Desynchronization of chaos in coupled logistic maps

Coupled Lorenz oscillators near the Hopf boundary: Multistability, intermingled basins, and quasiriddling

We investigate the dynamics of coupled identical chaotic Lorenz oscillators just above the subcritical Hopf bifurcation. In the absence of coupling, the motion is on a strange chaotic attractor and the fixed points of the system are all unstable. With the coupling, the unstable fixed points are converted into chaotic attractors, and the system can exhibit a multiplicity of coexisting attractors. Depending on the strength of the coupling, the motion of the individual oscillators can be synchronized (both in and out of phase) or desynchronized and in addition there can be mixed phases. We find that the basins have a complex structure: the state that is asymptotically reached shows extreme sensitivity to initial conditions. The basins of attraction of these different states are characterized using a variety of measures and depending on the strength of the coupling, they are intermingled or quasiriddled.

Formation and destruction of multilayered tori in coupled map systems

The paper first illustrates how multilayered tori can arise through one or more pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. The paper hereafter describes three different scenarios by which a multilayered torus can be destructed. One scenario involves a saddle-node bifurcation in which the middle layer of a three-layered torus disappears in an abrupt transition to chaos while the outer-layer manifolds and their associated saddle and unstable-focus cycles continue to exist and to control the transient dynamics. In a second scenario, the unstable focus cycles of the intermediate layers in a five-layered torus turn into unstable nodes, and closed loop connections are established between the unstable nodes and the points of the stable resonance node on the torus. Finally, a third scenario describes a transition in which homoclinic bifurcations destroy first the outer layers and thereafter also the inner layer. The paper also illustrates how the formation and destruction of multilayered tori can occur in the cluster dynamics of an ensemble of globally coupled maps. This leads to three additional scenarios for the destruction of multilayered tori.

Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats

We searched for synchronization among autoregulation mechanisms using wavelet transforms applied to tubular pressure recordings in nephron pairs from the surface of rat kidneys. Nephrons have two oscillatory modes in the regulation of their pressures and flows: a faster (100–200 mHz) myogenic mode, and a slower (20–40 mHz) oscillation in tubuloglomerular feedback (TGF). These mechanisms interact; the TGF mode modulates both the amplitude and the frequency of the myogenic mode. Nephrons also communicate with each other using vascular signals triggered by membrane events in arteriolar smooth muscle cells. In addition, the TGF oscillation changes in hypertension to an irregular fluctuation with characteristics of deterministic chaos. The analysis shows that, within single nephrons of normotensive rats, the myogenic mode and TGF are synchronized at discrete frequency ratios, with 5:1 most common. There is no distinct synchronization ratio in spontaneously hypertensive rats (SHR). In normotensive rats, full synchronization of both TGF and myogenic modes is the most probable state for pairs of nephrons originating in a common cortical radial artery. For SHR, full synchronization is less probable; most common in SHR is a state of partial synchronization with entrainment between neighboring nephrons for only one of the modes. Modulation of the myogenic mode by the TGF mode is much stronger in hypertensive than in normotensive rats. Synchronization among nephrons forms the basis for an integrated reaction to blood pressure fluctuations. Reduced synchronization in SHR suggests that the effectiveness of the coordinated response is impaired in hypertension.

Basin bifurcations in quasiperiodically forced coupled systems

We study the effect of quasiperiodic forcing on a system of coupled identical logistic maps. Upon a variation of system parameters, a variety of different dynamical regimes can be observed, including phenomena such as bistability and multistability. At the bifurcation to bistability, in a manner reminiscent of attractor expansion at interior crises, there is an abrupt change in the size of attractor basins. In the bistable region, attractor basins undergo additional bifurcations wherein holes and islands are created within the basins when system parameters change. These can be understood by examining critical surfaces for the coupled system.

Synchronized clusters in coupled map networks. I. Numerical studies

We study the synchronization of coupled maps on a variety of networks including regular one- and two-dimensional networks, scale-free networks, small world networks, tree networks, and random networks. For small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases. When nodes show synchronized behavior, we observe two interesting phenomena. First, there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently. Second, we identify two different ways of cluster formation, namely self-organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings. The synchronized clusters may be of dominant self-organized type, dominant driven type, or mixed type depending on the type of network and the parameters of the dynamics. We define different states of the coupled dynamics by considering the number and type of synchronized clusters. For the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant (epsilon) and the logistic map parameter (mu). For large coupling strengths and nonlinear coupling we find that the scale-free networks and the Caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity. For most of our study we use the number of connections of the order of the number of nodes. As the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase.

