Module dynamical_systems 

Dynamical systems analysis: ODE fixed points, Lyapunov exponents, bifurcations, phase plane, Poincaré sections, strange attractors, chaos indicators (RQA), Hamiltonian systems, KAM tori, and symplectic integration.

All arithmetic uses plain f64 and [f64; 3] arrays — no nalgebra.

Structs

BifurcationEvent

Result of a bifurcation scan along one parameter.

Brusselator

Brusselator chemical oscillator.

DoublePendulum

Double pendulum with equal masses m=1 and lengths l=1.

DuffingOscillator

Duffing oscillator:

FixedPointResult

Result of a fixed-point search.

HamiltonianOde

Wrap a HamiltonianSystem as a plain OdeSystem.

HarmonicOscillatorHam

1-DOF simple harmonic oscillator: H = p²/2 + ω²q²/2.

KuramotoModel

Kuramoto model of n coupled phase oscillators.

LimitCycleEstimate

Result of a limit-cycle period estimate.

LorenzSystem

The Lorenz strange attractor.

PoincareCrossing

Record of a single Poincaré crossing.

PoincareSectionConfig

Configuration for a Poincaré section.

RosslerSystem

The Rössler system, a simpler strange attractor.

RqaParams

Parameters for RQA computation.

RqaResult

Results of a recurrence quantification analysis.

VanDerPol

Van der Pol oscillator:

Enums

BifurcationType

Detected bifurcation type.

Traits

HamiltonianSystem

A Hamiltonian system with n degrees of freedom.

OdeSystem

A continuous-time ODE system ẋ = f(t, x).

Functions

add3

Add two 3-vectors.

cross3

Cross product of two 3-vectors.

delay_embed

Build the delay-embedded trajectory from a scalar time series.

detect_heteroclinic

Check whether two fixed points are connected by a heteroclinic orbit.

dot3

Dot product of two 3-vectors.

eigen2

Eigenvalues of a real 2×2 matrix \[\[a,b\\],\[c,d\]].

energy_drift

Monitor energy conservation in a Hamiltonian trajectory.

estimate_limit_cycle_period

Estimate the period of a limit cycle via successive Poincaré crossings.

estimate_winding_number

Estimate the winding number (frequency ratio) of a 2-DOF Hamiltonian orbit on a KAM torus using the mean frequency ratio from angle variables.

find_fixed_point

Find a fixed point of sys.rhs(0, x) = 0 using Newton–Raphson iteration starting from x0.

forest_ruth_step

4th-order symplectic Forest–Ruth integrator.

integrate_rk4

Integrate an ODE from t0 to t_end with fixed step h.

integrate_symplectic

Integrate a Hamiltonian system using the Störmer–Verlet method.

is_hopf_bifurcation

Check whether a 2-D system near a fixed point undergoes a Hopf bifurcation at the given Jacobian parameters.

kaplan_yorke_dim

Estimate the Kaplan–Yorke dimension from sorted Lyapunov exponents.

lyapunov_exponents_qr

Compute the n Lyapunov exponents of an n-dimensional ODE system using the QR-decomposition (Gram–Schmidt) method.

maximal_lyapunov_exponent

Compute the maximal Lyapunov exponent (MLE) using the Benettin algorithm.

norm3

Euclidean norm of a 3-vector.

numerical_jacobian

Compute the Jacobian matrix of sys.rhs at (t, x) numerically using central differences with step eps.

phase_plane_grid

Compute a vector field on a 2-D grid for phase-plane visualisation.

pitchfork_fixed_points

Analyse a 1-D pitchfork bifurcation (supercritical) ẋ = μ x − x³.

poincare_section

Compute a Poincaré section by integrating sys and recording crossings.

recurrence_matrix

Compute the recurrence matrix (as a flat bool vector, row-major).

rk4_step

Classic 4th-order Runge–Kutta step.

rqa

Perform full RQA on a scalar time series.

saddle_node_fixed_points

Analyse a 1-D saddle-node bifurcation ẋ = μ + x².

scale3

Scale a 3-vector by a scalar.

stability_2d

Stability classification of a 2-D fixed point given the Jacobian entries.

stormer_verlet_step

One Störmer–Verlet (leapfrog) step for a separable Hamiltonian H = T(p) + V(q).

sub3

Subtract two 3-vectors.