Generalized Van Der Pol Oscillator Research Articles

AN ANALYTIC PICTURE OF NEURON OSCILLATIONS

Current induced oscillations of a space clamped neuron action potential demonstrates a bifurcation scenario originally encapsulated by the four-dimensional Hodgkin–Huxley equations. These oscillations were subsequently described by the two-dimensional FitzHugh–Nagumo Equations in close agreement with the Hodgkin–Huxley theory. It is shown that the FitzHugh–Nagumo equations can to close approximation be reduced to a generalized van der Pol oscillator externally driven by the current. The current functions as an external constant force driving the action potential. As a consequence approximate analytic expressions are derived which predict the bifurcation scenario, the amplitudes of the oscillations and the oscillation periods in terms of the current and the physiological constants of the FitzHugh–Nagumo model. A second reduction permits explicit analytic solution and results in a spiking model which can be multiply coupled and extended to include the dynamics of phase locking, entrainment and chaos characteristic of time-dependent synaptic inputs.

Strongly non-linear oscillators with slowly varying parameters

A multiple scales method, which gives the approximate solution in terms of elliptic functions, is used for the study of strongly non-linear oscillators with slowly varying parameters. As an application, quadratic and cubic non-linear oscillators are studied in detail. Two examples are considered: a generalized Van der Pol oscillator and a pendulum with variable length. Comparisons are also made with numerical results to show the efficiency of the present method.

Stochastic analysis of a noisy oscillator

The paper deals with the stability behavior of a noisy generalized Van der Pol oscillator on the basis of complex stochastic averaging technique.

On the stationary distribution of self-sustained oscillators around bifurcation points

A double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones. Systems undergoing Hopf bifurcations belong to this class. As an illustrative example, the generalized van der Pol oscillator is considered around its bifurcation point. A calculation is carried out up to third order in both the noise intensity and the bifurcation parameter (damping coefficient).

Theory of noise in semiconductor laser emission

For the complex light field amplitude of a semiconductor laser an equation of a generalized van der Pol oscillator with a fluctuating driving term is derived from first principles. This equation is shown to be valid for optical band-to-band transitions with and withoutk-selection rule. Neglecting the nonlinearity in the saturation higher than of second order and also the intensity dependence of the dispersion this equation reduces to the standard van der Pol equation with a noisy driving term. From the general equation the linewidth and the noise of the intensity of the laser emission is calculated above and below threshold. The results are in agreement with the experimental data ofArmstrong andSmith.