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

Definition¶

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

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

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

import tsdynamics as ts

system = ts.systems.DoubleWell()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

Kramers (1940), Physica 7, 284-304
doi:10.1016/S0031-8914(40)90098-2

BibTeX

@misc{doublewell,
  title = {DoubleWell system},
  note = {Kramers (1940), Physica 7, 284-304},
  doi = {10.1016/S0031-8914(40)90098-2}
}