SSCPP: Stochastic system with a cylindric phase plane
Description
You can see from this simulation the stochastic system with a cylindric phase plane and the temporal graph and the phase portrait, and 3D plot for Fokker-Planck equation.
Initial conditions, position (rad), -pi < theta0 < pi.
theta1
Initial conditions, speed (rad/s).
a
Amortization (>= 0).
b
Constant (>= 0).
omega
Angular frequency (>= 0).
sigma
Dark random excitation (>= 0).
K0
Constant for Fokker-Planck equation (K0 > 0).
Prd
Period for plot 3D (Prd > 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 Rayleigh equation. 3D plot for Fokker-Planck equation.
Details
Stochastic perturbations of the system with a cylindric phase plane 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): $$x'' + a * x' + b +omega^2 * sin(x) = e(t)$$
where a,b,omega,sigma >= 0.
The Fokker-Planck equation of this system:
$$P(s,x,t,y) = exp( -a * (y^2 + 2 * b *x - 2 * omega^2 * cos(x)) / (2*pi*K0) )$$
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, Sosadd stochastic oscillator with additive noise.