^{1,a)}, G. Filatrella

^{2}, M. A. Aziz-Alaoui

^{3}and Hilda A. Cerdeira

^{4}

### Abstract

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.

The

*van der Pol*oscillator is a model of self-oscillating system that exhibits periodic oscillations. A modified version—essentially a higher order polynomial dissipation—has been proposed as a model equation for enzyme dynamics. This model is very interesting as a paradigm for birhythmicity, it contains multiple stable attractors with different natural frequencies, therefore, it can describe spontaneous switching from one attractor to another under the influence of noise. The noise induced transitions between different attractors depend upon the different stability properties of the attractors and are usually investigated by means of extensive

*Langevin*simulations. We show that the associated

*Fokker-Planck*equation, in the phase-amplitude approximation, is analytically solvable. The phase amplitude approximation requires a single frequency, and therefore fails when the two frequencies of the birhythmic system are significantly different. However, the approximation is not severe, for it explains the main features of the system when compared to the numerical simulations of the full model. The approximated

*Fokker-Planck*equation reveals the underlining structure of an effective potential that separates the different attractors with different frequency, thus explaining the remarkable differences of the stability between the coexisting attractors that give rise to birhythmicity. Moreover, it reveals that the noise can induce the stochastic suppression of the bifurcation that leads to birhythmicity. Finally, the approximated solution shows that the system is located with overwhelming probability in one attractor, thus being the dominant attractor. Which attractor is dominant depends upon the external control parameters. This is in agreement with the general expectation that in bistable systems the passage from an attractor to the other resembles phase transitions, since only in a very narrow interval of the external parameters it occurs in both directions with comparable probabilities.

R. Yamapi undertook this work with the support of the ICTP (International Centre for Theoretical Physics) in the framework of Training and Research in Italian Laboratories (TRIL) for AFRICA programme, Trieste, Italy and the CNPq-ProAfrica Project No. 490265/2010-3 (Brazil). He also acknowledges the hospitality of the Dipartimento di Fisica “E. R. Caianiello” of the Università di Salerno, Fisciano, Italy and the Institute of Theoretical Physics, UNESP, São Paulo, Brazil.

I. INTRODUCTION

II. THE BIRHYTHMIC PROPERTIES OF THE NOISY MODEL

A. The modified van der Pol oscillator with an additive noise

B. Birhythmic properties

III. ANALYTICAL ESTIMATES

IV. NUMERICAL SIMULATIONS AND RESULTS

V. CONCLUSIONS

### Key Topics

- Attractors
- 41.0
- Oscillators
- 20.0
- Probability theory
- 15.0
- Fokker Planck equation
- 14.0
- Langevin equation
- 7.0

## Figures

Parameters domain for the existence of a single limit cycle (white area) and three limit cycles (black area) solutions of Eq. (1) for μ = 0.1.

Parameters domain for the existence of a single limit cycle (white area) and three limit cycles (black area) solutions of Eq. (1) for μ = 0.1.

Effect of the noise intensity D on the boundary between the region of one and three limit-cycle solutions in the parametric (α,β)-plane of the Fokker-Planck Eq. (11) ) for μ = 0.1 as in Fig. 1 .

Behavior of the residence times in the parameter space. The solid line denotes the locus , while circles and crosses denote the situation where and , respectively. The dashed line is the border of existence of birhythmicity. The noise level is D = 0.1.

Behavior of the residence times in the parameter space. The solid line denotes the locus , while circles and crosses denote the situation where and , respectively. The dashed line is the border of existence of birhythmicity. The noise level is D = 0.1.

Residence times as a function of the parameter α for different values of the parameter β. The noise level is D = 0.1, the nonlinearity μ = 0.1.

Residence times as a function of the parameter α for different values of the parameter β. The noise level is D = 0.1, the nonlinearity μ = 0.1.

Residence times as a function of the parameter β for different values of the parameter α. The noise level is D = 0.1, the nonlinearity μ = 0.1.

Residence times as a function of the parameter β for different values of the parameter α. The noise level is D = 0.1, the nonlinearity μ = 0.1.

Residence times as a function of the noise intensity D for different values of the parameter α. The second dissipation parameter reads β = 0.0005, the nonlinearity μ = 0.1.

Residence times as a function of the noise intensity D for different values of the parameter α. The second dissipation parameter reads β = 0.0005, the nonlinearity μ = 0.1.

Asymmetric probability distributions for different values of the noise intensity D versus the amplitude A when the frequencies of both attractors are identical, i.e., . Parameters of the system are μ = 0.1 and α = 0.083, β = 0.0014.

Asymmetric probability distributions for different values of the noise intensity D versus the amplitude A when the frequencies of both attractors are identical, i.e., . Parameters of the system are μ = 0.1 and α = 0.083, β = 0.0014.

Probability distribution versus the amplitude A when the frequencies of the attractors are not identical i.e . Parameters of the system are D = 0.1, μ = 0.1, (i): α = 0.09, β = 0.0012, and (ii): α = 0.1, β = 0.014, .

Probability distribution versus the amplitude A when the frequencies of the attractors are not identical i.e . Parameters of the system are D = 0.1, μ = 0.1, (i): α = 0.09, β = 0.0012, and (ii): α = 0.1, β = 0.014, .

Variation of the amplitudes and the bandwidths versus the noise intensity D. Lines and symbols denote analytical and numerical results, respectively. The circles and dot-dashed lines refer to the inner attractor , solid lines and triangles to the outer attractor . The parameters used are μ = 0.1, α = 0.1, β = 0.002.

Variation of the amplitudes and the bandwidths versus the noise intensity D. Lines and symbols denote analytical and numerical results, respectively. The circles and dot-dashed lines refer to the inner attractor , solid lines and triangles to the outer attractor . The parameters used are μ = 0.1, α = 0.1, β = 0.002.

Variation of the amplitudes and the bandwidths versus the noise intensity D. Lines and symbols denote analytical and numerical results, respectively. The circles and dot-dashed lines refer to the inner attractor , solid lines and triangles to the outer attractor . The parameters used are μ = 0.1, α = 0.12, β = 0.003.

Variation of the amplitudes and the bandwidths versus the noise intensity D. Lines and symbols denote analytical and numerical results, respectively. The circles and dot-dashed lines refer to the inner attractor , solid lines and triangles to the outer attractor . The parameters used are μ = 0.1, α = 0.12, β = 0.003.

Behavior of energy barriers versus α. Solid lines denote the analytical results, while dashed lines with triangles denote numerical simulations. Parameters of the system are μ = 0.1 and β = 0.002.

Behavior of energy barriers versus α. Solid lines denote the analytical results, while dashed lines with triangles denote numerical simulations. Parameters of the system are μ = 0.1 and β = 0.002.

Behavior of energy barriers versus β. Solid lines denote the analytical results, while dashed lines with triangles denote numerical simulations. Parameters of the system are μ = 0.1 and α = 0.13.

Behavior of energy barriers versus β. Solid lines denote the analytical results, while dashed lines with triangles denote numerical simulations. Parameters of the system are μ = 0.1 and α = 0.13.

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