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Article http://dx.doi.org/10.26855/jamc.2026.06.007

Two-dimensional Unstable Manifold of Jerk System

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Suqi Ma^*, Huailei Wang, Jian Xu

Department of Mathematics, China Agricultural University, Beijing 100083, China.

*Corresponding author:Suqi Ma

Published: June 30,2026

Abstract

The two-dimensional unstable manifold of the equilibrium of a jerk system is investigated. Simultaneously, the focus on dynamical analysis and bifurcation is explored. alike periodicity doubly phenomena and the observed heteroclinic loop/orbit arising in the system. The period-doubling bifurcation of the periodic solutions is ubiquitous, which aids in understanding the periodic solution transition phenomena. The manifold surface is shaped by solving the BVPs problem, and a quick algorithm by taking a large arclength in the calculation of the frontier manifold further to set up the shaped surface is adopted. The unstable manifold of the jerk system is two-dimensional and drawn by solving the BVPs problem. As the free parameter varies, the unstable manifolds of the equilibrium solution are simulated and shown as the neighborhood regime of the related stable limit cycle, respectively. With the manifold computation method, the unstable manifold is drawn undertaken system has heteroclinc orbit or heteroclinic loop. The manifold surface actually manifests the boundary of the heteroclinic orbit. The unstable manifold of one equilibrium solution really attaches to the unstable manifold of another equilibrium and is immersed. The Mobius belt is viewed as the neighborhood regime of the stable limit cycle of double periodicity, and the Mobius manifold is computed as the unstable manifold of the related saddle cycle.

Keywords

Manifold of the equilibrium solution; manifold of the limit cycle; Mobius belt

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How to cite this paper

Two-dimensional Unstable Manifold of Jerk System

How to cite this paper: Suqi Ma, Huailei Wang, Jian Xu. (2026) Two-dimensional Unstable Manifold of Jerk System. Journal of Applied Mathematics and Computation, 10(2), 118-128.

DOI: http://dx.doi.org/10.26855/jamc.2026.06.007