Abstract
In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays, transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an extension of the Hopf bifurcation theorem by Eichmann (2006), along with an alternative proof.
Highlights
If a dynamical system is described by a differential equation where the derivative at the current time may depend on states in the past one speaks of delay differential or, more generally, functional differential equations (FDEs)
A reasonably general formulation of an autonomous dynamical system of this type looks like this: x(t) = f where τ > 0 is an upper bound for the delay
A central notion in the construction of the equivalent algebraic system for periodic orbits of FDEs are periodic boundary-value problems (BVPs) for FDEs on the interval [−π, π] with periodic boundary conditions
Summary
If a dynamical system is described by a differential equation where the derivative at the current time may depend on states in the past one speaks of delay differential or, more generally, functional differential equations (FDEs). Periodic orbits of (1) can be found as solutions of periodic BVPs. If one wants to make the equivalence result useful in practical applications, one has to find a regularity (smoothness) condition on the right-hand side f that includes the class of state-dependent delay equations reviewed in [12], while still ensuring that it is possible to prove the existence of an equivalent algebraic system. The equivalence theorem reduces statements about existence and smooth dependence of periodic orbits of FDEs to root-finding problems of smooth algebraic equations. Theorem to an application of the Algebraic Branching Lemma [1] This provides a complete proof for the Hopf Bifurcation Theorem for FDEs with state-dependent delays, including the regularity of the emerging periodic orbits. In numerical methods one typically has to increase the dimension of the algebraic system in order to get more and more accurate approximations of the true solution whereas the dimension of the algebraic system constructed in Section 2 is finite
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