lyapunov

experimental.network_dynamics.analysis.lyapunov

Functions

Name	Description
lyapunov	Estimate the maximum Lyapunov exponent for a network.
lyapunov_spectrum	Estimate the Lyapunov spectrum of a network.

lyapunov

experimental.network_dynamics.analysis.lyapunov.lyapunov(
    network,
    solver=None,
    t=1000.0,
    n=10,
    d0=1e-09,
    dt=0.1,
    t0=0.0,
)

Estimate the maximum Lyapunov exponent for a network.

Uses Benettin’s rescaling algorithm: two nearby trajectories are simulated for n segments of length t ms; their divergence rate is averaged to estimate the MLE.

Parameters

Name	Type	Description	Default
network	Network		required
solver	solver instance, optional (default: Heun())		None
t	float — segment duration in ms		1000.0
n	int — number of rescaling steps		10
d0	float — initial perturbation magnitude		1e-09
dt	float — integration timestep in ms		0.1
t0	float — simulation start time		0.0

Returns

Name	Type	Description
	float — estimated maximum Lyapunov exponent (1/ms)

Notes

For delayed coupling the history buffer is held fixed across segments (each rescaling step re-seeds only the point state), so the result is a good approximation only when t >> max_delay, not exact. The two-trajectory method is the natural route to a delay-correct MLE – each trajectory evolves its own history, so no explicit history-space tangents are needed – but that needs carrying the full final state (point + history) across segments, or running unsegmented. Run a warmup via network.update_history(result) first to initialise the delay buffer from a settled trajectory.

lyapunov_spectrum

experimental.network_dynamics.analysis.lyapunov.lyapunov_spectrum(
    network,
    solver=None,
    t=1000.0,
    n=10,
    k=None,
    dt=0.1,
    t0=0.0,
    mode='jvp',
    d0=1e-09,
)

Estimate the Lyapunov spectrum of a network.

Parameters

Name	Type	Description	Default
network	Network		required
solver	solver instance, optional (default: Heun())		None
t	float — segment duration in ms		1000.0
n	int — number of rescaling steps		10
k	int — number of exponents to compute (default: all D)		None
dt	float — integration timestep in ms		0.1
t0	float — simulation start time		0.0
mode	str — "jvp" (default) uses tangent-space propagation via	jax.linearize; exact and efficient for differentiable systems. “sim” uses finite-difference with d0-scaled perturbations; works for non-differentiable systems.	'jvp'
d0	float — perturbation magnitude (only used when mode="sim")		1e-09

Returns

Name	Type	Description
	jnp.ndarray — top k Lyapunov exponents sorted descending (1/ms)

Notes

Exact for instantaneous (non-delayed) coupling only. Both modes propagate a point state (perturb/linearize only initial_state.dynamics) and reset the delay history buffer each rescaling step, so for delayed coupling the history is held fixed across segments: a good approximation only when t >> max_delay, not the true DDE spectrum (which would need tangents spanning the augmented history-buffer state).