Synchronized clusters in coupled map networks. II. Stability analysis

We study self-organized and driven synchronization in some simple coupled map networks, namely globally coupled networks and complete bipartite networks, using both linear stability analysis and Lyapunov function approach and determine stability conditions for synchronization. The phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in Part I. Lyapunov function approach shows that when any two nodes are in driven synchronization, all the coupling terms in the difference between the variables of these two nodes cancel out, whereas when they are in self-organized synchronization, the direct coupling term between the two nodes adds an extra term while the other couplings cancel out. We also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional Lyapunov exponent about zero can be a criterion for its occurrence.

Crisis-induced intermittency in two coupled chaotic maps: towards understanding chaotic itinerancy

The present paper considers crisis-induced intermittency in a system composed of two coupled logistic maps. Its purpose is to clarify a bifurcation scenario generating such intermittent behaviors that can be regarded as a simple example of chaotic itinerancy. The intermittent dynamics appears immediately after an attractor-merging crisis of two off-diagonal chaotic attractors in a symmetrically coupled system. The scenario for the crisis is investigated through analyses of sequential bifurcations leading to the two chaotic attractors and successive changes in basin structures with variation of a system parameter. The successive changes of the basins are also characterized by variation of a dimension of a fractal basin boundary. A numerical analysis shows that simultaneous contacts between the attractors and the fractal basin boundary bring about the crisis and a snap-back repeller generated at the crisis produces the intermittent transitions. Furthermore, a modified scenario for intermittent behaviors in an asymmetrically coupled system is also discussed.

Torus breakdown in noninvertible maps

We propose a criterion for the destruction of a two-dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus. This mechanism is specific to noninvertible maps. The cusp points arise when the tangent to the torus at the point of intersection with the critical curve L(0) coincides with the eigendirection corresponding to vanishing eigenvalue for the noninvertible map. Further parameter changes lead typically to the generation of loops (self-intersections of the invariant manifolds) followed by the transformation of the torus into a complex chaotic set.

Experimental investigation of partial synchronization in coupled chaotic oscillators

The dynamical behavior of a ring of six diffusively coupled Rössler circuits, with different coupling schemes, is experimentally and numerically investigated using the coupling strength as a control parameter. The ring shows partial synchronization and all the five patterns predicted analyzing the symmetries of the ring are obtained experimentally. To compare with the experiment, the ring has been integrated numerically and the results are in good qualitative agreement with the experimental ones. The results are analyzed through the graphs generated plotting the y variable of the ith circuit versus the variable y of the jth circuit. As an auxiliary tool to identify numerically the behavior of the oscillators, the three largest Lyapunov exponents of the ring are obtained.

Synchronization and desynchronization under the influence of quasiperiodic forcing

We study the influence of quasiperiodic forcing on synchronization and desynchronization using two coupled quasiperiodically forced logistic maps as a paradigm. We show that due to the forcing the synchronization region in parameter space shrinks. The loss of transverse stability of the synchronized attractors leads to desynchronization. Two types of such blowout bifurcations are described, namely, the blowout bifurcations of synchronized quasiperiodic motion on invariant curves and synchronized strange nonchaotic attractors, both yielding desynchronized chaotic attractors.

Characterization of the noise effect on weak synchronization

We investigate the noise effect on weak synchronization in two coupled identical one-dimensional (1D) maps. Due to the existence of positive local transverse Lyapunov exponents, the weakly stable synchronous chaotic attractor (SCA) becomes sensitive with respect to the variation of noise intensity. To quantitatively characterize such noise sensitivity, we introduce a quantifier, called the noise sensitivity exponent (NSE). For the case of bounded noise, the values of the NSE are found to be the same as those of the exponent characterizing a parameter sensitivity of the weakly stable SCA in presence of a parameter mismatch between the two 1D maps. Furthermore, it is found that the scaling exponent for the average time spent near the diagonal for both the bubbling and riddling cases occurring in the regime of weak synchronization is given by the reciprocal of the NSE, as in the parameter-mismatching case. Consequently, both the noise and parameter mismatch have the same effect on the scaling behavior of the average characteristic time.

Population dynamics with a refuge: fractal basins and the suppression of chaos

We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.

Characterization of the parameter-mismatching effect on the loss of chaos synchronization

We investigate the effect of the parameter mismatch on the loss of chaos synchronization in coupled one-dimensional maps. Loss of strong synchronization begins with a first transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor (SCA), and then the SCA becomes weakly stable. Because of local transverse repulsion of the periodic repellers embedded in the weakly stable SCA, a typical trajectory may have segments of arbitrary length that have positive local transverse Lyapunov exponents. Consequently, the weakly stable SCA becomes sensitive with respect to the variation of the mismatching parameter. To quantitatively characterize such parameter sensitivity, we introduce a quantifier, called the parameter sensitivity exponent (PSE). As the local transverse repulsion of the periodic repellers strengthens, the value of the PSE increases. In terms of these PSEs, we also characterize the parameter-mismatching effect on the intermittent bursting and basin riddling occurring in the regime of weak synchronization.

