Dynamics of unidirectionally coupled bistable Hénon maps

We study dynamics of two bistable Hénon maps coupled in a master-slave configuration. In the case of coexistence of two periodic orbits, the slave map evolves into the master map state after transients, which duration determines synchronization time and obeys a −1/2 power law with respect to the coupling strength. This scaling law is almost independent of the map parameter. In the case of coexistence of chaotic and periodic attractors, very complex dynamics is observed, including the emergence of new attractors as the coupling strength is increased. The attractor of the master map always exists in the slave map independently of the coupling strength. For a high coupling strength, complete synchronization can be achieved only for the attractor similar to that of the master map.     

Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps

We study the peculiarities of the solitary state appearance in the ensemble of nonlocally coupled chaotic maps. We show that the nonlocal coupling and features of the partial elements lead to the emergence of multistability in the system. The existence of solitary state is caused by the formation of two attracting sets with different basins of attraction. Their evolution is analyzed depending on the coupling parameters.     

Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps

Area-preserving nontwist maps, i.e., maps that violate the twist condition, arise in the study of degenerate Hamiltonian systems for which the standard version of the Kolmogorov-Arnold-Moser (KAM) theorem fails to apply. These maps have found applications in several areas including plasma physics, fluid mechanics, and condensed matter physics. Previous work has limited attention to maps in 2-dimensional phase space. Going beyond these studies, in this paper, we study nontwist maps with many-degrees-of-freedom. We propose a model in which the different degrees of freedom are coupled through a mean-field that evolves self-consistently. Based on the linear stability of period-one and period-two orbits of the coupled maps, we construct coherent states in which the degrees of freedom are synchronized and the mean-field stays nearly fixed. Nontwist systems exhibit global bifurcations in phase space known as separatrix reconnection. Here, we show that the mean-field coupling leads to dynamic, self-consistent reconnection in which transport across invariant curves can take place in the absence of chaos due to changes in the topology of the separatrices. In the context of self-consistent chaotic transport, we study two novel problems: suppression of diffusion and breakup of the shearless curve. For both problems, we construct a macroscopic effective diffusion model with time-dependent diffusivity. Self-consistent transport near criticality is also studied, and it is shown that the threshold for global transport as function of time is a fat-fractal Cantor-type set.     

Electronically-implemented coupled logistic maps

The logistic map is a paradigmatic dynamical system originally conceived to model thediscrete-time demographic growth of a population, which shockingly, shows that discretechaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we designand characterize a simple, low-cost, easy-to-handle, electronic implementation of thelogistic map. In particular, our implementation allows for straightforwardcircuit-modifications to behave as different one-dimensional discrete-time systems. Also,we design a coupling block in order to address the behavior of two coupled maps, although,our design is unrestricted to the discrete-time system implementation and it can begeneralized to handle coupling between many dynamical systems, as in a complex system. Ourfindings show that the isolated and coupled maps’ behavior has a remarkable agreementbetween the experiments and the simulations, even when fine-tuning the parameters with aresolution of ~10^-3. We support these conclusions by comparing the Lyapunovexponents, periodicity of the orbits, and phase portraits of the numerical andexperimental data for a wide range of coupling strengths and map’s parameters.     

Bifurcations of coupled bistable maps

We show how to analytically determine the existence and stability properties of fixed points of piecewise-linear coupled map lattices, then use this technique to investigate the bifurcations undergone by systems of diffusively-coupled bistable maps. The behaviour of various piecewise-linear and smooth models is compared, and features peculiar to piecewise-linear models are highlighted. Some examples of counter-intuitive behaviour enforced by the bifurcation scenario are given.     

A hierarchy of coupled maps

A large number of logistic maps are coupled together as a mathematical metaphor for complex natural systems with hierarchical organization. The elementary maps are first collected into globally coupled lattices. These lattices are then coupled together in a hierarchical way to form a system with many degrees of freedom. We summarize the behavior of the individual blocks, and then explore the dynamics of the hierarchy. We offer some ideas that guide our understanding of this type of system. (c) 2002 American Institute of Physics.     

On the dynamics of ensemble averages in chaotic maps

The dynamics of mean values in two-dimensional Hénon-like maps and in their strongly dissipative limit is investigated by considering the direct product of n such maps represented in center-of-mass and relative coordinates. The existence of a well-defined mean value for n → ∞ shows up by a nontrivial mechanism on the projection of the center-of-mass variable. Numerical simulations up to n = 128 suggest that the deviation between the actual mean value at large n and the limiting one obeys a gaussian distribution in the chaotic regime.     

