pycont-lite 0.2.1

Lightweight matrix-free arclength continuation with bifurcation detection and branch switching.

PyCont-Lite

PyCont-Lite is a lightweight, matrix-free arclength continuation library for solving nonlinear parametric systems

$$ G(u, p) = 0 $$

with automatic bifurcation detection, branch switching, and stability analysis.

Key Features

• Matrix-free: only need to implement $G(u, p)$, no Jacobians required.
• Adaptive continuation: robust predictor–corrector with adjustable step sizes.
• Bifurcation detection: automatically localizes and classifies folds and branch points.
• Branch switching: continues all discovered branches recursively.
• Stability analysis: computes leading eigenvalues to distinguish stable vs. unstable segments.
• Lightweight plotting: quickly visualize bifurcation diagrams with plotBifurcationDiagram. Plots automatically distinguish stable vs. unstable branches.
• Structured output: branches (with stability) and events (folds, bifurcations) are returned for further analysis.”

Installation & Requirements

PyCont-Lite requires only:

• numpy
• scipy
• matplotlib

Install via PyPI (pip):

pip install pycont-lite

Quick Examples

Pitchfork Bifurcation

import numpy as np
import pycont

# Define the pitchfork function
def G(u, p):
    return u**3 - p*u

# Initial guess
u0 = np.array([-3.0])
p0 = 0.0

# Run continuation
ds_max = 0.01
ds_min = 1.e-6
ds = 0.001
n_steps = 1000
solver_parameters = {"tolerance": 1e-10}
continuation_result = pycont.arclengthContinuation(G, u0, p0, ds_min, ds_max, ds, n_steps, solver_parameters=solver_parameters)

# Plot the solution curve
pycont.plotBifurcationDiagram(continuation_result)

Fold Bifurcation

import numpy as np
import pycont

# Define the fold function
G = lambda x, r: r + x**2

# Initial guess
u0 = np.array([-5.0])
p0 = -u0[0]**2

# Run continuation
ds_max = 0.01
ds_min = 1.e-6
ds = 0.001
n_steps = 5000
solver_parameters = {"tolerance": 1e-10}
continuation_result = pycont.arclengthContinuation(G, u0, p0, ds_min, ds_max, ds, n_steps, solver_parameters=solver_parameters)

# Plot the curves
pycont.plotBifurcationDiagram(continuation_result)

Bratu: PDE Example

The nonlinear boundary value problem

$$ u_{xx} + \lambda e^u = 0, \quad u(0)=u(1)=0 $$

discretized with finite differences:

import numpy as np
import pycont

N = 101 # total number of points
x = np.linspace(0.0, 1.0, N)
dx = x[1] - x[0]

def G(u: np.ndarray, lam: float) -> np.ndarray:
    u_full = np.zeros(N, dtype=float)
    u_full[1:-1] = u
    
    u_xx = (u_full[:-2] - 2.0 * u_full[1:-1] + u_full[2:]) / (dx * dx)
    r = u_xx + lam * np.exp(u_full[1:-1])
    return r

# We know that u = 0 for lambda = 0 - otherwise we must solve G(u, lambda0) = 0.
lam0 = 0.0
u0 = np.zeros(N-2)

# Do continuation
ds_max = 0.01
ds_min = 1e-6
ds0 = 1e-4
n_steps = 2000
solver_parameters = {"tolerance": 1e-10}
continuation_result = pycont.arclengthContinuation(G, u0, lam0, ds_min, ds_max, ds0, n_steps, solver_parameters=solver_parameters)

# Plot the bifurcation diagram (lambda, max(u))
u_transform = lambda u : np.sign(u[50]) * np.max(np.abs(u))
pycont.plotBifurcationDiagram(continuation_result, u_transform=u_transform, p_label=r'$\lambda$', u_label=r'$\text{sign}(u) ||u||_{\infty}$')

This produces the classical S-shaped bifurcation curve with a fold near $\lambda \approx 3.51$.

Plotting Bifurcation Diagrams

PyCont-Lite includes a helper function plotBifurcationDiagram for quick visualization. Stable segments are shown as solid lines and unstable segments as dashed lines, just like in AUTO/MATCONT. By default, it plots the parameter value on the horizontal axis and the transformed state variable on the vertical axis.

import pycont

# After running continuation...
pycont.plotBifurcationDiagram(continuation_result)

For multi-dimensional systems, you can specify how to reduce $u$ to a single scalar for plotting via the u_transform argument.

• Default behavior:
  • If $u$ has dimension 1 → plot $u$ directly.
  • If $u$ has dimension >1 → plot the Euclidean norm $||u||$.
• Custom transform (example: plot the maximum component of $u$):

pycont.plotBifurcationDiagram(result, u_transform=lambda u: u.max())

Solver Parameters

You can fine-tune the solver by passing a dictionary:

solver_parameters = {
    "rdiff": 1e-8,              # finite-difference step
    "nk_maxiter": 20,           # Newton-Krylov iterations
    "tolerance": 1e-10,         # nonlinear tolerance
    "bifurcation_detection": True,
    "analyze_stability": True,  # compute leading eigenvalue
    "initial_directions": "both"  # 'both', 'increase_p', 'decrease_p'
}

Output Format

arclengthContinuation returns a ContinuationResult object with:

• branches: list of Branch objects (u_path, p_path, stability flag, etc.)
• events: list of Event objects (start points, folds, bifurcations)

This makes it easy to explore and plot bifurcation diagrams programmatically.

Planned Features

The following features are under active consideration for future releases:

• Hopf bifurcation detection
• Limit cycle continuation
• Improved tangent computation for high-dimensional PDE systems

License

This project is licensed under the MIT License — see the LICENSE file for details.

Acknowledgement

I started this project because most continuation software is either legacy Fortran (AUTO), Matlab-only (MATCONT, COCO), or heavyweight. PyCont-Lite is meant to be modern, lightweight, and useful both for industry and academia!

For feature requests or contributions, feel free to open an issue or reach out: hannesvdc[at]gmail[dot]com.