# 1D system birfurcation

[1]:

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.

[2]:

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


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

[3]:

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.

$\dot{x} = \mu+ \lambda x - x^3$
[4]:

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

[5]:

# 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)