1D system birfurcation

import brainpy as bp

Codimension1

Here we will show the birfurcation analysis of 1D system with dummy test neuronal model.

\[\dot{x} = x^3-x + I\]

First, let’s define the model.

@bp.odeint
def int_x(x, t, Iext):
    dx = x ** 3 - x + Iext
    return dx

Then, create a bifurcation analyzer with bp.symbolic.Bifurcation.

an = bp.symbolic.Bifurcation(
    int_x,
    target_pars={'Iext': [-0.5, 0.5]},
    target_vars={"x": [-2, 2]},
    numerical_resolution=0.0001)

_ = an.plot_bifurcation(show=True)

Codimension2

Here we define the following 1D model for codimension 2 bifurcation testing.

\[\dot{x} = \mu+ \lambda x - x^3\]

@bp.odeint
def int_x(x, t, mu, lambda_):
    dxdt = mu + lambda_ * x - x ** 3
    return dxdt

# please install numba!=0.54.x, because they have bugs

analyzer = bp.symbolic.Bifurcation(
    int_x,
    target_pars={'mu': [-4, 4], 'lambda_': [-1, 4]},
    target_vars={'x': [-3, 3]},
    numerical_resolution=0.1)
_ = analyzer.plot_bifurcation(show=True)