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Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems

Murilo R. Cândido, Douglas D. Novaes, Joan S. G. Rivera

TL;DR

This work develops an analytical framework to detect isolated invariant tori in non-smooth, $T$-periodic differential systems by analyzing the smooth time-$T$ map through Melnikov expansions and near-identity normal forms. It proves sufficient Neimark–Sacker bifurcation criteria for piecewise systems, establishing the existence and persistence of limit tori associated with normally hyperbolic closed curves of the time-$T$ map. The authors then apply the theory to a 3D piecewise linear family, providing the first analytic detection of a limit torus in such systems and showing its attracting/repelling nature and stability under perturbations. Collectively, the results extend averaging-based torus detection to non-smooth dynamics and offer a concrete 3D PL example with rigorous torus existence, suggesting broad applicability to higher-dimensional nonsmooth phenomena.

Abstract

This work investigates a class of non-autonomous $T$-periodic piecewise smooth differential systems and their associated time-$T$ maps. Our main result provides an analytical approach for detecting, within this class of piecewise differential systems, isolated invariant tori associated with normally hyperbolic invariant closed curves of the time-$T$ map. To achieve this, we derive sufficient conditions under which smooth near-identity maps undergo a Neimark--Sacker bifurcation. As an application of our main result, we present a family of 3D piecewise linear differential systems exhibiting attracting and repelling isolated invariant tori which, moreover, persist under small perturbations. To the best of our knowledge, this family provides the first examples in which limit tori are analytically detected in piecewise linear systems.

Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems

TL;DR

This work develops an analytical framework to detect isolated invariant tori in non-smooth, -periodic differential systems by analyzing the smooth time- map through Melnikov expansions and near-identity normal forms. It proves sufficient Neimark–Sacker bifurcation criteria for piecewise systems, establishing the existence and persistence of limit tori associated with normally hyperbolic closed curves of the time- map. The authors then apply the theory to a 3D piecewise linear family, providing the first analytic detection of a limit torus in such systems and showing its attracting/repelling nature and stability under perturbations. Collectively, the results extend averaging-based torus detection to non-smooth dynamics and offer a concrete 3D PL example with rigorous torus existence, suggesting broad applicability to higher-dimensional nonsmooth phenomena.

Abstract

This work investigates a class of non-autonomous -periodic piecewise smooth differential systems and their associated time- maps. Our main result provides an analytical approach for detecting, within this class of piecewise differential systems, isolated invariant tori associated with normally hyperbolic invariant closed curves of the time- map. To achieve this, we derive sufficient conditions under which smooth near-identity maps undergo a Neimark--Sacker bifurcation. As an application of our main result, we present a family of 3D piecewise linear differential systems exhibiting attracting and repelling isolated invariant tori which, moreover, persist under small perturbations. To the best of our knowledge, this family provides the first examples in which limit tori are analytically detected in piecewise linear systems.
Paper Structure (13 sections, 7 theorems, 133 equations, 2 figures)

This paper contains 13 sections, 7 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction and outline of the main results
  2. Torus bifurcation in a class of nonsmooth system
  3. Limit tori in 3D piecewise linear systems
  4. Organization of the paper
  5. Basic notions and preliminaries results
  6. Melnikov and Averaging methods
  7. Neimark-Sacker Bifurcation
  8. Regularity of the time-$T$ map
  9. Neimark-Sacker Bifurcation for Near-Identity Maps
  10. Torus bifurcation in nonsmooth differential equations
  11. Invariant torus in a 3D piecewise linear system
  12. Conclusions
  13. Appendix: Lyapunov Coefficients

Key Result

Theorem 1

kuznetsov Assume that for $|\alpha|$ sufficiently small $\mathbf{x}=0$ is a fixed point of the map two-map having eigenvalues $r(\alpha)\,e^{\pm i\varphi(\alpha)}$ with $r(0)=1$ and $\varphi(0)=\theta$, $0<\theta<\pi$. Suppose that the following assertions are satisfied Then, there exist an open set $U\subset \mathbb{R}^{2}$ around the fixed point $\mathbf{x}=0$ and an open interval $I\subset \ma

Figures (2)

  • Figure 1: Numerical simulation of systems \ref{['ds0intro']}--\ref{['ds0intro1']} for $b=-5$, $\varepsilon=1/40$, and $\alpha=\varepsilon(\pi^{2}/8-2)$. The left panel shows an invariant closed curve on the Poincaré section $y=0$, $x>0$, of the Poincaré map, corresponding to an invariant torus. The right panel presents two views of a trajectory in phase space, close to the associated invariant torus, obtained by numerically integrating the system with initial condition $(3.669234340877,\,0,\,0.48488236396962971)$ over the time interval $t\in[0,10000]$.
  • Figure 2: Schematic representation of a crossing solution $\varphi$ of the piecewise smooth system \ref{['PSDE']}.

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Remark 2
  • Theorem A
  • proof
  • ...and 3 more