Characteristic relations of type-I intermittency in the presence of noise

Type III intermittency without characteristic relation

Chaotic intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories from the chaotic region into the laminar one. The RPD classically was taken as a constant. This hypothesis is behind the classically reported characteristic relations, a tool describing how the mean value of the laminar length goes to infinity as the control parameter goes to zero. Recently, a generalized non-uniform RPD has been observed in a wide class of 1D maps; hence, the intermittency theory has been generalized. Consequently, the characteristic relations were also generalized. However, the RPD and the characteristic relations observed in some experimental Poincaré maps still cannot be well explained in the actual intermittency framework. We extend the previous analytical results to deal with the mentioned class of maps. We found that in the mentioned maps, there is not a well-defined RPD in the sense that its shape drastically changes depending on a small variation of the parameter of the map. Consequently, the characteristic relation classically associated to every type of intermittency is not well defined and, in general, cannot be determined experimentally. We illustrate the results with a 1D map and we develop the analytical expressions for every RPD and its characteristic relations. Moreover, we found a characteristic relation going to a constant value, instead of increasing to infinity. We found a good agreement with the numerical simulation.

Jump intermittency as a second type of transition to and from generalized synchronization

The transition from asynchronous dynamics to generalized chaotic synchronization and then to completely synchronous dynamics is known to be accompanied by on-off intermittency. We show that there is another (second) type of the transition called jump intermittency which occurs near the boundary of generalized synchronization in chaotic systems with complex two-sheeted attractors. Although this transient behavior also exhibits intermittent dynamics, it differs sufficiently from on-off intermittency supposed hitherto to be the only type of motion corresponding to the transition to generalized synchronization. This type of transition has been revealed and the underling mechanism has been explained in both unidirectionally and mutually coupled chaotic Lorenz and Chen oscillators. To detect the epochs of synchronous and asynchronous motion in mutually coupled oscillators with complex topology of an attractor a technique based on finding time intervals when the phase trajectories are located on equal or different sheets of chaotic attractors of coupled oscillators has been developed. We have also shown that in the unidirectionally coupled systems the proposed technique gives the same results that may obtained with the help of the traditional method using the auxiliary system approach.

Separation of coexisting dynamical regimes in multistate intermittency based on wavelet spectrum energies in an erbium-doped fiber laser

We propose a method for the detection and localization of different types of coexisting oscillatory regimes that alternate with each other leading to multistate intermittency. Our approach is based on consideration of wavelet spectrum energies. The proposed technique is tested in an erbium-doped fiber laser with four coexisting periodic orbits, where external noise induces intermittent switches between the coexisting states. Statistical characteristics of multistate intermittency, such as the mean duration of the phases for every oscillation type, are examined with the help of the developed method. We demonstrate strong advantages of the proposed technique over previously used amplitude methods.

Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series

A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.

Theory of intermittency applied to classical pathological cases

The classical theory of intermittency developed for return maps assumes uniform density of points reinjected from the chaotic to laminar region. Though it works fine in some model systems, there exist a number of so-called pathological cases characterized by a significant deviation of main characteristics from the values predicted on the basis of the uniform distribution. Recently, we reported on how the reinjection probability density (RPD) can be generalized. Here, we extend this methodology and apply it to different dynamical systems exhibiting anomalous type-II and type-III intermittencies. Estimation of the universal RPD is based on fitting a linear function to experimental data and requires no a priori knowledge on the dynamical model behind. We provide special fitting procedure that enables robust estimation of the RPD from relatively short data sets (dozens of points). Thus, the method is applicable for a wide variety of data sets including numerical simulations and real-life experiments. Estimated RPD enables analytic evaluation of the length of the laminar phase of intermittent behaviors. We show that the method copes well with dynamical systems exhibiting significantly different statistics reported in the literature. We also derive and classify characteristic relations between the mean laminar length and main controlling parameter in perfect agreement with data provided by numerical simulations.

Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise

We consider a behavior of the zero Lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics. We analytically deduce the laws for the dependence of this Lyapunov exponent on the control parameter both above and below the bifurcation point. The developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. We also discuss how the revealed regularities are expected to take place in other relevant physical circumstances.