Chaotic synchronization of coupled ergodic maps

With few exceptions, studies of chaotic synchronization have focused on dissipative chaos. Though less well known, chaotic systems that lack dissipation may also synchronize. Motivated by an application in communication systems, we couple a family of ergodic maps on the N-torus and study the global stability of the synchronous state. While most trajectories synchronize at some time, there is a measure zero set that never synchronizes. We give explicit examples of these asynchronous orbits in dimensions two and four. On more typical trajectories, the synchronization error reaches arbitrarily small values and, in practice, converges. In dimension two we derive bounds on the average synchronization time for trajectories resulting from randomly chosen initial conditions. Numerical experiments suggest similar bounds exist in higher dimensions as well. Adding noise to the coupling signal destroys the invariance of the synchronous state and causes typical trajectories to desynchronize. We propose a modification of the standard coupling scheme that corrects this problem resulting in robust synchronization in the presence of noise.     

Synchronization learning of coupled chaotic maps

We study the dynamics of an ensemble of globally coupled chaotic logistic maps under the action of a learning algorithm aimed at driving the system from incoherent collective evolution to a state of spontaneous full synchronization. Numerical calculations reveal a sharp transition between regimes of unsuccessful and successful learning as the algorithm stiffness grows. In the regime of successful learning, an optimal value of the stiffness is found for which the learning time is minimal.     

Finite-size-induced transition in ensemble of globally coupled oscillators

The collective behavior of overdamped nonlinear noise-driven oscillators coupled via mean field is investigated numerically. When a coupling constant is increased, a transition in the dynamics of the mean field is observed. This transition scales with the number of oscillators and disappears when this number tends to infinity. Analytical arguments explaining the observed scaling are presented.     

Bursting synchronization in non-locally coupled maps

Neuron activity presents two timescales, a fast one related to action-potential spiking, and a slow timescale in which bursting takes place. Bursting activity in neuron ensembles can be synchronized, meaning the adjustment of the bursting phases due to coupling. We investigated bursting synchronization in a non-locally coupled lattice using a two-dimensional map to describe neuron activity. The coupling involves all sites in a lattice, the corresponding strength decreasing with the lattice distance in a power-law fashion. We observed bursting synchronization for wide intervals of the coupling parameters. We also investigated the bursting synchronization of the ensemble with an external time-periodic signal applied to one or more selected neurons.     

Globally coupled maps with sequential updating

A system of globally coupled logistic maps with sequential updating is analyzed numerically. It is found that deterministic asynchronous updating schemes may have dramatic influences on the dynamical behaviors of globally coupled systems. Transitions from spatio–temporal chaos to spatially organized states are observed as the coupling parameter varies. It is shown that the model system may exhibit a variety of collective properties such as the clustering, traveling wave patterns, and spatial bifurcation cascades.     

Analysing local observations of weakly coupled maps

We analyse dimension and entropy estimates from local measurements in a weakly coupled extensively chaotic system. We observe a staircase-like signature related to the coupling strength. This behaviour can be explained by studying the transformation between the spatial states and the states reconstructed by delay embedding, using the concept of the continuous entropy.     

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

Controlling synchronization in an ensemble of globally coupled oscillators

We propose a technique to control coherent collective oscillations in ensembles of globally coupled units (self-sustained oscillators or maps). We demonstrate numerically and theoretically that a time delayed feedback in the mean field can, depending on the parameters, enhance or suppress the self-synchronization in the population. We discuss possible applications of the technique.     

Synchronized family dynamics in globally coupled maps

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many stable solutions, except for sufficiently large coupling strengths, where the lattice elements always synchronize, and for small map parameter, where only simple fixed points are found. For smaller varepsilon and larger a, there is a portion of the parameter plane in which chaotic, asynchronous lattices are found. Over much of the parameter plane, lattices converge to states in which the maps are partitioned into a number of synchronized families. The dynamics and stability of two-family states (solutions partitioned into two families) are explored in detail. (c) 1999 American Institute of Physics.     

Dynamics of simple one-dimensional maps under perturbation

It is known that the one-dimensional discrete maps having single-humped nonlinear functions with the same order of maximum belong to a single class that shows the universal behaviour of a cascade of period-doubling bifurcations from stability to chaos with the change of parameters. This paper concerns studies of the dynamics exhibited by some of these simple one-dimensional maps under constant perturbations. We show that the “universality” in their dynamics breaks down under constant perturbations with the logistic map showing different dynamics compared to the other maps. Thus these maps can be classified into two types with respect to their response to constant perturbations. Unidimensional discrete maps are interchangeably used as models for specific processes in many disciplines due to the similarity in their dynamics. These results prove that the differences in their behaviour under perturbations need to be taken into consideration before using them for modelling any real process.     

Image encryption with chaotically coupled chaotic maps

We present a novel secure cryptosystem for direct encryption of color images, based on chaotically coupled chaotic maps. The proposed cipher provides good confusion and diffusion properties that ensures extremely high security because of the chaotic mixing of pixels’ colors. Information is mixed and distributed over a complete image using a complex strategy that makes known plaintext attack unfeasible. The encryption algorithm guarantees the three main goals of cryptography: strong cryptographic security, short encryption/decryption time, and robustness against noise and other external disturbances. Due to the high speed, the proposed cryptosystem is suitable for application in real-time communication systems.