Formal Setting for Period Doubling Bifurcation of Limit Cycles
| Authors | |
|---|---|
| Year of publication | 2023 |
| Type | Article in Proceedings |
| Conference | 15th Chaotic Modeling and Simulation International Conference |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.1007/978-3-031-27082-6_27 |
| Doi | http://dx.doi.org/10.1007/978-3-031-27082-6_27 |
| Keywords | Limit cycle; Period doubling; Fredholm operator; Lyapunov-Schmidt reduction; Pitchfork bifurcation |
| Description | A rigorous description of period doubling bifurcation of limit cycles in autonomous systems of first order differential equations based on tools of functional analysis and singularity theory is presented. It is an alternative approach which is independent of the theory of discrete-time dynamical systems, especially Poincaré sections. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. Also, stability of solutions is analysed and it is related to particular derivatives of the operator. Our approach is an adjustment of techniques used by Golubitsky and Schaeffer (Singularities and Groups in Bifurcation Theory: Volume 1. Springer, New York, 1985) in the study of Hopf bifurcation and it can be considered as a theoretical background for calculations presented in Kuznetsov et al. (SIAM J. Numer. Anal. 43:1407–1435, 2006). The normal form of a vector field derived in Iooss (J. Differ. Equ. 76:47–76, 1988) is not needed, since a given differential equation is considered as an algebraic equation. The theory used here concerns Fredholm operators, Lyapunov-Schmidt reduction and recognition problem for pitchfork bifurcation. |
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