Length distribution of laminar phases for type-I intermittency in the presence of noise

We consider a type of intermittent behavior that occurs as the result of the interplay between dynamical mechanisms giving rise to type-I intermittency and random dynamics. We analytically deduce the laws for the distribution of the laminar phases, with the law for the mean length of the laminar phases versus the critical parameter deduced earlier [W.-H. Kye and C.-M. Kim, Phys. Rev. E 62, 6304 (2000)] being the corollary fact of the developed theory. We find a very good agreement between the theoretical predictions and the data obtained by means of both the experimental study and numerical calculations. We discuss also how this mechanism is expected to take place in other relevant physical circumstances.

Current reversal with type-I intermittency in deterministic inertia ratchets

The intermittency is investigated when the current reversal occurs in a deterministic inertia ratchet system. To determine which type the intermittency belongs to, we obtain the return map of velocities of particle by using stroboscopic recordings, and by numerically calculating the distribution of the average laminar length <l>. The distribution follows the scaling law of <l> proportional to epsilon(-1/2), the characteristic relation of type-I intermittency.

Experimental observation of characteristic relations of type-III intermittency in the presence of noise in a simple electronic circuit

We investigate the characteristic relations of type-II and -III intermittencies in the presence of noise. The theoretically predicted characteristic relation is that <l> approximately exp[/epsilon/(2)] for a negative regime of epsilon and <l> approximately epsilon(-nu) for the positive regime of epsilon (1/2</=nu<1), where <l> is the average laminar length and (1+epsilon) is the slope of the local Poincaré map around the tangent point. We experimentally confirm these relations in a simple electronic circuit.

Periodic phase synchronization in coupled chaotic oscillators

We investigate the characteristics of temporal phase locking states observed in the route to phase synchronization. It is found that before phase synchronization there is a periodic phase synchronization state characterized by periodic appearance of temporal phase-locking state and that the state leads to local negativeness in one of the vanishing Lyapunov exponents. By taking a statistical measure, we present the evidences of the phenomenon in unidirectionally and mutually coupled chaotic oscillators, respectively. And it is qualitatively discussed that the phenomenon is described by a nonuniform oscillator model in the presence of noise.

Transition from phase synchronization to complete synchronization in mutually coupled nonidentical Nd:YAG lasers

Using mutually coupled nonidentical continuous-wave Nd:YAG lasers, we experimentally confirmed the recently proposed transition route from phase synchronization to complete synchronization. As evidence of this transition we obtained the probability distribution of the intermittent synchronization time near the threshold of the complete synchronization transition.

Generalized phase synchronization in unidirectionally coupled chaotic oscillators

We investigate phase synchronization between two identical or detuned response oscillators coupled to a slightly different drive oscillator. Our result is that phase synchronization can occur between response oscillators when they are driven by correlated (but not identical) inputs from the drive oscillator. We call this phenomenon generalized phase synchronization and clarify its characteristics using Lyapunov exponents and phase difference plots.

Persistent Na+ current modifies burst discharge by regulating conditional backpropagation of dendritic spikes

The estimation and detection of stimuli by sensory neurons is affected by factors that govern a transition from tonic to burst mode and the frequency characteristics of burst output. Pyramidal cells in the electrosensory lobe of weakly electric fish generate spike bursts for the purpose of stimulus detection. Spike bursts are generated during repetitive discharge when a frequency-dependent broadening of dendritic spikes increases current flow from dendrite to soma to potentiate a somatic depolarizing afterpotential (DAP). The DAP eventually triggers a somatic spike doublet with an interspike interval that falls inside the dendritic refractory period, blocking spike backpropagiation and the DAP. Repetition of this process gives rise to a rhythmic dendritic spike failure, termed conditional backpropagation, that converts cell output from tonic to burst discharge. Through in vitro recordings and compartmental modeling we show that burst frequency is regulated by the rate of DAP potentiation during a burst, which determines the time required to discharge the spike doublet that blocks backpropagation. DAP potentiation is magnified through a positive feedback process when an increase in dendritic spike duration activates persistent sodium current (I(NaP)). I(NaP) further promotes a slow depolarization that induces a shift from tonic to burst discharge over time. The results are consistent with a dynamical systems analysis that shows that the threshold separating tonic and burst discharge can be represented as a saddle-node bifurcation. The interaction between dendritic K(+) current and I(NaP) provides a physiological explanation for a variable time scale of bursting dynamics characteristic of such a bifurcation.

