Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

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.

[1]  Gábor Stépán,et al.  Criticality of Hopf bifurcation in state-dependent delay model of turning processes , 2008 .

[2]  Gábor Stépán,et al.  Continuation of Bifurcations in Periodic Delay-Differential Equations Using Characteristic Matrices , 2006, SIAM J. Sci. Comput..

[3]  Qingwen Hu,et al.  Global Hopf bifurcation for differential equations with state-dependent delay , 2010 .

[4]  J. Baumgarte.  Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[5]  Tibor Krisztin,et al.  A local unstable manifold for differential equations with state-dependent delay , 2003 .

[6]  Radakovič.  The theory of approximation , 1932 .

[7]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[8]  H. Walther,et al.  Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays , 2004 .

[9]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[10]  Jianhong Wu,et al.  S1-degree and global Hopf bifurcation theory of functional differential equations , 1992 .

[11]  Hans-Otto Walther,et al.  Stable periodic motion of a system with state dependent delay , 2002, Differential and Integral Equations.

[12]  O. Arino,et al.  Existence of Periodic Solutions for Delay Differential Equations with State Dependent Delay , 1998 .

[13]  Markus Eichmann,et al.  A local Hopf Bifurcation Theorem for difierential equations with state - dependent delays , 2006 .

[14]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[15]  Uniqueness of the zero solution for delay differential equations with state dependence , 1970 .

[16]  Ferenc Hartung,et al.  Chapter 5 Functional Differential Equations with State-Dependent Delays: Theory and Applications , 2006 .

[17]  A. R. Humphries,et al.  Dynamics of a delay differential equation with multiple state-dependent delays , 2012 .

[18]  John Mallet-Paret,et al.  Periodic solutions for functional differential equations with multiple state-depend time lags , 1994 .

[19]  Shangjiang Guo,et al.  Equivariant Hopf bifurcation for functional differential equations of mixed type , 2011, Appl. Math. Lett..

[20]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[21]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[22]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[23]  Jan Sieber,et al.  Characteristic Matrices for Linear Periodic Delay Differential Equations , 2011, SIAM J. Appl. Dyn. Syst..

[24]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[25]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[26]  Dirk Roose,et al.  Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL , 2002, TOMS.

[27]  Jianhong Wu,et al.  Theory of Degrees with Applications to Bifurcations and Differential Equations , 1997 .