Periodic Center Manifolds and Normal Forms for DDEs in the Light of Suns and Stars

Abstract

Bifurcation theory has been very successful in the study of qualitative changes in nonlinear dynamical systems. An important tool of this theory is the existence of a center manifold near nonhyperbolic equilibria and limit cycles or homoclinic orbits. The existence has already been proven for many kinds of different systems, but not fully for limit cycles in delay differential equations (DDEs). In this paper, we prove the existence of a smooth finite-dimensional periodic center manifold near a nonhyperbolic cycle in DDEs and the existence of a special coordinate system on this manifold. This allows us to describe the local dynamics on the center manifold in terms of the standard normal forms. These results are based on the rigorous functional analytic perturbation framework for dual semigroups, also called sun-star calculus.

Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

“…where τ and ξ are coordinates on the center manifold with ξ being transverse to the orbit of u0 , Lentjes et al [12,Preprint]. On the other hand, normal form coefficients derived in this work correspond to Taylor coefficients of the reduced mapping φ.…”

“…The existence of a smooth periodic center manifold in a neighbourhood of a limit cycle together with a suitable coordinate system is a base of the investigation of period-doubling bifurcation in Iooss [8], Kuznetsov et al [11] and Lentjes et al [12,Preprint]. We have not needed such information, since the main idea of the presented theory lies in considering a given differential equation as an algebraic equation.…”

“…Since Im L ⊆ [v 1 , v 2 ] ⊥ , the last summands are equal to zero in both cases. The value W x (0, λ 0 , 0) can be deduced from the defining equations ( 8) or (12):…”

“…Their procedure depend on the normal form of a vector field in a neighbourhood of a limit cycle deduced in Iooss [8]. Recently, the existence of a smooth periodic center mani-fold in a neighbourhood of a non-hyperbolic periodic orbit in retarded functional differential equations (RFDE) has been proved, Lentjes et al [12,Preprint]. Moreover, the existence of a suitable coordinate system on the manifold enables calculation of normal form coefficients of codimension one bifurcations of limit cycles in RFDE.…”

“…The computation of normal form coefficients of period-doubling bifurcation in this text can be considered as a demonstration of the robustness of these tools and a prospect for the successful application of this method in other and more general contexts. Although Lentjes et al [12,Preprint], announced the computation of these coefficients and Szalai and Stépán [16], derived them for the special case where delay and period are equal, we provide explicit formulae in full generality.…”