NDLI: Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kuznetsov, Yu. A. Govaerts, W. Witte, V. De Rossa, F. Della
Copyright Year	2013
Abstract	Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval $[0,T]$, where $T$ is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time.
Starting Page	722
Ending Page	788
Page Count	67
File Format	PDF
DOI	10.1137/120874904
e-ISSN	15360040
Issue Number	2
Volume Number	12
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-05-14
Access Restriction	Subscribed
Subject Keyword	normal forms Computational methods for bifurcation problems bifurcations Transformation and reduction of equations and systems, normal forms Numerical investigation of stability of solutions Bifurcations of limit cycles and periodic orbits codimension 2 limit cycles
Content Type	Text
Resource Type	Article

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