DoubleWell

Systems / SDEs / Noise-driven systems

A particle in a bistable potential driven by noise — noise-induced switching between the two wells.

stochastic · SDE1 dimensionswitchingbistable

sample path

Definition

\[ \begin{aligned} dx &= \left(a x - b x^{3}\right)\,dt + \left(\sigma\right)\,dW_{x} \end{aligned} \]

Parameters

Symbol	Default	Role
a	1	curvature of the potential at the origin (barrier / well locations)
b	1	quartic stiffness confining the wells
sigma	0.5	noise amplitude (sets the escape rate)

State variables: x

Properties

Lyapunov spectrum

TODO — SDE Lyapunov not computed at build time

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a - 3 b x^{2}$
state-dependent

Equilibria

TODO — equilibria of a stochastic system not enumerated

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.DoubleWell()
traj = sys.integrate(final_time=100.0, dt=0.01)

Reference

Kramers (1940), Physica 7, 284-304

BibTeX

@misc{doublewell,
  title = {DoubleWell system},
  note = {Kramers (1940), Physica 7, 284-304}
}