2D bifurcation for 3+ D dynamical systems

[nosratullah]

Hi, Thanks for the awesome software. I was wondering if there's a way to do 2D bifurcation on more than 2D dynamical systems in your software.

Please:

• [x] Check for duplicate requests. I checked that there's an example for FastSlow2D systems which indeed it provided bifurcation analyzing for 3D system, but the problem is that it's still reduces the dimension of the system to 2D (probably with time-scale separation methods). And more importantly, (I think) it only allows you to set one of the variable (dimensions) as the control parameter which is not desired.
• [x] Describe your goal, and if possible provide a code snippet with a motivating example.

  
  @bp.odeint
  def int_x(x, t, y, z, Iext, a):
  return F(x,y,z, Iext, a)

@bp.odeint def int_y(y, t, x, z, Iext, a): return G(x,y,z, Iext, a) @bp.odeint def int_z(x, t, x, y, Iext, a): return H(x,y,z, Iext, a)

analyzer = bp.analysis.Bifurcation2D( model = [int_x, int_y, int_z], target_vars=dict(x=[-3, 3], y=[-3., 3.], z=[-3, 3]), target_pars=dict(a=[0.5, 1.], Iext=[0., 1.]), resolutions={'a': 0.01, 'Iext': 0.01}, ) analyzer.plot_bifurcation(num_rank=10, tol_aux=1e-9) analyzer.show_figure()

Another example which I guess it's more common to use is analyzing the 2D bifurcation diagram of 4D Hodgkin-Huxley model . Maybe with this example I can rephrase my question which is can we perform low-dimensional bifurcation analysis on high-dimensional dynamical system with your software?
Many thanks.

[chaoming0625]

This is a good question. I will show you inspirational examples, and their limitations.

[nosratullah]

Great! Maybe a good example is provided in this link with PyDSTool. It is a 4D predator-prey system with 2D bifurcation analysis.

[chaoming0625]

Currently, brainpy does not implement the continuation analysis. However, we can do bifurcation analysis using fast-slow bifurcation analysis: fixing one variable first, then do bifurcation analysis of two remaining variables. For your cases, we can use:

@bp.odeint
def int_x(x, t, y, z, Iext, a):
    return F(x,y,z, Iext, a)

@bp.odeint
def int_y(y, t, x, z, Iext, a):
    return G(x,y,z, Iext, a)
@bp.odeint
def int_z(x, t, x, y, Iext, a):
    return H(x,y,z, Iext, a)

analyzer = bp.analysis.FastSlow2D(
  [int_x, int_y, int_z],
  fast_vars={'x': [-3, 3], 'y': [-3., 3.]},
  slow_vars={'z': [-3., 3.]},
  pars_update={'a': 0.5, 'Iext': 0.5},
  resolutions={'a': 0.01, 'Iext': 0.01}
)
analyzer.plot_bifurcation(num_rank=20)
analyzer.show_figure()

Hoping this message can help you.