Fronts Between Periodic Patterns for Bistable Recursions on Lattices
Bistable space–time discrete systems commonly possess a large variety of stable stationary solutions with periodic profile. In this context, it is natural to ask about the fate of trajectories composed of interfaces between steady configurations with periodic pattern and in particular, to study their propagation as traveling fronts. Here, we investigate such fronts in piecewise affine bistable recursions on the one-dimensional lattice. By introducing a definition inspired by symbolic dynamics, we prove the existence of front solutions and uniqueness of their velocity, upon the existence of their ground patterns. Moreover, the velocity dependence on parameters and the co-existence of several fronts with distinct ground patterns are also described. Finally, robustness of the results to small \(C^1\)-perturbations of the piecewise affine map is argued by mean continuation arguments.
KeywordsPeriodic fronts Bistable recursions Space–time discrete systems Symbolic dynamics.
- 4.Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial Differential Equations and Related Topics, Lecture Notes Mathematics, vol. 446, pp. 5–49. Springer, Berlin (1975)Google Scholar
- 8.Chazottes, J.-R., Fernandez, B. (eds.): Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, vol. 671. Springer, Berlin (2005)Google Scholar
- 11.Collet, P., Eckmann, J.-P.: Instabilities and fronts in extended systems. Princeton University Press, Princeton (1990)Google Scholar
- 14.Coutinho, R., Fernandez, B.: http://www.cpt.univ-mrs.fr/~bastien/Fronts.html. Accessed Jan 2012
- 17.Coutinho, R., Fernandez, B.: Fronts and interfaces in bistable extended mappings. Nonlinearity 11, 1407–1433 (1998)Google Scholar
- 21.Eckmann, J.-P., Proccacia, I.: Onset of defect-mediated turbulence. Phys. Rev. Lett., 66, 891 (1991)Google Scholar
- 22.Elaydi S.: An introduction to difference equations. Springer, New York (1996)Google Scholar
- 25.Jin, Y., Zhao, X.-Q.: Spatial dynamics of a discrete-time population model in a periodic lattice habitat. J. Dyn. Differ. Equ. 21, 501–525 (2009)Google Scholar
- 27.Kocic, V., Ladas, G.: Global behaviour of nonlinear difference equations of higher order with applications. Kluwer, Dordrecht (1993)Google Scholar
- 30.Weinberger, H.F.: Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982)Google Scholar