Spontaneous Emergence of Spatio-Temporal Order in Class 4 Automata

We report surprisingly regular behaviors observed for a class 4 cellular automaton, the totalistic rule 20: starting from disordered initial configurations the automaton produces patterns which are periodic not only in time but also in space. This is the first evidence that different types of spatio-temporal order can emerge under specific conditions out of disorder in the same discrete rule based algorithm.Comment: 5 pages, 6 color figures, Proceedings Medyfinol 2004, Physica A in prin

Naimark-Sacker Bifurcations in Linearly Coupled Quadratic Maps

We report exact analytical expressions locating the $0\to1$, $1\to2$ and $2\to4$ bifurcation curves for a prototypical system of two linearly coupled quadratic maps. Of interest is the precise location of the parameter sets where Naimark-Sacker bifurcations occur, starting from a non-diagonal period-2 orbit. This result is the key to understand the onset of synchronization in networks of quadratic maps.Comment: 6 pages, 3 figures (1 in color), submitted to Physica

Method for extracting arbitrarily large orbital equations of the Pincherle map

We report an algorithm to extract equations of motion for orbits of arbitrarily high periods generated by iteration of the Pincherle map, the operational kernel used in the so-called chaotic computers. The performance of the algorithm is illustrated explicitly by extracting expeditiously, among others, an orbit buried inside a polynomial cluster of equations with degree exceeding one billion, out of reach by ordinary brute-force factorization. Large polynomial clusters are responsible for the organization of the phase-space and knowledge of this organization requires decomposing such clusters. Keywords: Algebraic dynamics, Preperiodic points, Orbital decomposition, Chaotic computer

Distribution of chaos and periodic spikes in a three-cell population model of cancer

We study complex oscillations generated by the de Pillis-Radunskaya model of cancer growth, a model including interactions between tumor cells, healthy cells, and activated immune system cells. We report a wide-ranging systematic numerical classification of the oscillatory states and of their relative abundance. The dynamical states of the cell populations are characterized here by two independent and complementary types of stability diagrams: Lyapunov and isospike diagrams. The model is found to display stability phases organized regularly in old and new ways: Apart from the familiar spirals of stability, it displays exceptionally long zig-zag networks and intermixed cascades of two- and three-doubling flanked stability islands previously detected only in feedback systems with delay. In addition, we also characterize the interplay between continuous spike-adding and spike-doubling mechanisms responsible for the unbounded complexification of periodic wave patterns. This article is dedicated to Prof. Hans Jürgen Herrmann on the occasion of his 60th birthday

Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang-Kobayashi model

We report a numerical characterization of the stability of semiconductor lasers with delayed feedback under the simultaneous variation of the delay time τ and the pump current P. Changes in the number of External Cavity Modes are studied as a function of the delay time while the Regular Pulse Package regime is characterized as a function of the pump current. In addition, we describe some remarkable structures observed in the τ × P control plane, delimiting where these and other complex regimes of laser operation exist

Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows

Infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow. We show that spirals associated with periodicity hubs emerge/accumulate at the folding of certain fractal-like sheaves of Shilnikov homoclinic bifurcations of a common saddle-focus equilibrium. The specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor