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.     

Nonequilibrium critical behavior in unidirectionally coupled stochastic processes

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.     

Exponential transient rotating waves in a bistable ring of unidirectionally coupled maps

Properties of transient rotating waves in a bistable ring of unidirectionally coupled antisymmetric cubic maps are studied. A kinematical model shows that the duration of rotating waves increases exponentially with the number of elements. The probability density function of the duration of rotating waves generated under random initial conditions has a power law form up to a cut-off. In addition, spatiotemporal noise of intermediate intensity makes the duration of rotating waves increase. Further, rotating waves are stabilized through bifurcations of steady states as coupling strength increases.     

Phase synchronization in unidirectionally coupled chaotic ratchets

We study chaotic phase synchronization of unidirectionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states and perfect phase locking was observed as the coupling was gradually increased. We identified the region of phase synchronization for the ratchets and show that the transition to chaotic phase synchronization is via an interior crisis transition to strange attractor in the phase space.     

Synchronization of spectral components in unidirectionally coupled systems

The object of investigation is synchronization of spectral components in unidirectionally coupled chaotic systems. It is shown that the behavior of the spectral components in this case is governed by the amount of detuning between interacting systems. Universal laws of spectral component synchronization are discovered.     

Phase synchronization in unidirectionally coupled Ikeda time-delay systems

Phase synchronization in unidirectionally coupled Ikeda time-delay systems exhibiting non-phase-coherent hyperchaotic attractors of complex topology with highly interwoven trajectories is studied. It is shown that in this set of coupled systems phase synchronization (PS)does exist in a range of the coupling strength which is preceded by a transition regime (approximate PS)and a nonsynchronous regime. However, exact generalized synchronization does not seem to occur in the coupled Ikeda systems (for the range of parameters we have studied)even for large coupling strength, in contrast to our earlier studies in coupled piecewise-linear and Mackey-Glass systems [27,28]. The above transitions are characterized in terms of recurrence based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence (CPR), joint probability of recurrence (JPR)and similarity of probability of recurrence (SPR). The existence of phase synchronization is also further confirmed by typical transitions in the Lyapunov exponents of the coupled Ikeda time-delay systems and also using the concept of localized sets.     

Homoclinic connections and period doublings of a ship advancing in quartering waves

The large-amplitude motions of a ship running in waves are analyzed with a mathematical model reduced to a system of eight coupled nonlinear ordinary differential equations. Bifurcation analysis in relation to the surf-riding condition, with control parameter the angle of the rudder, shows the existence of a region of oscillatory behavior, containing also a chaotic domain. This region ends with a homoclinic connection and a dangerous jump toward the overtaking-wave mode, which can incur ship capsize. Addition of linear control removes the chaotic domain while giving rise to new regions of oscillation. (c) 1996 American Institute of Physics.     

Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers

Synchronization of chaos is achieved experimentally in unidirectionally coupled external-cavity vertical-cavity surface-emitting semiconductor lasers operating in an open-loop regime. Synchronization is observed when the polarization of the transmitter is perpendicular to the polarization (x polarization) of the free-running receiver. The ratio of transmitter output to y-polarized receiver output power shows normal (positive-slope) synchronization. However, inverse (negative-slope) synchronization is found to arise between the transmitter output and the x-polarized receiver output power.     

Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers

Synchronization dynamics of mutually coupled chaotic semiconductor lasers are investigated experimentally and compared to identical synchronization of unidirectionally coupled lasers. Mutual coupling shows high quality synchronization in a broad range of self-feedback and coupling strengths. It is found to be tolerant to significant parameter mismatch which for unidirectional coupling would result in loss of synchronization. The advantages of mutual coupling are emphasized in light of its potential use in chaos communications.     

Characteristics of generation in a chain of unidirectionally coupled low-voltage vircators

The output power of generation in a chain of unidirectionally coupled low-voltage vircators upon variation in controlling parameters of a system is studied. It is shown that at a certain set of controlling parameters, the output power of the generator chain grows sharply. Physical processes occurring in the system are analyzed by plotting space-time diagrams of the motion of charged particles of the electron flow with a virtual cathode. The obtained results are in good agreement with the data from a previous experimental study on signal amplification in a low-voltage vircator.     

Duration of transient oscillations in ring networks of unidirectionally coupled neurons

Properties of the duration of long lasting transient oscillations in ring networks of unidirectionally coupled sigmoidal neurons are derived with a kinematical model of traveling waves in the network. The duration of the transient oscillations occurring from random initial conditions increases exponentially as the number of neurons. The distribution of the duration is approximated by a power-law function when the number of neurons is large. Further, transient oscillations which oscillate about one thousand cycles before ceasing are observed in a network of forty neurons in circuit experiments though the duration decreases owing to random biases.     