Phase synchronization in coupled chaotic oscillators with time delay

The phase synchronization (PS) of two Rössler oscillators with time-delayed signal coupling is studied. We find that time delay can always lead to PS even when the delay is very long. Moreover, with the increase of time delay, the coupling strength at the transition to PS undergoes a nearly periodic wave distribution. At some fixed time-delayed signal coupling, a PS region is followed by a non-PS region when the coupling strength increases. However, an increase of the coupling leads to the PS state again. This phenomenon occurs in systems with a relatively large PS transition point.

Phase synchronization in bidirectionally coupled optothermal devices

We present the experimental observation of phase synchronization transitions in the bidirectional coupling of chaotic and nonchaotic oscillators. A variety of transitions are characterized and compared to numerical simulations of a time delayed model. The characteristic 2pi phase jumps usually appear during the transitions, specially in those clearly associated with a saddle-node bifurcation. The study is done with pairs of optothermal oscillators linearly coupled by heat transfer.

Routes to complete synchronization via phase synchronization in coupled nonidentical chaotic oscillators

We study the transition route to complete synchronization through phase synchronization in generic coupled nonidentical chaotic oscillators. Through numerical studies, two routes are found, i.e., one, via lag synchronization, the other, via the intermittent chaotic burst state without lag synchronization. We claim that these two routes are universal. As evidence, we analyze several examples on the basis of the conventional theory of intermittency in the presence of noise.

Experimental observation of the characteristic relations of type-I intermittency in the presence of noise

Recently, it has been reported that the characteristic relation of type-I intermittency in the presence of noise is deformed nontrivially as the channel width epsilon changes from the positive region to the negative. In order to verify it experimentally as a real phenomenon, we study the characteristic relations both for epsilon<0 and for epsilon>0 in a simple inductor-resistor-diode circuit that is under noisy circumstances. The experimental results agree well with the theoretical expectation that the characteristic relations are <l> proportional to epsilon(-1/4) for epsilon>0 and <l> proportional to exp(alpha/epsilon/(3/2)) for epsilon<0.

Phase synchronization in coupled Nd:YAG lasers

We investigate experimentally the transition to phase synchronization in coupled Nd:YAG lasers. As the coupling strength increases, the phase difference of two chaotic laser outputs develops from nonsynchronous to a phase-synchronous state via +/-2pi phase jumps. We analyze this transition.

Intermittent phase synchronization of coupled spatiotemporal chaotic systems

Phase synchronization is studied with a discrete system formed by two coupled map lattices, in which phases are measured in two-dimensional vectors. Simulation results show that by imposing external coupling between the two lattices, phase synchronization can be found in all two-dimensional phase planes between them. When the system is approaching the phase synchronizing state, unstable phase synchronization is observed. This is referred to as intermittent phase synchronization that appears when the trajectories on two interacting phase planes have opposite directions of rotation but with only a small phase difference. The intermittent phase synchronization could also be observed in coupled autonomous systems with diffusive attractors although their phase concepts are inconsistent. Our results show that the intermittent phase synchronization of both discrete and autonomous systems relates to the diffusion or the complexity of the attractors.

Attractor bifurcation and on-off intermittency

We analyze the bifurcation of an attractor in a system of multiplicatively coupled maps. It is shown that the phenomenon can be characterized by on-off intermittency and it is rigorously described by the on-off intermittency in the presence of noise through the concept of a virtually invariant manifold. Presenting numerical results which conform our claims, we elucidate the role of on-off intermittency in bifurcation phenomena.

Phase synchronization with type-II intermittency in chaotic oscillators

We study the phase synchronization (PS) with type-II intermittency showing +/-2pi irregular phase jumping behavior before the PS transition occurs in a system of two coupled hyperchaotic Rossler oscillators. The behavior is understood as a stochastic hopping of an overdamped particle in a potential which has 2pi-periodic minima. We characterize it as type-II intermittency with external noise through the return map analysis. In epsilon(t)<epsilon<epsilon(c) (where epsilon(t) is the bifurcation point of type-II intermittency and epsilon(c) is the PS transition point in coupling strength parameter space), the average length of the time interval between two successive jumps follows <l> approximately exp(|epsilon(t)-epsilon|(2)), which agrees well with the scaling law obtained from the Fokker-Planck equation.