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Dynamics of polynomial generalized Liénard system near the origin and infinity

Jun Zhang

TL;DR

This work advances the understanding of polynomial generalized Liénard systems by providing a complete local-global phase-portrait theory. It develops a framework based on quasi-homogeneous blow-ups and Newton polygon techniques to classify origin dynamics into 9 portraits, and extends to infinity to obtain 26 portraits, with precise monodromy conditions (M1)-(M2) at the origin and (W1)-(W3) at infinity. It further characterizes global centers under the no-extra-equilibria scenario, yielding 5 global portraits and explicit (G1)-(G4) criteria, and proves that global centers are non-isochronous. Collectively, the results solve isochronicity-related questions for generalized Liénard systems and enrich the center and monodromy theory for high-degree polynomial planar systems.

Abstract

We classify all topological phase portraits of the polynomial generalized Liénard system, determined by three arbitrary polynomials, at the origin and the infinity. This yields a complete characterization of monodromy at the origin and the infinity. Moreover, we obtain a necessary and sufficient condition for the local center via Cherkas' method. When the origin is the only equilibrium and it is a center, there are 5 global phase portraits, including two types of global center. Further, we prove that the global center is not isochronous.

Dynamics of polynomial generalized Liénard system near the origin and infinity

TL;DR

This work advances the understanding of polynomial generalized Liénard systems by providing a complete local-global phase-portrait theory. It develops a framework based on quasi-homogeneous blow-ups and Newton polygon techniques to classify origin dynamics into 9 portraits, and extends to infinity to obtain 26 portraits, with precise monodromy conditions (M1)-(M2) at the origin and (W1)-(W3) at infinity. It further characterizes global centers under the no-extra-equilibria scenario, yielding 5 global portraits and explicit (G1)-(G4) criteria, and proves that global centers are non-isochronous. Collectively, the results solve isochronicity-related questions for generalized Liénard systems and enrich the center and monodromy theory for high-degree polynomial planar systems.

Abstract

We classify all topological phase portraits of the polynomial generalized Liénard system, determined by three arbitrary polynomials, at the origin and the infinity. This yields a complete characterization of monodromy at the origin and the infinity. Moreover, we obtain a necessary and sufficient condition for the local center via Cherkas' method. When the origin is the only equilibrium and it is a center, there are 5 global phase portraits, including two types of global center. Further, we prove that the global center is not isochronous.

Paper Structure

This paper contains 4 sections, 4 theorems, 104 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Phase portraits at the origin
  3. Phase portraits at the infinity
  4. Global center and its non-isochronicity

Key Result

Theorem 2.1

System GL:pfg with $a_p=-1$ has 9 different topological phase portraits at the origin, see Fig. fig:O-phase. More concretely, the relation between the phase portraits and parameters is listed in Table tab:O, in which $\hat{c}:=q|\frac{b_q}{p+1}|^{\frac{p+1}{p}}$.

Figures (18)

  • Figure 1: Topological phase portrait of system \ref{['GL:pfg']} at the origin.
  • Figure 2: Three cases of Newton polygons of system \ref{['GL:pfg']}.
  • Figure 3: Phase portraits of systems \ref{['equ:OI+']} and \ref{['equ:OI-']} along the $v$-axis.
  • Figure 4: Phase portraits of system \ref{['equ:OII4+']} along the $v$-axis.
  • Figure 5: Phase portraits of system \ref{['equ:OII4-']} along the $v$-axis.
  • ...and 13 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.1
  • Example 4.1