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Traveling waves in lattice dynamical systems. (English) Zbl 0911.34050
Summary: The authors study the existence and stability of travelling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODEs) and in coupled map lattices (CMLs). Instead of employing the moving coordinate approach as for partial differential equations they construct a local coordinate system around a traveling wave solution to a lattice ODE, analogous to the local coordinate system around a periodic solution to an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution to this equation. The authors prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. It is shown the existence of traveling waves in CMLs which arise as time-discretizations of lattice ODEs. Finally, these results are applied to the discrete Nagumo equation. $©$ 1998 Academic Press.

##### MSC:
 34D30 Structural stability of ODE and analogous concepts 37-99 Dynamic systems and ergodic theory (MSC2000) 34C37 Homoclinic and heteroclinic solutions of ODE 34D10 Stability perturbations of ODE