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Computation of the normal form as well as the unfolding of the vector field with zero-zero-Hopf bifurcation at the origin

Author

Listed:
  • Xue, Miao
  • Gou, Junting
  • Xia, Yibo
  • Bi, Qinsheng

Abstract

The normal forms of the vector fields with local bifurcations at the equilibrium points can be employed to describe the topological structure in the neighborhood of the critical points. Furthermore, the relationship between the coefficients of the normal form and the original system is very important to understand the behaviors of the practical dynamics. Though many results related to low co-dimensional local bifurcations are presented, the normal forms as well as the computation with high co-dimensional local bifurcations still remain an open problem to be investigated. The main purpose of this paper is to derive the normal form of a vector field with codimension-3 zero-zero-Hopf bifurcation at the origin and develop an uniform program to compute the coefficients of the normal form from a general system. By employing the central manifold theory and the normal form theory, all the expressions of the coefficients of the nonlinear transformations and the normal form up to any desired order related to the local bifurcation are presented, which can be computed via a software program based on the symbolic language Maple, attached in the appendix. Perturbation of the vector field at the bifurcation point can also be derived accordingly, which can be used to explore the topological property of the bifurcation.

Suggested Citation

  • Xue, Miao & Gou, Junting & Xia, Yibo & Bi, Qinsheng, 2021. "Computation of the normal form as well as the unfolding of the vector field with zero-zero-Hopf bifurcation at the origin," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 377-397.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:377-397
    DOI: 10.1016/j.matcom.2021.05.032
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    References listed on IDEAS

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    1. Qin, Bo-Wei & Chung, Kwok-Wai & Algaba, Antonio & Rodríguez-Luis, Alejandro J., 2020. "Analytical approximation of cuspidal loops using a nonlinear time transformation method," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    2. Algaba, A. & Fuentes, N. & Gamero, E. & García, C., 2020. "Orbital normal forms for a class of three-dimensional systems with an application to Hopf-zero bifurcation analysis of Fitzhugh–Nagumo system," Applied Mathematics and Computation, Elsevier, vol. 369(C).
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    Cited by:

    1. Weipeng Lyu & Shaolong Li & Zhenyang Chen & Qinsheng Bi, 2023. "Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation," Mathematics, MDPI, vol. 11(11), pages 1-20, May.

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