Inducing coherence in networks of bistable maps by varying the interaction range

Non-trivial generation and transmission of information in electronically designed logistic-map networks

In this work, we carry out a critical analysis of the information generated and transmitted in an electronic implementation of diffusively coupled logistic maps. Our implementation allows one to change the coupling configuration (i.e., the network) and fine-tune the coupling strength and map parameters, but has minimal electronic noise and parameter heterogeneity, which generates collective behaviors that differ from numerical simulations. In particular, we focus on analyzing two dynamical regimes and their dependence on the coupling configuration: one where there is a maximum of information generated and transmitted-corresponding to synchronization of chaotic orbits-and another where information is generated but (practically) not transmitted-corresponding to spatiotemporal chaos. We use Shannon entropy to quantify information generation and mutual information to quantify information transmission. To characterize the two dynamical regimes, we introduce a conditional joint entropy that uses both quantities (entropy and mutual information) and analyze its values for 60 different coupling configurations involving 6 and 12 coupled maps. We find that 90% of the configurations exhibit chaotic synchronization and 92% spatiotemporal chaos, which emerges preceding the chaotic synchronous regime that requires strong coupling strengths. Our results also highlight the coupling configurations that maximize the conditional joint entropy in these regimes without requiring a densely coupled system, which has practical implications (since introducing couplings between units can be costly). Overall, our work contributes to understand the relevance that the network structure has on the generation and transmission of information in complex systems.

From synchronous to one-time delayed dynamics in coupled maps

We study the completely synchronized states (CSSs) of a system of coupled logistic maps as a function of three parameters: interaction strength (ɛ), range of the interaction (α), that can vary from first neighbors to global coupling, and a parameter (β) that allows one to scan continuously from nondelayed to one-time delayed dynamics. In the α-ɛ plane we identify periodic orbits, limit cycles, and chaotic trajectories, and describe how these structures change with delay. These features can be explained by studying the bifurcation diagrams of a two-dimensional nondelayed map. This allows us to understand the effects of one-time delays on CSSs, e.g., regularization of chaotic orbits and synchronization of short-range coupled maps, observed when the dynamics is moderately delayed. Finally, we substitute the logistic map with cubic and logarithmic maps, in order to test the robustness of our findings.

Complete synchronization equivalence in asynchronous and delayed coupled maps

Coupled map lattices are paradigmatic models of many collective phenomena. However, quite different patterns can emerge depending on the updating scheme. While in early versions, maps were updated synchronously, there has been in recent years a concern to consider more realistic updating schemes where elements do not change all at once. Asynchronous updating schemes and the inclusion of time delays are seen as more realistic than the traditional parallel dynamics, and, in diverse works, either one or the other has been implemented separately. But are they actually distinct cases? For coupled map lattices with adjustable range of interactions, we prove, using both numerical and analytical tools, that an adequate delayed dynamics leads to the same completely synchronized states as an asynchronous update, providing a unified framework to identify the stability conditions for complete synchronization.

Phase ordering in coupled noisy bistable systems on scale-free networks

We study a system consisting of diffusively coupled noisy bistable elements on a scale-free random network. This system exhibits an order-disorder phase transition as the noise intensity is varied. The phase ordering process takes place consecutively and in order of the degrees, reflecting strong degree heterogeneity of the scale-free network. A nonlinear Fokker-Planck equation describing the network dynamics is derived under mean-field approximation of the network, and is used to explain the phase ordering dynamics of the system.

Scaling behavior of nonhyperbolic coupled map lattices

Coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of Nth order which exhibit strongest possible chaotic behavior for small coupling constants a. We prove that the expectations of arbitrary observables scale with sqrt of a in the low-coupling limit, contrasting the hyperbolic case which is known to scale with a . Moreover we prove that there are log-periodic oscillations of period ln N2 modulating the sqrt of a dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant one-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.

Dynamical advantages of scale-free networks

A dynamical analysis of common network topologies is given and it is reported that a scale-free structure has two vital and distinctive features. First, complex but nevertheless reproducible states exist and, second, single-site induced state switching reminiscent of gene-expression control exists also. This indicates that scale-free networks have key dynamical advantages over other network topologies that could have contributed to their evolutionary success and thus may provide another reason for their prevalence in nature.

Some aspects of the synchronization in coupled maps

We numerically study the synchronization behavior of a coupled map lattice consisting of a chain of chaotic logistic maps exhibiting power law interactions. We report two main results. First, we find a practical lower bound in the lattice size in order that this system could be considered in the thermodynamic limit in numerical simulations. Second, we observe the existence of a strong correlation between the Lyapunov dimension and the averaged synchronization time.

Basin size evolution between dissipative and conservative limits

Recent methods for stabilizing systems like, e.g., loss-modulated CO2 lasers, involve inducing controlled monostability via slow parameter modulations. However, such stabilization methods presuppose detailed knowledge of the structure and size of basins of attraction. In this Brief Report, we numerically investigate basin size evolution when parameters are varied between dissipative and conservative limits. Basin volumes shrink fast as the conservative limit is approached, being well approximated by Gaussian profiles, independently of the period. Basin shrinkage and vanishing is due to the absence of bounded motions in the Hamiltonian limit. In addition, we find basin volume to remain essentially constant along a peculiar parameter path along which it is possible to recover the dissipation rate solely from metric properties of self-similar structures in phase-space.

Coherence in scale-free networks of chaotic maps

We study fully synchronized states in scale-free networks of chaotic logistic maps as a function of both dynamical and topological parameters. Three different network topologies are considered: (i) a random scale-free topology, (ii) a deterministic pseudofractal scale-free network, and (iii) an Apollonian network. For the random scale-free topology we find a coupling strength threshold beyond which full synchronization is attained. This threshold scales as k(-mu) , where k is the outgoing connectivity and mu depends on the local nonlinearity. For deterministic scale-free networks coherence is observed only when the coupling strength is proportional to the neighbor connectivity. We show that the transition to coherence is of first order and study the role of the most connected nodes in the collective dynamics of oscillators in scale-free networks.

Pattern formation in diffusive-advective coupled map lattices

We investigate pattern formation and evolution in coupled map lattices when advection is incorporated, in addition to the usual diffusive term. All patterns may be suitably grouped into five classes: three periodic, supporting static patterns and traveling waves, and two nonperiodic. Relative frequencies are determined as a function of all model parameters: diffusion, advection, local nonlinearity, and lattice size. Advection plays an important role in coupled map lattices, being capable of considerably altering pattern evolution. For instance, advection may induce synchronization, making chaotic patterns evolve periodically. As a byproduct we describe a practical algorithm for classifying generic pattern evolutions and for measuring velocities of traveling waves.