Transition to intermittent chaotic synchronization

Synchronization transitions in a hyperchaotic SQUID trimer

The phenomena of intermittent and complete synchronization between two out of three identical, magnetically coupled Superconducting QUantum Interference Devices (SQUIDs) are investigated numerically. SQUIDs are highly nonlinear superconducting oscillators/devices that exhibit strong resonant and tunable response to applied magnetic field(s). Single SQUIDs and SQUID arrays are technologically important solid-state devices, and they also serve as a testbed for exploring numerous complex dynamical phenomena. In SQUID oligomers, the dynamic complexity increases considerably with the number of SQUIDs. The SQUID trimer, considered here in a linear geometrical configuration using a realistic model with experimentally accessible control parameters, exhibits chaotic and hyperchaotic behavior in wide parameter regions. Complete chaos synchronization as well as intermittent chaos synchronization between two SQUIDs of the trimer is identified and characterized using the complete Lyapunov spectrum of the system and appropriate measures. The passage from complete to intermittent synchronization seems to be related to chaos-hyperchaos transitions as has been conjectured in the early days of chaos synchronization.

Multi-branched resonances, chaos through quasiperiodicity, and asymmetric states in a superconducting dimer

A system of two identical superconducting quantum interference devices (SQUIDs) symmetrically coupled through their mutual inductance and driven by a sinusoidal field is investigated numerically with respect to dynamical properties such as its multibranched resonance curve, its bifurcation structure and transition to chaos as well as its synchronization behavior. The SQUID dimer is found to exhibit a hysteretic resonance curve with a bubble connected to it through Neimark-Sacker (torus) bifurcations, along with coexisting chaotic branches in their vicinity. Interestingly, the transition of the SQUID dimer to chaos occurs through a torus-doubling cascade of a two-dimensional torus (quasiperiodicity-to-chaos transition). Periodic, quasiperiodic, and chaotic states are identified through the calculated Lyapunov spectrum and illustrated using Lyapunov charts on the parameter plane of the coupling strength and the frequency of the driving field. The basins of attraction for chaotic and non-chaotic states are determined. Bifurcation diagrams are constructed on the parameter plane of the coupling strength and the frequency of the driving field, and they are superposed to maps of the three largest Lyapunov exponents on the same plane. Furthermore, the route of the system to chaos through torus-doubling bifurcations and the emergence of Hénon-like chaotic attractors are demonstrated in stroboscopic diagrams obtained with varying driving frequency. Moreover, asymmetric states that resemble localized synchronization have been detected using the correlation function between the fluxes threading the loop of the SQUIDs.

Complex networks exhibit intermittent synchronization

The path toward the synchronization of an ensemble of dynamical units goes through a series of transitions determined by the dynamics and the structure of the connections network. In some systems on the verge of complete synchronization, intermittent synchronization, a time-dependent state where full synchronization alternates with non-synchronized periods, has been observed. This phenomenon has been recently considered to have functional relevance in neuronal ensembles and other networked biological systems close to criticality. We characterize the intermittent state as a function of the network topology to show that the different structures can encourage or inhibit the appearance of early signs of intermittency. In particular, we study the local intermittency and show how the nodes incorporate to intermittency in hierarchical order, which can provide information about the node topological role even when the structure is unknown.

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.

Transition to complete synchronization and global intermittent synchronization in an array of time-delay systems

We report the nature of transitions from the nonsynchronous to a complete synchronization (CS) state in arrays of time-delay systems, where the systems are coupled with instantaneous diffusive coupling. We demonstrate that the transition to CS occurs distinctly for different coupling configurations. In particular, for unidirectional coupling, locally (microscopically) synchronization transition occurs in a very narrow range of coupling strength but for a global one (macroscopically) it occurs sequentially in a broad range of coupling strength preceded by an intermittent synchronization. On the other hand, in the case of mutual coupling, a very large value of coupling strength is required for local synchronization and, consequently, all the local subsystems synchronize immediately for the same value of the coupling strength and, hence, globally, synchronization also occurs in a narrow range of the coupling strength. In the transition regime, we observe a type of synchronization transition where long intervals of high-quality synchronization which are interrupted at irregular times by intermittent chaotic bursts simultaneously in all the systems and which we designate as global intermittent synchronization. We also relate our synchronization transition results to the above specific types using unstable periodic orbit theory. The above studies are carried out in a well-known piecewise linear time-delay system.

Detecting the temporal structure of intermittent phase locking

This study explores a method to characterize the temporal structure of intermittent phase locking in oscillatory systems. When an oscillatory system is in a weakly synchronized regime away from a synchronization threshold, it spends most of the time in parts of its phase space away from the synchronization state. Therefore characteristics of dynamics near this state (such as its stability properties and Lyapunov exponents or distributions of the durations of synchronized episodes) do not describe the system's dynamics for most of the time. We consider an approach to characterize the system dynamics in this case by exploring the relationship between the phases on each cycle of oscillations. If some overall level of phase locking is present, one can quantify when and for how long phase locking is lost, and how the system returns back to the phase-locked state. We consider several examples to illustrate this approach: coupled skewed tent maps, the stability of which can be evaluated analytically; coupled Rössler and Lorenz oscillators, undergoing through different intermittency types on the way to phase synchronization; and a more complex example of coupled neurons. We show that the obtained measures can describe the differences in the dynamics and temporal structure of synchronization and desynchronization events for the systems with a similar overall level of phase locking and similar stability of the synchronized state.

Chaotic phase synchronization and desynchronization in an oscillator network for object selection

Object selection refers to the mechanism of extracting objects of interest while ignoring other objects and background in a given visual scene. It is a fundamental issue for many computer vision and image analysis techniques and it is still a challenging task to artificial visual systems. Chaotic phase synchronization takes place in cases involving almost identical dynamical systems and it means that the phase difference between the systems is kept bounded over the time, while their amplitudes remain chaotic and may be uncorrelated. Instead of complete synchronization, phase synchronization is believed to be a mechanism for neural integration in brain. In this paper, an object selection model is proposed. Oscillators in the network representing the salient object in a given scene are phase synchronized, while no phase synchronization occurs for background objects. In this way, the salient object can be extracted. In this model, a shift mechanism is also introduced to change attention from one object to another. Computer simulations show that the model produces some results similar to those observed in natural vision systems.

Intermittency transition to generalized synchronization in coupled time-delay systems

We report the nature of the transition to generalized synchronization (GS) in a system of two coupled scalar piecewise linear time-delay systems using the auxiliary system approach. We demonstrate that the transition to GS occurs via an on-off intermittency route and that it also exhibits characteristically distinct behaviors for different coupling configurations. In particular, the intermittency transition occurs in a rather broad range of coupling strength for the error feedback coupling configuration and in a narrow range of coupling strength for the direct feedback coupling configuration. It is also shown that the intermittent dynamics displays periodic bursts of periods equal to the delay time of the response system in the former case, while they occur in random time intervals of finite duration in the latter case. The robustness of these transitions with system parameters and delay times has also been studied for both linear and nonlinear coupling configurations. The results are corroborated analytically by suitable stability conditions for asymptotically stable synchronized states and numerically by the probability of synchronization and by the transition of sub-Lyapunov exponents of the coupled time-delay systems. We have also indicated the reason behind these distinct transitions by referring to the unstable periodic orbit theory of intermittency synchronization in low-dimensional systems.