We study the behaviour of solutions to nonlinear autonomous functional differential equations of mixed type in the neighbourhood of an equilibrium. We show that all solutions that remain sufficiently close to an equilibrium can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. In addition, we provide a Hopf bifurcation theorem for such equations. We illustrate the application range of our results by discussing an economic life-cycle model that gives rise to functional differential equations of mixed type.
Similar content being viewed by others
References
Abell K.A., Elmer C.E., Humphries A.R., Vleck E.S.V. (2005). Computation of mixed type functional differential boundary value problems. SIAM J. Appl. Dyn. Sys. 4, 755–781
Bart H., Gohberg I., Kaashoek M.A. (1986). Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators. J. Funct. Anal. 68, 1–42
Bates P.W., Chen X., Chmaj A. (2003). Traveling waves of bistable dynamics on a lattice. SIAM J. Math. Anal. 35, 520–546
Bates P.W., Chmaj A. (1999). A discrete convolution model for phase transitions. Arch. Rational Mech. Anal. 150, 281–305
Bell J. (1981). Some threshold results for models of myelinated nerves. Math. Biosci. 54, 181–190
Benhabib J., Nishimura K. (1979). The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth. J. Econ. Theory 21, 421–444
Benzoni-Gavage S. (1998). Semi-discrete shock profiles for hyperbolic systems of conservation laws. Physica D 115, 109–124
Benzoni-Gavage S., Huot P. (2002). Existence of semi-discrete shocks. Discrete Contin. Dyn. Syst. 8, 163–190
Benzoni-Gavage S., Huot P., Rousset F. (2003). Nonlinear stability of semidiscrete shock waves. SIAM J. Math. Anal. 35, 639–707
Cahn J.W. (1960). Theory of crystal growth and interface motion in crystalline materials. Acta Met. 8, 554–562
Cahn J.W., Mallet-Paret J., Van Vleck E.S. (1999). Traveling wave solutions for systems of ODE’s on a two-dimensional spatial lattice. SIAM J. Appl. Math 59, 455–493
Chi H., Bell J., Hassard B. (1986). Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory. J. Math. Biol. 24, 583–601
Chow S.N., Mallet-Paret J., Shen W. (1998). Traveling waves in lattice dynamical systems. J. Diff. Eq. 149, 248–291
Chua L.O., Roska T. (1993). The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156
Crandall M.C., Rabinowitz P.H. (1978). The Hopf bifurcation theorem in infinite dimensions. Arch. Rat. Mech. Anal. 67, 53–72
d’Albis, H. and Augeraud-Veron, E. (2004). Competitive growth in a life-cycle model: existence and dynamics. Preprint.
Diekmann O., van Gils S.A., Verduyn-Lunel S.M., Walther H.O. (1995). Delay Equations. Springer-Verlag, New York
Elmer C.E., Van Vleck E.S. (2002). A variant of Newton’s method for the computation of traveling waves of bistable differential-difference equations. J. Dyn. Diff. Eq. 14, 493–517
Elmer C.E., Van Vleck E.S. (2005). Dynamics of monotone travelling fronts for discretizations of Nagumo PDEs. Nonlinearity 18, 1605–1628
Erneux T., Nicolis G. (1993). Propagating waves in discrete bistable reaction-diffusion systems. Physica D 67, 237–244
Fife P., McLeod J. (1977). The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Rat. Mech. Anal. 65, 333–361
Frasson M.V.S., Verduyn-Lunel S.M. (2003). Large time behaviour of linear functional differential equations. Integral Eq. Operat. Theory 47, 91–121
Hankerson D., Zinner B. (1993). Wavefronts for a cooperative tridiagonal system of differential equations. J. Dyn. Diff. Eq. 5, 359–373
Härterich J., Sandstede B., Scheel A. (2002). Exponential dichotomies for linear non-autonomous functional differential equations of mixed type. Indiana Univ. Math. J. 51, 1081–1109
Hewitt E., Stromberg K. (1965). Real and Abstract Analysis, Springer-Verlag, Berlin
Hupkes H.J., Verduyn-Lunel S.M. (2005). Analysis of Newton’s method to compute travelling waves in discrete media. J. Dyn. Diff. Eq. 17, 523–572
Iooss G. (2000). Traveling waves in the Fermi–Pasta–Ulam lattice. Nonlinearity 13, 849–866
Iooss G., Kirchgässner K. (2000). Traveling waves in a chain of coupled nonlinear oscillators. Comm. Math. Phys. 211, 439–464
Kaashoek M., Verduyn-Lunel S.M. (1994). An integrability condition on the resolvent for hyperbolicity of the semigroup. J. Diff. Eq. 112, 374–406
Keener J., Sneed J. (1998). Mathematical Physiology, Springer-Verlag, New York
Keener J.P. (1987). Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572
Laplante J.P., Erneux T. (1992). Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem. 96:4931–4934
Mallet-Paret, J. (1996). Spatial patterns, spatial chaos and traveling waves in lattice differential equations. In Stochastic and Spatial Structures of Dynamical Systems, Royal Netherlands Academy of Sciences. Proceedings, Physics Section. Series 1, Vol. 45. Amsterdam, pp. 105–129.
Mallet-Paret J. (1999). The Fredholm alternative for functional differential equations of mixed type. J. Dyn. Diff. Eq. 11, 1–48
Mallet-Paret J. (1999). The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Diff. Eq. 11, 49–128
Mallet-Paret, J. (2001). Crystallographic pinning: direction dependent pinning in lattice differential equations. Preprint.
Mallet-Paret, J. and Verduyn-Lunel, S. M., (to appear) Exponential dichotomies and Wiener–Hopf factorizations for mixed-type functional differential equations. J. Diff. Eq.
Mielke A. (1986). A reduction principle for nonautonomous systems in infinite-dimensional spaces. J. Diff. Eq. 65, 68–88
Mielke A. (1994). Floquet theory for, and bifurcations from spatially periodic patterns. Tatra Mountains Math. Publ. 4, 153–158
Rustichini A. (1989). Hopf bifurcation for functional-differential equations of mixed type. J. Dyn. Diff. Eq. 1, 145–177
Vanderbauwhede A., Iooss G. (1992). Center manifold theory in infinite dimensions. Dyn. Reported: Expositions Dyn. Sys. 1, 125–163
Vanderbauwhede A., van Gils S.A. (1987). Center manifolds and contractions on a scale of banach spaces. J. Funct. Anal. 72, 209–224
Widder D.V. (1946). The Laplace Transform. Princeton Univ. Press, Princeton NJ
Wu J., Zou X. (1997). Asymptotic and periodic boundary value problems of mixed FDE’s and wave solutions of lattice differential equations. J. Diff. Eq. 135, 315–357
Zinner B. (1992). Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Diff. Eq. 96, 1–27
Zinner B., Harris G., Hudson W. (1993). Traveling wavefronts for the discrete Fisher’s equations. J. Diff. Eq. 105, 46–62
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hupkes, H.J., Lunel, S.M.V. Center Manifold Theory for Functional Differential Equations of Mixed Type. J Dyn Diff Equat 19, 497–560 (2007). https://doi.org/10.1007/s10884-006-9055-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-006-9055-9