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

Abstract

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 boundaryvalue 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.

Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

“…To describe the normal forms of (2) on the critical center manifold W c (Γ) for these codim 2 cases, we parameterize W c (Γ) near Γ by (n c −1) transverse coordinates and τ ∈ [0, kT ] for k ∈ {1, 2}, depending on the bifurcation. The 8 cases with n c ≤ 3 were treated in [17]. Based on [14] we showed in Appendix A in [18] that the restriction of (2) to the corresponding critical center manifold W c (Γ) with n c = 4 or n c = 5 will take one of the following Iooss normal forms.…”

“…and the complex conjugate of the last equation, which merely reflect the definition of u 0 , (33) and (17). By collecting the ξ 2 1 -terms we obtain h 200 as the unique solution of the BVP…”

“…Recently, we have extended this algorithm to codim 2 bifurcations of limit cycles with center manifold dimension at most 3 [17]. Here we consider the three remaining and most difficult cases, LPNS, PDNS, and NSNS, that are characterized by a center manifold of the critical cycle of dimension 4 or 5.…”

“…First, our aim is to derive coefficients of a periodic critical normal form. We present these normal forms in Section 2 using (contrary to [16,17]) the original Iooss [14] representation. Remark that these normal forms are closely related to the normal forms for the Zero-Hopf and Hopf-Hopf bifurcations of equilibria.…”

“…We extensively discuss the LPNS case but omit details in the PDNS and NSNS cases (the complete discussion can be found in [18]). Here we also comment on the implementation which is similar to [17]. Finally in Section 4, we consider several examples that involve tori bifurcations: a laser model, a model from population biology, and one for mechanical vibrations.…”