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Dynamics near a class of nonhyperbolic fixed points

Meihua Jin, Shihao Meng, Yunhua Zhou

TL;DR

This work analyzes local dynamics near nonhyperbolic fixed points for a planar map $F=(f,g)$ with $P$ and $Q$ homogeneous of odd degree and higher-order perturbations $X,Y$. It develops a stable manifold theorem via cone conditions, proves a degenerate Hartman-type result giving a topological conjugacy to a decoupled map $F_1\times F_2$, and establishes finite shadowing near the fixed point under positive definiteness of associated coefficient tensors. The results extend classical hyperbolic-type conclusions to a broader nonhyperbolic class, yielding Lipschitz local stable manifolds, a product-type local model, and shadowing guarantees that enhance robustness of trajectories near degenerate equilibria.

Abstract

In this paper, we investigate some dynamical properties near a nonhyperbolic fixed point. Under some conditions on the higher nonlinear terms, we establish a stable manifold theorem and a degenerate Hartman theorem. Furthermore, the finite shadowing property also be discussed.

Dynamics near a class of nonhyperbolic fixed points

TL;DR

This work analyzes local dynamics near nonhyperbolic fixed points for a planar map with and homogeneous of odd degree and higher-order perturbations . It develops a stable manifold theorem via cone conditions, proves a degenerate Hartman-type result giving a topological conjugacy to a decoupled map , and establishes finite shadowing near the fixed point under positive definiteness of associated coefficient tensors. The results extend classical hyperbolic-type conclusions to a broader nonhyperbolic class, yielding Lipschitz local stable manifolds, a product-type local model, and shadowing guarantees that enhance robustness of trajectories near degenerate equilibria.

Abstract

In this paper, we investigate some dynamical properties near a nonhyperbolic fixed point. Under some conditions on the higher nonlinear terms, we establish a stable manifold theorem and a degenerate Hartman theorem. Furthermore, the finite shadowing property also be discussed.

Paper Structure

This paper contains 5 sections, 9 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stable manifolds
  4. Degenerate Hartman theorem
  5. Finite shadowing property

Key Result

Theorem 1.1

If $\mathscr{A}$, $\mathscr{B}$, $\mathscr{C}$, $\mathscr{D}$, $\mathscr{E}$, and $\mathscr{H}$ are positive definite, then there is a neighborhood $N$ of $\mathbf 0$ and $\delta>0$ such that the stable and unstable sets $W^s(F,N)$ and $W^u(F,N)$ are the graphs of two Lipschitz functions $\varphi^s$

Figures (1)

  • Figure 1: The boundary of N. We use four dashed lines, from $\mathbf{p}_2$ to $\mathbf{q}_2$, from $\mathbf{q}_1$ to $\overline{\mathbf{p}}_2$, from $\overline{\mathbf{p}}_1$ to $\overline{\mathbf{q}}_1$, and from $\overline{\mathbf{q}}_2$ to $\mathbf{p}_1$, to represent the corresponding iterated curves.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Example 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.1
  • ...and 10 more