Oscillations, period doublings, and chaos in CO oxidation and catalytic mufflers

Early experimental observations of chaotic behavior arising via the period-doubling route for the CO catalytic oxidation both on Pt(110) and Ptgamma-Al(2)O(3) porous catalyst were reported more than 15 years ago. Recently, a detailed kinetic reaction scheme including over 20 reaction steps was proposed for the catalytic CO oxidation, NO(x) reduction, and hydrocarbon oxidation taking place in a three-way catalyst (TWC) converter, the most common reactor for detoxification of automobile exhaust gases. This reactor is typically operated with periodic variation of inlet oxygen concentration. For an unforced lumped model, we report results of the stoichiometric network analysis of a CO reaction subnetwork determining feedback loops, which cause the oscillations within certain regions of parameters in bifurcation diagrams constructed by numerical continuation techniques. For a forced system, numerical simulations of the CO oxidation reveal the existence of a period-doubling route to chaos. The dependence of the rotation number on the amplitude and period of forcing shows a typical bifurcation structure of Arnold tongues ordered according to Farey sequences, and positive Lyapunov exponents for sufficiently large forcing amplitudes indicate the presence of chaotic dynamics. Multiple periodic and aperiodic time courses of outlet concentrations were also found in simulations using the lumped model with the full TWC kinetics. Numerical solutions of the distributed model in two geometric coordinates with the CO oxidation subnetwork consisting of several tens of nonlinear partial differential equations show oscillations of the outlet reactor concentrations and, in the presence of forcing, multiple periodic and aperiodic oscillations. Spatiotemporal concentration patterns illustrate the complexity of processes within the reactor.     

Phase diagrams in unidirectionally coupled map lattice for open traffic flow

The jamming transition from the free traffic to the oscillatory traffic is investigated with the unidirectionally coupled map lattice model which has the hyperbolic tangent local map. Spatio-temporal structures in the jamming transition are found with the use of numerical simulation. The traffic states are studied for both constant and noisy boundary conditions. We show the phase diagrams of different kinds of congested traffic. It is found that the noise at the boundary has an important effect on the traffic states. The traffic behavior in the coupled map lattice model exhibits a jamming transition similar to that found in the car-following model.     

Generalized synchronization of two unidirectionally coupled discrete stochastic dynamical systems

The existence of two kinds of generalized synchronization manifold in two unidirectionally coupled discrete stochastic dynamical systems is studied in this paper. When the drive system is chaotic and the modified response system collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbit, under certain conditions, the existence of the generalized synchronization can be converted to the problem of a Lipschitz contractive fixed point or Schauder fixed point. Moreover, the exponential attractive property of generalized synchronization manifold is strictly proved. In addition, numerical simulations demonstrate the correctness of the present theory. The physical background and meaning of the results obtained in this paper are also discussed.%vspace1mm     

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.     

Nonlinear resonance and synchronization in the ring of unidirectionally coupled Toda oscillators

In the ring of unidirectionally coupled Toda oscillators the nonlinear resonance and the synchronization are investigated. It is shown how the nonlinear resonance affects the structure of the main synchronization region. As a result of nonlinear resonance we observe the coexistence of two stable limit cycles near the resonant frequency, which leads to coexistence of periodic and quasi-periodic regimes within the synchronization region.     

Local bifurcations of synchronization in self-excited and forced unidirectionally coupled micromechanical resonators

A unidirectionally coupled system of Microelectromechanical Systems (MEMS) resonators is investigated. Local bifurcations of both the single resonator and the coupled system are analyzed in the design of the coupled system. We discuss the behavior of the coupled system according to the change in the coupling parameter and the excitation force. Phenomena such as quasiperiodic oscillations and synchronization are shown by numerical simulations and studied analytically through perturbation methods. The coupled system shows oscillatory behavior without the excitation force. In the presence of the excitation force the coupled system exhibits complex behavior which can be beneficial in the design of the coupled system as a sensor.     

Chaotic synchronization of unidirectionally coupled electron-wave media with interacting counterpropagating waves

Chaotic synchronization of two electron-wave media with interacting counterpropagating waves and cubic phase nonlinearity (transverse-field backward-wave oscillators) is studied. Analysis is based on considering a continuous set of the phases of a chaotic signal. The parameters of chaotic synchronization in a system of unidirectionally coupled backward-wave oscillators are found, and the complex dynamics of establishing the chaotic synchronization conditions in an active medium is investigated.     

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.