In this paper, we provide sufficient conditions for the existence of two limit cycles bifurcating from the unique zero–Hopf equilibrium of the differential system <Equation ID="Equ11"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_726_Article_Equ11.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="335" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \dot{x}=y+a,\quad \dot{y}=-x+z,\quad \dot{z}=-bx^2+z^2+c, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mspace width="1em" /> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>,</mo> <mspace width="1em" /> <mover accent="true"> <mi>z</mi> <mo>˙</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>b</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>c</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where a, b, and c are real arbitrary parameters. Our study uses the averaging theory. This differential system has been studied previously for some authors, because it can exhibit chaotic motion when it has no equilibrium points.

Analytic study of two limit cycles bifurcating from a zero–Hopf equilibrium

期刊论文
发表日期：

2025年3月3日

查看全文

Jaume Llibre,
Jaime R. de Moraes

摘要

In this paper, we provide sufficient conditions for the existence of two limit cycles bifurcating from the unique zero–Hopf equilibrium of the differential system $\begin{aligned} \dot{x}=y+a,\quad \dot{y}=-x+z,\quad \dot{z}=-bx^2+z^2+c, \end{aligned}$ $\begin{matrix} \dot{x} = y + a, \dot{y} = - x + z, \dot{z} = - b x^{2} + z^{2} + c, \end{matrix}$ where a, b, and c are real arbitrary parameters. Our study uses the averaging theory. This differential system has been studied previously for some authors, because it can exhibit chaotic motion when it has no equilibrium points.