Front propagation in extended dynamical systems

I am interested on the mechanisms of propagation of coherent fronts in extended dynamical systems. In particular, I focus on the generic (yet usually microscopic and therefore rarely reported) phenomenon of velocity mode-locking of propagating fronts in discrete media. By means of this mode-locking, the velocity of the propagating front gains structural stability (i.e. remains constant) with respect to small perturbations of the system's parameters. In its most acute form, this mode-locking produces a front velocity profile, with respect to the coupling parameters, described by a series of plateaus (or superstable regions) arranged in a staircase with fractal properties (a Devil's staircase).

One-dimensional interfaces [b.2,a.3]
By careful choice of a coupled map lattice, with a bistable piece-wise linear local map, one obtains a one-dimensional front dynamics for certain parameter values. This one-dimensional front corresponds to a single lattice site belonging to the interfacial region between the two stable points. The velocity of the traveling front is then given by the rotation number of a circle map that describes the dynamics of the interfacial site. The mode-locking of the propagating velocity with respect to system parameters follows from the generic mode-locking of the rotation number in nonlinear maps of the circle.

Low-dimensional interfaces [a.4]
When more than one sites, let us say $N$ ($N<\infty$), enter the interfacial region it is still possible to give a finite-dimensional description of the interface. In this case the dynamics is prescribed by a $N$-toral map (generalization of a circle map in $N$ dimensions) whose rotation number describes the propagating front velocity and exhibits mode-locking.

Infinite-dimensional interfaces [a.8]
For the general case of a smooth local map, the front (usually exponentially localized) typically involves an infinite number of interfacial sites. In this case a description using an infinite-dimensional toral map is not convenient. Nonetheless, provided some translational shape invariance of the front, it is still possible to reduce the dynamics of the infinite-dimensional front to a one-dimensional map of the circle. The mode-locking is then attributed to the resonant tongues (Arnold tongues) of this one-dimensional circle map.

Future directions: the continuum limit
The mode-locking described above originates from the discrete spatio-temporal nature. Continuous models do not, in principle, exhibit mode-locking. However, when numerically implementing a continuous spatio-temporal system one has to discretize time and space. This discretization indeed leads to spurious mode-locking of the front velocity for coarse grids. I would like to better understand the dynamical implications of space-time discretizations on continuous models.