Theory

The equations behind each method, with the notation used throughout the package: a scalar observable \(x(t)\) sampled at \(\Delta t\) (so \(x_i = x(t_i)\), \(N\) samples), a model state \(\psi \in \mathbb{R}^{N_\phi}\), and delay \(\zeta\) expressed in samples.

At a glance

Section	What it covers
Phase-space reconstruction	Delay embedding (Takens) and its two free parameters — AMI lag, FNN dimension — plus classical MDS from the full state trajectory.
Signal and attractor diagnostics	PSD peaks, first-return map and peak clusters, recurrence plot, Poincaré section, and the correlation dimension \(D_2\).
Lyapunov exponents	Benettin QR spectrum, perturbation-growth leading exponent, the data-only Rosenstein estimator, and the Kaplan–Yorke dimension.
Regime classification	The decision tree, the Lorenz-96 F-route bifurcation diagram, measured \(\lambda_1\) tables, defaults, and notes.

The Lorenz96 Ruelle–Takens–Newhouse route ties the four together in the worked example.

References

• Takens (1981). Detecting strange attractors in turbulence. Lecture Notes in Math. 898.
• Fraser & Swinney (1986). Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134.
• Kennel, Brown & Abarbanel (1992). Determining embedding dimension for phase-space reconstruction. Phys. Rev. A 45, 3403.
• Eckmann, Kamphorst & Ruelle (1987). Recurrence plots of dynamical systems. Europhys. Lett. 4, 973.
• Grassberger & Procaccia (1983). Characterization of strange attractors. Phys. Rev. Lett. 50, 346.
• Benettin, Galgani, Giorgilli & Strelcyn (1980). Lyapunov characteristic exponents for smooth dynamical systems. Meccanica 15, 9.
• Rosenstein, Collins & De Luca (1993). A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65, 117.
• Kaplan & Yorke (1979). Chaotic behavior of multidimensional difference equations. Lecture Notes in Math. 730.
• Kantz & Schreiber (2004). Nonlinear Time Series Analysis, 2nd ed., Cambridge University Press.