New characteristic relations in type-I intermittency

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.

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.

Horizontal visibility graphs generated by type-I intermittency

The type-I intermittency route to (or out of) chaos is investigated within the horizontal visibility (HV) graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct their associated HV graphs. We show how the alternation of laminar episodes and chaotic bursts imprints a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values for several network parameters. In particular, we predict that the characteristic power-law scaling of the mean length of laminar trend sizes is fully inherited by the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of block entropy functionals defined on the graph. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization-group framework, where the fixed points of its graph-theoretical renormalization-group flow account for the different types of dynamics. We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibits extremal entropic properties.

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.

The phase-modulated logistic map

We study the logistic mapping with the nonlinearity parameter varied through a delayed feedback mechanism. This history dependent modulation through a phaselike variable offers an enhanced possibility for stabilization of periodic dynamics. Study of the system as a function of nonlinearity and modulation parameters reveals new phenomena: In addition to period-doubling and tangent bifurcations, there can be bifurcations where the period increases by unity. These are extensions of crises that arise in nonlinear dynamical systems. Periodic orbits in this system can be systematized via the kneading theory, which in the present case extends the analysis of Metropolis, Stein, and Stein for unimodal maps.

Chaotic behaviors of operational amplifiers

We investigate nonlinear dynamical behaviors of operational amplifiers. When the output terminal of an operational amplifier is connected to the inverting input terminal, the circuit exhibits period-doubling bifurcation, chaos, and periodic windows, depending on the voltages of the positive and the negative power supplies. We study these nonlinear dynamical characteristics of this electronic circuit experimentally.

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.

Scaling properties of saddle-node bifurcations on fractal basin boundaries

We analyze situations where a saddle-node bifurcation occurs on a fractal basin boundary. Specifically, we are interested in what happens when a system parameter is slowly swept in time through the bifurcation. Such situations are known to be indeterminate in the sense that it is difficult to predict the eventual fate of an orbit that tracks the prebifurcation node attractor as the system parameter is swept through the bifurcation. In this paper we investigate the scaling of (1) the fractal basin boundary of the static (i.e., unswept) system near the saddle-node bifurcation, (2) the dependence of the orbit's final destination on the sweeping rate, (3) the dependence of the time it takes for an attractor to capture a swept orbit on the sweeping rate, and (4) the dependence of the final attractor capture probability on the noise level. With respect to noise, our main result is that the effect of noise scales with the 5/6 power of the parameter drift rate. Our approach is to first investigate all these issues using one-dimensional map models. The simplification of treatment inherent in one dimension greatly facilitates analysis and numerical experiment, aiding us in obtaining the new results listed above. Following our one-dimensional investigations, we explain that these results can be applied to two-dimensional systems. We show, through numerical experiments on a periodically forced second-order differential equation example, that the scalings we have found also apply to systems that result in two-dimensional maps.

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.

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.

Universal scaling properties of type-I intermittent chaos in isolated resistance arteries are unaffected by endogenous nitric oxide synthesis

Spontaneous fluctuations in flow in isolated rabbit ear resistance arteries may exhibit almost-periodic behavior interrupted by chaotic bursts that can be classified as type-I Pomeau-Manneville intermittency. This conclusion was supported by the construction of parabolic return maps and identification of the characteristic probability distributions for the number of oscillations per laminar segment (n) associated with the type-I scenario. Pharmacological inhibition of nitric oxide (NO) synthesis by the vascular endothelium modulated the dynamics of the reinjection mechanism, and thus the generic shape of the probability distribution for n. Nevertheless, average laminar length was related to a derived bifurcation parameter epsilon according to power-law scaling of the form <n> approximately epsilon(beta), where the estimated critical exponent beta was close to the theoretical value of -0.5 both in the presence and absence of NO synthesis.

Two-dimensional type-I intermittency

The general structure of two-dimensional intermittency is discussed. The structure of channel and the trajectory in the return map are compared with those of one-dimensional intermittency and the scaling relations are obtained according to the trajectory. We illustrate the temporal behavior and scaling relations in a coupled map. The numerical results agree well with the theoretical predication of <l> approximately equal 1/sqrt[epsilon].

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.

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

Near the point of tangent bifurcation, the scaling properties of the laminar length of type-I intermittency are investigated in the presence of noise. Based on analytic and numerical studies, we show that the scaling relation of the laminar length is dramatically deformed from 1/sqrt[epsilon] for epsilon>0 to exp(1/D)|epsilon|(3/2) for epsilon<0 as epsilon passes the bifurcation point (epsilon=0). The results explain why two coupled Rossler oscillators exhibit deformation of the scaling relation of the synchronous length in the nearly synchronous regime.