Multi-rhythmic oscillations and correlated noise effects of a self-sustaining biological system

In this paper, we study a high-order polynomial variation of the van der Pol oscillator interpreted as a self-sustaining biological system driven by correlated noise. This particular model gives rise to multi-rhythmic oscillations for which it is possible to find an approximate expression of the amplitudes and frequencies of the oscillatory states. Depending on the physical parameters, it is possible through bifurcation diagrams to those for which the system exhibits monorhythmicity, birhythmicity, or even trirhythmicity with different amplitudes and frequencies of the oscillatory states. However, the phase portraits are used to determine the stable and unstable amplitudes of oscillatory states. Correlated noise induces escape of the system from one attractor to another and influences the depth of the effective potential wells associated with the corresponding Langevin equation. The Fokker–Planck equation allows one to obtain analytically the probability density function which, depending on the values of the noise amplitude and the correlation time, presents drastic changes in the probability distribution function. Indeed, by varying the amplitude of the noise and the correlation time when system changes from three limit cycles to five limit cycles, correspondingly, the probability density function changes from two peaks to three peaks. The comparison between analytical and numerical results is acceptable. The Lyapunov exponent of the noisy system signals the appropriate noise amplitude for which the system is in a stable or in chaotic states.