This is a toy stochastic gradient system which can have bistability in some conditions. Model specification: $$U = x^4 + y^4 + axy + bx + cy$$ $$dx/dt = - \partial U/ \partial x + \sigma dW/dt = - 4x^3 - ay - b + \sigma dW/dt$$ $$dy/dt = - \partial U/ \partial y + \sigma dW/dt = - 4y^3 - ax - c + \sigma dW/dt$$

sim_fun_grad(
  initial = list(x = 0, y = 0),
  parameter = list(a = -4, b = 0, c = 0, sigmasq = 1),
  length = 1e+05,
  stepsize = 0.01,
  seed = NULL
)

Arguments

initial, parameter

Two sets of parameters. initial contains the initial value of x and y; parameter contains a,b,c, which control the shape of the potential landscape, and sigmasq, which is the square of \(\sigma\) and controls the amplitude of noise.

length

The length of simulation.

stepsize

The step size used in the Euler method.

seed

The initial seed that will be passed to set.seed() function.

Value

A matrix of simulation results.