Epsilon Continuation

Reconstruct stable orbits by stepping system parameters and following continuation paths.

The Epsilon Continuation workflow is a numerical method to track limit cycles and attractors under slow parameter variations. By introducing a scaling factor $\epsilon \in [0, 1]$, the workflow starts from a known harmonic seed (computed via describing-function methods) and gradually steps the system parameters towards the full chaotic regime, using the final state of each step as the initial state of the next.

Concept & Methodology

For a parameter continuation path from $\epsilon_{\rm start}$ to $\epsilon_{\rm end}$:

1. Initial Seed: Start with a high-precision limit cycle initial condition at a parameter state where the orbit is stable or analytically known.
2. Step Epsilon: Slightly increase the continuation parameter: $\epsilon_{k+1} = \epsilon_k + \Delta\epsilon$
3. Transient Burn: Integrate the system for a transient duration ($t_{\rm burn}$) to let the trajectory settle on the shifted attractor.
4. Capture State: Save the final state $x(t_{\rm final})$ to serve as the initial seed $(x_0)$ for the next parameter step.
5. Repeat: Continue until $\epsilon = 1.0$ is reached, mapping out the continuation branch.

This is highly effective for locating hidden attractors because it avoids manual coordinate searching and follows the natural continuation curves of the dynamical system.

Usage

Epsilon continuation is natively used inside the Lur’e integer continuation workflow:

from hidden_attractors.workflows.integer_lure import continue_integer_lure_seed

# Step system parameters systematically
steps = continue_integer_lure_seed(
    system,
    initial_seed,
    eps_values=(0.20, 0.40, 0.60, 0.80, 1.0),
    t_transient=20.0,
    t_keep=10.0,
    h=0.01
)