Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Chebyshev criterion for at most two non-zero limit cycles in Abel equations

Jianfeng Huang, Renhao Tian, Yulin Zhao

TL;DR

The paper resolves the Smale-Pugh variant for the reduced Abel equation $\dot{x}=A(t)x^{3}+B(t)x^{2}$ by proving a Chebyshev criterion: if $A,B$ lie in the span of an ET-system $\{f_{0},f_{1},f_{2}\}$ with $f_{0}>0$, then the equation has at most two non-zero limit cycles (counted with multiplicity). The approach combines a rotated-differential-equations framework, a second-order Poincaré map derivative formula, and a quasi-dichotomy for ET-systems to bound possible limit-cycle configurations; Lyapunov constants for $x=0$ are computed to distinguish centers from limit cycles. The theory is applied to concrete coefficient classes, including linear trig coefficients and quadratic polynomials, recovering and sharpening existing results (e.g., at most three limit cycles including $x=0$ for linear trig cases) and extending to trinomial coefficients. Overall, the work links Chebyshev-system structure to finite upper bounds on limit cycles, providing sharp, class-dependent Hilbert numbers and a unified method for Smale-Pugh-type problems in Abel equations.

Abstract

In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation $\dot{x}=A(t)x^{3}+B(t)x^{2}$ on an interval $[0,T]$. The Smale-Pugh problem asks whether this maximum number is bounded in terms of a given class of coefficients. We establish for the first time a Chebyshev criterion, providing a positive answer to the problem when this class spanned by an extended Chebyshev system (ET-system) $\mathcal{F}=\{f_{0},f_{1},f_{2}\}$ on $[0,T)$ with $f_{0}\not=0$. As an application, we prove that the equation has at most three limit cycles (including $x=0$) when the coefficients $A$ and $B$ are both linear trigonometric functions or quadratic polynomials. This reestablishes the result of Yu et al. (J. Differ. Equ., 2024) and improves the work of Bravo et al. (Disc. Cont. Dyn. Syst., 2015 \& J. Differ. Equ., 2024). We also obtain the same maximum number of limit cycles for the equation with trinomial coefficients.

A Chebyshev criterion for at most two non-zero limit cycles in Abel equations

TL;DR

The paper resolves the Smale-Pugh variant for the reduced Abel equation by proving a Chebyshev criterion: if lie in the span of an ET-system with , then the equation has at most two non-zero limit cycles (counted with multiplicity). The approach combines a rotated-differential-equations framework, a second-order Poincaré map derivative formula, and a quasi-dichotomy for ET-systems to bound possible limit-cycle configurations; Lyapunov constants for are computed to distinguish centers from limit cycles. The theory is applied to concrete coefficient classes, including linear trig coefficients and quadratic polynomials, recovering and sharpening existing results (e.g., at most three limit cycles including for linear trig cases) and extending to trinomial coefficients. Overall, the work links Chebyshev-system structure to finite upper bounds on limit cycles, providing sharp, class-dependent Hilbert numbers and a unified method for Smale-Pugh-type problems in Abel equations.

Abstract

In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation on an interval . The Smale-Pugh problem asks whether this maximum number is bounded in terms of a given class of coefficients. We establish for the first time a Chebyshev criterion, providing a positive answer to the problem when this class spanned by an extended Chebyshev system (ET-system) on with . As an application, we prove that the equation has at most three limit cycles (including ) when the coefficients and are both linear trigonometric functions or quadratic polynomials. This reestablishes the result of Yu et al. (J. Differ. Equ., 2024) and improves the work of Bravo et al. (Disc. Cont. Dyn. Syst., 2015 \& J. Differ. Equ., 2024). We also obtain the same maximum number of limit cycles for the equation with trinomial coefficients.

Paper Structure

This paper contains 13 sections, 13 theorems, 98 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A formula for the second-order derivative of the Poincaré map
  4. Rotated differential equations
  5. A quasi-dichotomy for ET-systems with accuracy one
  6. Multiplicity and stability of limit cycles of equation \ref{['eq0']}
  7. Maximum multiplicity of non-zero limit cycles
  8. The Lyapunov constants of solution $x=0$
  9. Maximum number of limit cycles of equation \ref{['eq0']}
  10. Preliminary estimates for the number of limit cycles
  11. Proof of Theorem \ref{['theorem0']}
  12. Proofs of Theorems \ref{['app1']}, \ref{['main theorem1']} and \ref{['main theorem']}
  13. Appendix

Key Result

Theorem 1

Consider the Abel equation eq0. Assume that the coefficients $A,B\in \text{span}(\mathcal{F})$, where $\mathcal{F}=\{f_{0},f_{1},f_{2}\}$ is an ET-system on $[0,T)$ (therefore naturally each $f_i\in C^2\left([0, T)\right)$), with each $f_i$ continuous on $[0, T]$ and $f_{0}|_{[0,T]}>0$. Then the equ

Figures (4)

  • Figure 1: Geometric interpretations of $\bm{(D.1)}$ and $\bm{(D.2)}$. The black curve represents the graph for $x={B(t)}/{A(t)}$, the blue curve represents the graph for $x=-{\lambda}/{\mu}$, and the dashed lines denote the vertical lines passing through the zeros of $A(t)$. In subfigure $(A)$, the horizontal line $x=-{\lambda}/{\mu}$ does not cross the graph of the function $x={B(t)}/{A(t)}$. In subfigure $(B)$, the function $x={B(t)}/{A(t)}$ is strictly increasing on each connected components of $\{t\in (0,2\pi)|A(t)\not =0\}$. (For interpretation of the colors in the figures, the reader is referred to the web version of this article.)
  • Figure 2: Case where $x(t)-\varphi(t)$ has exactly one zero $t=t_{1}$ in $[0,T)$. The black solid curve represents the limit cycle $x(t)$, the black dashed line represents the vertical line passing through the zero of $A(t)$, and the blue dashed curve represents $\varphi(t)$. The two subfigures illustrate the subcases where the limit cycle intersects $\varphi(t)$ exactly once at $(t_{1},x_{1})$ when $A(t)$ has either a single simple zero or two simple zeros.
  • Figure 3: Case where $x(t)-\varphi(t)$ has exactly two zeros $t=t_{1}$ and $t=t_{2}$ in $[0,T)$. The black solid curve represents the limit cycle $x(t)$, the black dashed line represents the vertical line passing through the zero of $A(t)$, and the blue dashed curve represents $\varphi(t)$. The two subfigures illustrate the subcases where the limit cycle intersects $\varphi(t)$ twice at $(t_{1},x_{1})$ and $(t_{2},x_{2})$ with $t_{1}<t_{2}$ when $A(t)$ has either a single simple zero or two simple zeros.
  • Figure 4: Direction of the movement of the stable and unstable limit cycles as parameter $\lambda$ decreases. The blue, red and black lines represent stable limit cycles, unstable limit cycles and nearby non-periodic solutions, respectively.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2: YHL
  • Theorem 3
  • Theorem 4: Generalization of Theorem \ref{['main theorem1']}
  • Proposition 5
  • Definition 6: Han
  • Proposition 7: Han
  • Proposition 8
  • proof
  • Remark 9
  • ...and 17 more