Unexpected coherence and conservation

The effects of migration in a network of patch populations, or metapopulation, are extremely important for predicting the possibility of extinctions both at a local and a global scale. Migration between patches synchronizes local populations and bestows upon them identical dynamics (coherent or synchronous oscillations), a feature that is understood to enhance the risk of global extinctions. This is one of the central theoretical arguments in the literature associated with conservation ecology. Here, rather than restricting ourselves to the study of coherent oscillations, we examine other types of synchronization phenomena that we consider to be equally important. Intermittent and out-of-phase synchronization are but two examples that force us to reinterpret some classical results of the metapopulation theory. In addition, we discuss how asynchronous processes (for example, random timing of dispersal) can paradoxically generate metapopulation synchronization, another non-intuitive result that cannot easily be explained by the standard theory.

Loss of coherence in a system of globally coupled maps

We study the formation of symmetric (i.e., equally sized) or nearly symmetric clusters in an ensemble of globally coupled, identical chaotic maps. It is shown that the loss of synchronization for the coherent state and the emergence of subgroups of oscillators with synchronized behavior are two distinct processes, and that the type of behavior that arises after the loss of total synchronization depends sensitively on the dynamics of the individual map. For our system of globally coupled logistic maps, symmetric two-cluster formation is found to proceed through a periodic state associated with the stabilization either of an asynchronous period-2 cycle or of an asynchronous period-4 cycle. With further reduction of the coupling strength, each of the principal clustering states undergoes additional bifurcations leading to cycles of higher periodicity or to quasiperiodic and chaotic dynamics. If desynchronization of the coherent chaotic state occurs before the formation of stable clusters becomes possible, high-dimensional chaotic motion is observed in the intermediate parameter interval.

Resolving clusters in chaotic ensembles of globally coupled identical oscillators

Clustering in ensembles of globally coupled identical chaotic oscillators is reconsidered using a twofold approach. Stability of clusters towards "emanation" of the elements is described with the evaporation Lyapunov exponents. It appears that direct numerical simulations of ensembles often lead to spurious clusters that have positive evaporation exponents, due to a numerical trap. We propose a numerical method that surmounts the spurious clustering. We also demonstrate that clustering can be very sensitive to the number of elements in the ensemble.

Effect of asymmetry on the loss of chaos synchronization

We investigate the effect of asymmetry of coupling on the bifurcation mechanism for the loss of synchronous chaos in coupled systems. It is found that only when the symmetry-breaking pitchfork bifurcations take part in the process of the synchronization loss for the case of symmetric coupling, the asymmetry changes the bifurcation scenarios of the desynchronization. For the case of weak coupling, pitchfork bifurcations of asynchronous periodic saddles are replaced by saddle-node bifurcations, while for the case of strong coupling, pitchfork bifurcations of synchronous periodic saddles transform to transcritical bifurcations. The effects of the saddle-node and transcritical bifurcations for the weak asymmetry are similar to those of the pitchfork bifurcations for the symmetric-coupling case. However, with increasing the "degree" of the asymmetry, their effects change qualitatively, and eventually become similar to those for the extreme case of unidirectional asymmetric coupling.

Transcritical riddling in a system of coupled maps

The transition from fully synchronized behavior to two-cluster dynamics is investigated for a system of N globally coupled chaotic oscillators by means of a model of two coupled logistic maps. An uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system. While the transverse period-doubling bifurcation remains essentially unaffected by this asymmetry, the transverse pitchfork bifurcation is turned into a saddle-node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability. We show that the transcritical riddling transition is always hard. For this, we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle-node bifurcation undergo, and show how the manifolds of these cycles control the magnitude of asynchronous bursts. In the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters, the transcritical riddling will be replaced by two subsequent saddle-node bifurcations, or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold. In this way, the transcritical riddling bifurcation is substituted by a symmetry-breaking bifurcation, which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state.

Partial synchronization and spontaneous spatial ordering in coupled chaotic systems

A model of many symmetrically and locally coupled chaotic oscillators is studied. Partial chaotic synchronizations associated with spontaneous spatial ordering are demonstrated. Very rich patterns of the system are revealed, based on partial synchronization analysis. The stabilities of different partially synchronous spatiotemporal structures and some dynamical behaviors of these states are discussed both numerically and analytically.

Mechanism for the riddling transition in coupled chaotic systems

We investigate the loss of chaos synchronization in coupled chaotic systems without symmetry from the point of view of bifurcations of unstable periodic orbits embedded in the synchronous chaotic attractor (SCA). A mechanism for direct transition to global riddling through a transcritical contact bifurcation between a periodic saddle embedded in the SCA and a repeller on the boundary of its basin of attraction is thus found. Note that this bifurcation mechanism is different from that in coupled chaotic systems with symmetry. After such a riddling transition, the basin becomes globally riddled with a dense set of repelling tongues leading to divergent orbits. This riddled basin is also characterized by divergence and uncertainty exponents, and thus typical power-law scaling is found.