Maximal Lyapunov Exponent (MLE) in Dynamics

• Maximal Lyapunov Exponent (MLE) is a measure that quantifies the average exponential divergence of nearby trajectories, serving as a clear indicator of chaotic behavior.
• MLE aids in constructing dynamical phase diagrams by classifying systems into stable, marginal, or chaotic regimes through precise numerical techniques like the Benettin–QR algorithm.
• In classical–quantum correspondence, MLE determines the time window for agreement between quantum and mean-field trajectories, marking the onset of quantum chaos.

The maximal Lyapunov exponent (MLE) quantifies the exponential rate at which infinitesimal perturbations to a dynamical trajectory grow or decay under evolution. Its sign and magnitude provide a rigorous criterion for detecting chaos in both classical and quantum dynamical systems. In periodically driven, many-body, and open systems—as typified by macrospin ensembles subject to time-dependent fields—the MLE characterizes the transition from regular to chaotic behavior, the structure of phase diagrams, and the correspondence between quantum and classical dynamics.

1. Formal Definition and Calculation

For a dynamical system with state vector $\mathbf{x}(t)$ evolving under a flow or map, initial infinitesimal deviations $\delta\mathbf{x}(0)$ obey, to linear order, the equation

$$\frac{d}{dt}\delta\mathbf{x}(t) = D\mathbf{F}(\mathbf{x}(t)) \, \delta\mathbf{x}(t)$$

where $D\mathbf{F}$ is the Jacobian of the flow. The maximal Lyapunov exponent $\lambda_{\max}$ is defined as

$$\lambda_{\max} = \lim_{t\rightarrow\infty} \frac{1}{t}\ln\frac{|\delta\mathbf{x}(t)|}{|\delta\mathbf{x}(0)|}$$

This measures the average rate (per unit time) of local phase-space separation for the most unstable direction.

For periodically driven non-autonomous systems, the MLE is computed stroboscopically, i.e., at integer multiples of the drive period. In mean-field models with constraints such as spin length conservation, the dynamics of the perturbation $\delta\mathbf{m}(t)$ is projected onto the tangent plane of the unit sphere and evolved concurrently with the base trajectory, with periodic orthonormalization (e.g., via the Benettin–QR algorithm) to ensure numerical stability (Fan et al., 31 Dec 2025).

2. Diagnostic Role in Chaos and Bifurcation Analysis

The sign and magnitude of $\lambda_{\max}$ uniquely diagnose the nature of the system’s attractors:

• $\lambda_{\max} < 0$: Stable periodic orbits or fixed points (perturbations decay exponentially; dynamics is regular).
• $\lambda_{\max} = 0$: Marginally stable, typically associated with quasiperiodic tori.
• $\lambda_{\max} > 0$: Chaos; exponential separation of initially close orbits and sensitive dependence on initial conditions.

MLEs are central to constructing dynamical phase diagrams. In periodic macrospin systems with anisotropic interactions and dissipation, regions in parameter space (e.g., drive amplitude, dissipation rate) can be classified as stable, marginal, or chaotic by mapping $\lambda_{\max}(A, \kappa)$—forming a color-coded stability chart with, respectively, blue, white, and red regions (Fan et al., 31 Dec 2025).

MLEs also enable identification of period-doubling cascades, onset of chaos, and the appearance of fractal basin boundaries. The approach characterizes attractor transitions and critical phenomena, such as Feigenbaum scaling in period-doubling sequences (Fan et al., 31 Dec 2025).

3. MLE in Classical–Quantum Correspondence

In many-body quantum systems with a classical limit (e.g., large-$N$ macrospin ensembles on the Dicke manifold), the MLE calculated in the classical limit governs the time window over which quantum and classical trajectories agree:

• When $\lambda_{\max} < 0$, quantum expectation values converge to the classical mean-field result as $N \to \infty$ for all accessible times.
• For $\lambda_{\max} > 0$, quantum-classical correspondence holds only for times $t \lesssim t_L=1/\lambda_{\max}$ (the Lyapunov time), beyond which quantum fluctuations are exponentially amplified, leading to statistically distinct orbits and diffusive exploration of Hilbert space, i.e., quantum chaos (Fan et al., 31 Dec 2025).

The quantum-classical convergence criterion can be monitored via the localization properties of the quantum density matrix: sharp localization ($\sim N^{-1/2}$ width) around the classical mean-field value corresponds to valid mean-field dynamics; delocalization signals onset of quantum chaos (Fan et al., 31 Dec 2025).

4. MLE in Bifurcation, Basin Structure, and Attractor Classification

By tracking $\lambda_{\max}$ across parameter sweeps, MLEs provide a quantitative bifurcation analysis. In stroboscopically sampled macrospin systems, period-doubling bifurcations can be detected by changes in the largest Floquet multiplier crossing unity in magnitude, which signals a loss of stability of a period-$p$ orbit to a period-$2p$ orbit.

MLEs also help reveal the structure of basins of attraction. Sampling many initial conditions on the phase space sphere and labeling each by the final attractor, one observes that the boundaries between different attractors often develop a fractal structure, particularly as the system approaches chaotic regimes (where $\lambda_{\max}>0$) (Fan et al., 31 Dec 2025). Fourier spectra of magnetization observables distinguish periodic (sharp peaks), quasiperiodic (multiple discrete peaks), and chaotic (broad continuous) dynamics—each associated with characteristic MLE values.

5. Methods for Numerical and Analytical Determination

Practically, MLEs are computed via tangent-space evolution—either as variational equations integrated alongside the system or by continuous reorthonormalization (e.g., QR with Gram–Schmidt) of a set of tangent vectors. In periodically driven systems, the analysis is performed stroboscopically, and the exponent is averaged over many periods to reach asymptotic convergence (Fan et al., 31 Dec 2025).

In some cases, especially for systems admitting Floquet analysis, the maximal Lyapunov exponent can be related to the logarithm of the largest Floquet multiplier, i.e.,

$$\lambda_{\max} = \frac{1}{T} \ln|\Lambda_{\mathrm{max}}|$$

where $T$ is the period, and $\Lambda_{\mathrm{max}}$ is the dominant eigenvalue of the monodromy matrix over one period.

6. Broader Significance, Limitations, and Interpretation

The MLE is a robust, system-agnostic measure of chaoticity and phase-space mixing. However, while a positive MLE signals chaos, it does not capture the full multifractal spectrum of Lyapunov exponents nor does it distinguish between different types of high-dimensional chaos (such as extensive or hyperchaos).

In the context of quantum-classical correspondence, the breakdown of agreement for $t > t_L$ in chaotic regimes does not signal a failure of mean-field theory per se, but rather a transition to quantum chaos characterized by delocalization and maximal mixing in the quantum state (Fan et al., 31 Dec 2025).

The concise interplay between chaos (diagnosed by the MLE), system size, decoherence, and other forms of dissipation is central to understanding emergent statistical mechanics and non-equilibrium dynamics in complex many-body systems.

7. Summary Table: MLE Regimes and Dynamical Structures

$\lambda_{\max}$	Dynamical Regime	Observable Behavior
$< 0$	Stable/Periodic	Regular motion, fixed points, tori
$= 0$	Marginal/Quasiperiodic	Incommensurate frequencies, tori
$> 0$	Chaotic	Exponential divergence, mixing, chaos

In periodically driven macrospin systems with collective dissipation and anisotropic interactions, the MLE enables comprehensive dynamical classification, the detection of critical transitions, and a nuanced understanding of quantum-classical correspondence in chaotic regimes (Fan et al., 31 Dec 2025).