^{1,2}

^{2}

^{1}

^{1}

^{1}

^{2}

The coupled system of two forced Liénard-type oscillators has applications in diode-based electric circuits and phenomenological models for the heartbeat. These systems typically exhibit intermittent transitions between laminar and chaotic states; what affects their performance and, since noise is always present in such systems, dynamical models should include these effects. Accordingly, we investigated numerically the effect of noise in two intermittent phenomena: the intermittent transition to synchronized behavior for identical and unidirectionally coupled oscillators, and the intermittent transition to chaos near a periodic window of bidirectionally coupled oscillators. We found that the transition from a nonsynchronized to a synchronized state exhibits a power-law scaling with exponent

Intermittency is a ubiquitous phenomenon in nonlinear dynamics. It consists of the intermittent switching between a laminar phase of regular behavior and irregular bursts. In one-dimensional quadratic-type maps it was first associated with a saddle-node bifurcation by Pomeau and Manneville, who also described its scaling characteristics [

Synchronization of nonlinear oscillators is a subject with a venerable history dating back from the early observation by Huygens that two pendula suspended from the same frame—which provides the mechanical coupling—can synchronize their librations so as to become antiphase [

A paradigmatic example of a nonlinear oscillator is the van der Pol equation [

The question of how two or more Liénard-type oscillators can synchronize their motions arises in the study of coupled vacuum-tube circuits [

In this work we focus on the influence of parametric noise in intermittency phenomena numerically observed in two Liénard-type forced oscillators. The first case is related to the intermittent transition to synchronized behavior in such systems in the presence of noise. Bearing in mind the usefulness of Liénard-type oscillators to model vacuum-tube circuits, we can regard the presence of parametric noise as unavoidable, since virtually each circuit component has a fluctuating magnitude (like resistances, capacitances, or inductances) within a given noise level [

The rest of the paper is organized as follows. Section

The triode circuit is a standard textbook example of the Van der Pol equation [

Moreover, we can rewrite (

We consider two such circuits, of which only one is driven by an AC-voltage, and they are supposed to be almost identical, except for their natural frequencies

The rationale for using such coupling schemes lies in the modeling of the interaction between the heart pacemakers, the sino-atrial (SA), and atrio-ventricular (AV) nodes [

We will use throughout this work the following values for the system parameters:

For all the cases studied in this paper the system asymptotic behavior will consist of a periodic or chaotic orbit which encircles the points

The winding number, defined as

By way of contrast, the strongest type of synchronization is complete synchronization (CS), for which the positions and velocities themselves (and not only the phases) are equal:

Finally, if the oscillator parameters are widely different, as in the case we are investigating here, there is no longer LS because the oscillator positions and velocities differ by a large amount [

We will introduce extrinsic noise on the driving oscillator, by adding to (

We have considered also this type of noisy term applied on the response oscillator, when both systems are identical, but the results do not differ appreciably from the case we consider here. On the other hand, the inclusion parametric noise (i.e., noise terms applied to system parameters like normal mode frequencies or coupling constants) leads to qualitatively different phenomena, like GLS states, and which we have considered in a recent paper [

We initiate our analysis by the case of unidirectional coupling (

A useful numerical diagnostic of CS is the order parameter

Order parameter for two values of

The approach to the CS state is characterized by the appearance of weaker forms of synchronization, like PS, and their breakdown. This is a particularly interesting point to investigate the role of noise on synchronization properties of our model. The temporal evolution of the phase difference

Time evolution of the phase difference between the coupled oscillator with and without noise, for two values of

If we add a noise level

In the absence of noise, we observed that the oscillators present FS, characterized by

Winding number difference between the coupled oscillator with and without noise, as a function of the coupling strength.

Another dynamical feature observed in the neighborhood of the synchronization transition at

On-off intermittency has a numerical signature, which is the scaling obeyed by the statistical distribution

Probability distribution for the duration of laminar synchronized intervals between consecutive bursts of nonsynchronized behavior, for (a)

The two scalings are roughly separated by a shoulder which, according to the general theory of noisy on-off intermittency, defines a crossover time whose value depends on the noise level [

Now we present results considering the case where

The dependence of

Bifurcation diagram for

In the neighborhood of the period-12 window, that is, at

Return maps for

Let

Average duration of interburst laminar intervals versus the difference

We can now investigate the role of a noise level on this average duration of laminar intervals. Figure

In the context of the heartbeat model described in [

It is possible, at least in principle, to record the durations of the laminar (interburst) intervals and make a histogram of the laminar times

We have studied the effect of parametric noise in the coupled system of two Liénard-type oscillators with external periodic forcing, focusing on two different intermittent phenomena exhibited by the system under distinct types of coupling. In the unidirectional coupling or master-slave configuration, we have analyzed the occurrence of complete synchronization of identical oscillators and have determined the necessary coupling strength for a transition from a nonsynchronized to a synchronized state. Near this transition there is an intermittent switching between laminar phases of synchronized (albeit chaotic) behavior and bursts of nonsynchronized dynamics. We verified the universal

We also verified the presence of other types of synchronization, like phase and frequency synchronization, and observed that the latter is robust in the sense that it is not likely to cease with addition of white noise. In the bidirectional coupling of nonidentical oscillators (because of a mismatch of their natural frequencies), we no longer have synchronization, and the intermittent phenomenon of interest is the transition to chaos in the beginning of a periodic window for a parameter range where chaos is the dominant feature. We verified that this transition obeys the Pomeau-Manneville type-I intermittency scenario, by considering the statistical properties of the average laminar durations as well as evidencing the saddle-node bifurcation which is the mechanism underlying the phenomenon. The addition of noise affects these properties in the way predicted for one-dimensional maps. Finally, the results of this paper can be applied to a number of physical systems described by Liénard-type oscillators. Two representative examples are electronic circuits using tunnel diodes, like Zener diode, and models of the heartbeat. The statistical nature of our numerical results makes them amenable to further comparisons with experimental investigations of intermittent behavior.

The authors thank Drs. E. Macau, J. A. C. Gallas, and M. W. Beims for useful discussions and suggestions. This work was made possible by partial financial support from the following Brazilian government agencies: CNPq, CAPES, and FINEP. The numerical computations were performed in the NAUTILUS cluster of the Universidade Federal do Paraná.