Sim.DiffProc (version 2.5)

Srayle: Stochastic Rayleigh oscillator

Description

The stochastic Rayleigh oscillator is much like the stochastic van Der Pol oscillator save one key difference: as voltage increases, the van Der Pol oscillator increases in frequency while the Rayleigh oscillator increases in amplitude. You can see from this simulation the stochastic Rayleigh oscillator, the temporal graph and the phase portrait.

Usage

Srayle(N, T, x0, v0, a, 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

Amortization (1/s).

omega

Angular frequency (rad/s).

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 Rayleigh equation.

Details

Here is the second order differential equation for the stochastic Rayleigh oscillator : $$x'' - a * ( 1 - x'^2 ) * x' + omega^2 * x = e(t)$$ where a,omega > 0 and sigma >= 0. Like the stochastic van Der Pol oscillator Svandp, omega controls how much voltage is injected into the system. a controls the way in which voltage flows through the system.

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, SSCPP stochastic system with a cylindric phase plane, Sosadd stochastic oscillator with additive noise.

Examples

## a= 4 , omega= 1, sigma =0.1
Srayle(N=5000, T=50, x0=3, v0=0, a=4, omega=1, sigma=0.1)