Sim.DiffProc (version 2.5)

Svandp: Stochastic Van der Pol oscillator

Description

The stochastic Van Der Pol equation is used to model oscillators maintained. It is not linear and has no explicit solution, you can see from this simulation the stochastic Van der Pol equation, the temporal graph and the phase portrait.

Usage

Svandp(N, T, x0, v0, a, b, omega, sigma, Step = FALSE, Output = FALSE)

Arguments

N

size of process.

T

final time.

x0

Initial conditions, position (mm).

v0

Initial conditions, speed (mm/s).

a

reaction parameter (>= 0).

b

control parameter (> 0).

omega

Angular frequency (>= 0).

sigma

Dark random excitation (>= 0).

Step

if Step = TRUE ploting step by step.

Output

If Output = yes write a output to an Excel (.csv).

Value

• data.frame(time,X(t)), plot of process X(t) in the phase portrait (2D) and temporal evolution of stochastic van der Pol equation.

Details

The stochastic equation of Van Der pol, is used to model maintained oscillators. It is not linear and does not have an explicit solution. In this simulation makes it possible to vary these parameters (cursors), as well as the initial conditions x0 and v0 (ringed points). Stochastic perturbations of the Van Der pol equation, and random excitations force of such systems by White noise e(t), with delta-type correlation functions E(e(t)e(t+h))=sigma*deltat(h): $$x'' + a * x' * ( x^2 /b - 1 ) + omega^2 * x = e(t)$$ where a,omega,sigma >= 0 and b > 0.

• Influence initial conditions: the oscillations occur even with low values, then are stabilized. The portrait of phase shows a limiting cycle, which does not depend on the initial conditions.
• Influence reaction: whena=0one obtains the stochastic harmonic oscillatorSharosc; the amplitude of the oscillations depends on the initial conditions. By increasingaone notes an increasingly important deformation of the oscillations and portrait of phase.
• Influence control: the coefficientbdetermines the amplitude of the oscillations: when|x|, the reaction is positive and the amplitude increases. When|x|>bit is the reverse which occurs. The amplitude is stabilized around2b.

References

Fima C Klebaner. Introduction to stochastic calculus with application (Second Edition), Imperial College Press (ICP), 2005.

See Also

Spendu stochastic pendulum, Sharosc stochastic harmonic oscillator, Srayle stochastic Rayleigh oscillator, SSCPP stochastic system with a cylindric phase plane, Sosadd stochastic oscillator with additive noise.

Examples

## a = 0, b = 0.3, omega= 2.5, sigma=0.1
 Svandp(N=10000, T=100, x0=1, v0=0, a=0, b=0.3, omega=2.5, sigma=0.1)
## a = 3
 Svandp(N=10000, T=100, x0=1, v0=0, a=3, b=0.3, omega=2.5, sigma=0.1)