2011/11/18 by Fabio Della Rossa, Della Rossa, Fabio, Virginie De Witte +6 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #34C20 #37G15 #37M20 #65L07 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1111.4445
openalex publication_date 2011/11/18 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Periodic normal forms for the codim 2 bifurcations of limit cycles up to a\n3-dimensional center manifold in generic autonomous ODEs and computational\nformulas for their coefficients are derived. The formulas are independent of\nthe dimension of the phase space and involve solutions of certain\nboundary-value problems on the interval [0, T ], where T is the period of the\ncritical cycle, as well as multilinear functions from the Taylor expansion of\nthe right-hand sides near the cycle. The formulas allow us to distinguish\nbetween various bifurcation scenarios near codim 2 bifurcations. Our\nformulation makes it possible to use robust numerical boundary-value algorithms\nbased on orthogonal collocation, rather than shooting techniques, which greatly\nexpands its applicability. The actual implementation is described in detail\nwith numerical examples.\n