A pair of centro-symmetric heteroclinic orbits coined
Haijun Wang, Jun Pan, Guiyao Ke, Feiyu Hu
Abstract
Abstract Although the axis-symmetric heteroclinic orbits of Lorenz-like systems have been intensively studied in the past decades, scholars seem to pay scant attention to the centro-symmetric ones. To achieve this target, the present paper introduces a new subquadratic centro-symmetric three-dimensional Lorenz-like system: $\dot{x}=a(y - x)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> , $\dot{y}=cx - \sqrt[3]{x^{2}}z$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mroot> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mn>3</mml:mn> </mml:mroot> <mml:mi>z</mml:mi> </mml:math> , $\dot{z}= -bz + \sqrt[3]{x^{2}}y$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi>b</mml:mi> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mroot> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mn>3</mml:mn> </mml:mroot> <mml:mi>y</mml:mi> </mml:math> , and proves the existence of a pair of centro-symmetric to $E_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and $E_{\pm}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mo>±</mml:mo> </mml:msub> </mml:math> combining the definitions of α -limit and ω -limit set, Lyapunov functions. The effectiveness and correctness of the theoretical conclusions are verified via a few numerical examples. Not only does the study provide new ideas for finding heteroclinic orbits, but also it poses an interesting question that the existence of heteroclinic orbits may depend on the degrees of the considered models.