We analyze global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit cycles variation of the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients α and β. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time τ from one limit-cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation of the escape time τ versus the inverse noise-intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters. Some models employed to describe natural systems, such as for instance glycolysis reactions and circadian proteins rhythmics, exhibit spontaneous oscillations at two distinct frequencies. The phenomenon is known as birhythmicity, and the underlying dynamical structure is characterized by the coexistence of two stable attractors, each displaying a different frequency. Being the attractors locally stable, the system would however stay at a single frequency, the one selected by the choice of the initial conditions, unless an external source disturbs the evolution and causes a switch to the other attractor. To investigate such process, we have focused on a particular system of biological interest, a modified van der Pol oscillator (that displays birhythmicity), to determine the global stability properties of the attractors under the influence of noise. More specifically, we have characterized the stability of the attractors with the escape times, or the average time that the system requires to switch from an attractor to the other under the influence of random fluctuations. Such analysis reveals that the two attractors can possess very different properties, with very different relative residence times. Even excluding the most asymmetric cases, the system can spend something like 10 years on one attractor for each second spent on the other. We conclude that although a system can be structurally biorhythmic for the contemporary presence of two locally stable attractors at two different frequencies, actual switch from one frequency to the other could be very difficult to observe. A global stability analysis can therefore help to determine the region of the parameter space in which birhythmic behavior will be genuinely observed.

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We analyze the stability and optimization of the synchronization process between two coupled self-excited systems modeled by the multi-limit cycles van der Pol oscillators through the case of an enzymatic substrate reaction with ferroelectric behavior in brain waves model. The one-way and two-way couplings synchronization are considered. The stability boundaries and expressions of the synchronization time are obtained using the properties of the Hill equation. Numerical simulations validate and complement the results of analytical investigations.

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We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated with switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise, the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.

We study the effect of noise on a Josephson junction that, coupled to a linear RLC resonator, can oscillate at two frequencies. To establish the global stability of the attractors, we estimate the position of the separatrix, essential information to establish the stability of the attractor for this multidimensional system, from the analysis of the mean first passage time. We find that the frequency locked to the resonator is most stable at low bias and less stable at high bias, where the resonator exhibits the largest oscillations. The change in the birhythmic region is dramatic for the effective barrier changes of an order of magnitude and the corresponding lifetime of about seven decades.

This paper deals with the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The amplitude of the oscillatory states both in the autonomous and nonautonomous case are derived. The interaction between the amplitudes of the external excitation and the limit-cycles are also analyzed. The domain of the admissible values on the amplitude for the external excitation is found. The effects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model are analyzed and it appears that with the appropriate selection of the coupling parameter, the quenching of chaotic vibrations takes place.