Sim.DiffProc (version 2.5)

Sosadd: Stochastic oscillator with additive noise

Description

You can see from this simulation the stochastic oscillator with additive noise and the temporal graph and the phase portrait, and 3D plot for Fokker-Planck equation.

Usage

Sosadd(N, T, x0, v0, a, omega, sigma, K0 = 1, Step = FALSE, Output = FALSE)

Arguments

N

size of process.

T

final time.

x0

Initial conditions, position.

v0

Initial conditions, speed.

a

Constant (>= 0).

omega

Angular frequency (>= 0).

sigma

Dark random excitation (>= 0).

K0

Constant for Fokker-Planck equation (K0 > 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 oscillator with additive noise. 3D plot for Fokker-Planck equation.

Details

Stochastic perturbations of oscillator with additive noise, 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): $$x'' - a * (1 - x^2 - x'^2) * x' + omega^2 * x = e(t)$$ where a,omega,sigma >= 0. The Fokker-Planck equation of this system: $$P(s,x,t,y) = exp( - a * (x^2 + y^2 )^2 / (2*pi*K0) )$$

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, Svandp stochastic Van der Pol oscillator, Srayle stochastic Rayleigh oscillator, SSCPP Stochastic system with a cylindric phase plane.

Examples

## a = 0.1, omega= 1, sigma = 0.2 ,K0 = 0.5.
 Sosadd(N=5000, T=50, x0=3, v0=0, a=0.1, omega=1, sigma=0.2, K0 = 0.5)
## a = 3
 Sosadd(N=5000, T=50, x0=3, v0=0, a=3, omega=1, sigma=0.2, K0 = 0.5)