Noise-induced oscillations in system near Hopf bifurcation.

Van der Pol oscillator is a paradigmatic model of the system with a potential to demonstrate non-damped self-sustained oscillations which is realized at positive values of its single critical parameter epsilon (see equations below). If this parameter is negative, the system does not self-oscillate and its attractor is a fixed point. The transition between the fixed point (no self-oscillations) and periodic self-oscillations occurs through Hopf bifurcation at zero value of epsilon. In this Section we will be interested in Van der Pol system at negative values of epsilon, i.e. below Hopf bifurcation. This means that we consider the system with the potential to self-oscillate that is not realized.

Proximity to an instability (bifurcation) is a typical feature of e.g. sensory systems in living organisms. In particular, we are able to hear in many respects due to the presence in our inner ear of special sensor cells, which transform acoustic stimuli into an electrical signal that afterwards goes to the brain. It is shown that the dynamics of such sensors is characterized by the proximity to the Hopf bifurcation (see more details here). As a simplified model of sound detection one can consider the following stochastic Van der Pol equations, where random fluctuations xi(t) might represent a complex sound:

Pictures are taken from the home page of Dr. A. J. Aranyosi

Displacement of a hair bundle due to the sound stimuli

Sensor cell of inner ear (hair cell)

Here the parameter epsilon is a critical parameter, and omega₀ defines the frequency of oscillations. Let omega₀=1 and epsilon =-0.01. At this parameters the system is just below the Hopf bifurcation, and its only stable stationary solution is a fixed point. Addition of noise can make the system fluctuate with the frequency close to omega_0. Let us assume noise xi(t) to be a Gaussian white noise with zero mean and variance equal to 1, whose intensity is defined by the parameter D.

The regularity of noise-induced oscillations in Van der Pol system falls as noise intensity increases. With this, the period of oscillations is almost unchanged. One can chzaracterize the regularity of oscillations for example by correlation time t_corr that is introduced as follows:

Here, Psi(tau) is autocorrelation function, and angular brackets denote average over time.

Figure shows the correlation time of noise-induced oscillations vs noise intensity D.