Statistical cycling in coupled map lattices

In memory of Viktor Iosifovich Belinicher

“…Typically, synchronized chaotic behaviour results from coupling chaotic and regular systems, if the chaotic contribution is strong enough. When chaotic systems are coupled, however, synchronized chaotic behaviour as well as macroscopically synchronized regular behaviour may be the result (Bunimovich and Sinai 1988, Losson and Mackey 1994, Bunimovich 1995. This simplified analysis suggests that macroscopic chaos and synchronous behaviour have their origin in, and can be explained by, the corresponding mesoscopic properties.…”

“…Relying only on very general mechanisms of coupling, we will argue that, as a consequence of the second property, (a) there are good reasons to expect that the observed mesoscopic chaos translates into macroscopic chaos and that (b) there is a non-zero probability for strong phase-locked synchronization among inhibitory connections. In the latter case, formerly independently chaotically spiking neurons are brought into stable spiking patterns only by the application of a small coupling (Losson and Mackey 1994).…”

Effect of switching links in networks of piecewise linear maps

“…At the outset, note that, while coupled smooth dynamical systems are widespread in the literature, very few results are available for coupled non-smooth systems, with the research in this direction being focused mostly on coupled piecewise linear onedimensional maps [21,22,32], such as tent maps [24,25] or generalizations of piecewise linear Markov maps [23]. So, although non-smooth systems are widespread in engineering [33,34], economics [35], ecology [36,37] and biology [38,39], the exploration of such systems under coupling in wide ranging topologies has been limited.…”

Information capacity and pattern formation in a tent map network featuring statistical periodicity

“…To properly understand the significance of the two quantities that we intend to calculate (information capacity and locking time), and understand how these two quantities can compete with each other, it is instructive to review the properties of the isolated tent map S(x), and, by simple a generalization, of the CML with = 0 (more information can be found in [5][6][7]). More specifically, we have that the dynamics of the tent map alone are characterized by a Lyapunov exponent equal to ln a.…”

“…Thus, for 0 < a < 1, all orbits of the map converge to the unique fixed point x * = 0 independent of the value of the initial conditions, while for a = 1, all initial points of the map are fixed points. When 1 < a ≤ 2, the dynamics of the tent map switches abruptly to a chaotic regime (see Behind the chaotic behaviour of the temporal trajectory, the underlying regularity of the dynamics apparent in the banded structure of the bifurcation diagram is the signature of a property called statistical periodicity which can be observed in the density distribution of an ensemble of tent maps [2,6,7,9]. Specifically, for 2 1/2 1/(n+1) < a ≤ 2 1/2 1/n , the densities of the tent map have periodicity with period T = (n + 1) for n = 1, 2, · · · .…”