mathematical interest, these results can be thought of as a first step in the analysis of several interesting problems, such as the stability of a homoclinic trajectory of a non-autonomous ODE and a possible extension of Melnikov chaos to a discontinuous setting.

On the dynamics of non-autonomous systems in a~neighborhood of a~homoclinic trajectory

TL;DR

This work analyzes the local dynamics near a homoclinic trajectory of a 2D piecewise‑smooth nonautonomous system , establishing a time‑dependent Poincaré map to track loops close to . Through a four‑leg loop decomposition and a combination of exponential dichotomy and fixed‑point methods, the authors derive precise asymptotics for fly time and space displacement, with forward/backward estimates tied to the system’s hyperbolic eigenvalues and logarithmic corrections in the loop distance . They formulate two main results (Theorems key and keymissed) under no‑sliding assumptions, linking persistence of the homoclinic orbit to nondegenerate zeros of the Melnikov function and outlining when Melnikov chaos may occur in discontinuous settings. The paper further discusses future directions, including subharmonic Melnikov theory, safe regions near to avoid chaos, and extensions to settings where sliding obstructions are absent.

Abstract

This article is devoted to the study of a -dimensional piecewise smooth (but possibly) discontinuous dynamical system, subject to a non-autonomous perturbation; we assume that the unperturbed system admits a homoclinic trajectory . Our aim is to analyze the dynamics in a neighborhood of as the perturbation is turned on, by defining a Poincaré map and evaluating fly time and space displacement of trajectories performing a loop close to . Besides their intrinsic mathematical interest, these results can be thought of as a first step in the analysis of several interesting problems, such as the stability of a homoclinic trajectory of a non-autonomous ODE and a possible extension of Melnikov chaos to a discontinuous setting.

Paper Structure

This paper contains 11 sections, 25 theorems, 267 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminary constructions and notation
  3. Construction of the Poincaré map
  4. Statement of the main results
  5. Proofs of Theorems \ref{['key']} and \ref{['keymissed']}
  6. The loop: from $\boldsymbol{L^0}$ to $\boldsymbol{S^+(\tau)}$ and from $\boldsymbol{S^-(\tau)}$ to $\boldsymbol{L^0}$
  7. The loop: from $\boldsymbol{S^+(\tau)}$ to $\boldsymbol{L^\textrm{in}}$
  8. The loop: from $\boldsymbol{S^-(\tau)}$ to $\boldsymbol{L^\textrm{in}}$
  9. Final step: estimate of time and space displacement
  10. Some remarks and future developments
  11. Sketch of the proofs of Lemmas \ref{['lemma0s']} and \ref{['lemma0u']}

Key Result

Lemma 3.3

Assume F0, F1, F2, K, G. We can choose $\varepsilon_0$ and $\beta_0$ such that for any $0< \varepsilon \le \varepsilon_0$, any $0< \beta \le \beta_0$ with $\beta \ge \varepsilon^{\frac{\sigma^{\textrm{fb}}}{2}}$ we can define $Z^{\textrm{fwd},\textrm{in}}$, $Z^{\textrm{fwd},\textrm{out}}$ so that th Finally we can assume that $K^{\textrm{fwd}} \subset B({\boldsymbol{\Gamma}}, \beta^{ \underline

Figures (8)

  • Figure 1: Stable and unstable leaves (the superscript "$^{\textrm{out}}$" is denoted as "$^{\textrm{o}}$" for short).
  • Figure 2: Scenarios 1 and 2. In these two settings there is no sliding close to the origin. Further Melnikov theory guarantees persistence of the homoclinic trajectories CaFr, and we conjecture that we may have chaotic phenomena.
  • Figure 3: Scenarios 3 and 4: in these settings we have persistence of the homoclinic trajectories but sliding might occur close to the origin. Here our analysis does not apply directly. Further Melnikov theory guarantees persistence of the homoclinic trajectories CaFr, but chaos is forbidden FrPoin general.
  • Figure 4: Construction of the set $K^{\textrm{fwd}}$ (the superscripts "$^{\textrm{fwd}}$", "$^{\textrm{in}}$", "$^{\textrm{out}}$" are denoted as "$^{\textrm{f}}$", "$^{\textrm{i}}$", "$^{\textrm{o}}$", respectively, for short). This picture enables us to control the trajectories of \ref{['eq-disc']} in a neighborhood of ${\boldsymbol{\Gamma}}$ in forward time.
  • Figure 5: Construction of the set $K^{\textrm{bwd}}$ (the the superscripts "$^{\textrm{bwd}}$", "$^{\textrm{in}}$", "$^{\textrm{out}}$" are denoted as "$^{\textrm{b}}$", "$^{\textrm{i}}$", "$^{\textrm{o}}$", respectively, for short). This picture enables us to control the trajectories of \ref{['eq-disc']} in a neighborhood of ${\boldsymbol{\Gamma}}$ in backward time.
  • ...and 3 more figures

Theorems & Definitions (63)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Remark 3.7
  • ...and 53 more