Fractional Scaling of Lyapunov Exponents

• Fractional Scaling of Lyapunov Exponents is characterized by non-integer power laws relating system parameters to stability indicators in complex, high-dimensional systems.
• Detailed numerical and analytical studies reveal fractional exponents (e.g., 0.53, 1/3) that capture universality in turbulence, chaos, and random matrix products.
• These insights have practical implications for modeling phenomena ranging from turbulent flows and fractional differential dynamics to SPDEs and gravitational phase transitions.

Fractional scaling of Lyapunov exponents encompasses a broad family of scaling laws, phenomenology, and universality classes where Lyapunov exponents exhibit non-integer (fractional) dependence on system parameters such as temporal/spatial scale, system size, strength of noise, or control parameters. Such scaling arises in deterministic and stochastic dynamical systems, turbulence, spatially extended chaos, random matrix products, fractional-order differential systems, and stochastic PDEs with subordinate stable processes or fractional noise. The analysis of this fractional scaling provides direct insights into predictability, instability, fluctuations, and universality in high-dimensional and complex systems.

1. Fractional Scaling in Turbulent and Chaotic Dissipative Systems

Direct numerical simulation studies of homogeneous isotropic turbulence reveal that the maximal Lyapunov exponent $\lambda$ scales with Reynolds number as a power law $\lambda \sim Re^\alpha$ with $\alpha=0.53 \pm 0.02$ when normalized by the large eddy turnover time $T_0 = L/U$ (Ho et al., 2019). This is a fractional scaling, contrasting with dimensional predictions ($\alpha=1/2$ from Kolmogorov theory), and arises due to sub-Kolmogorov scale instabilities. The scaling is robust under changes in lattice size, numerical integration parameters, and persists for large $Re$.

Similarly, Lyapunov exponents in turbulence scale as $\lambda_{max} \tau_\eta \sim Re_\lambda^\beta$ with $\beta \in [0.25,0.33]$ ($95\%$ confidence), as shown by direct measurements and Bayesian inference (Mohan et al., 2017). Here, $\tau_\eta$ is the Kolmogorov time scale. The implication is that "Lyapunov time" decreases faster than the Kolmogorov time as $Re$ increases, confirming the development of instabilities at sub-Kolmogorov scales.

In spatially-extended deterministic chaos, the diffusion coefficient $D(L)$ of fluctuations of finite-time Lyapunov exponents decays as $D(L)\sim L^{-\gamma}$, where the wandering exponent $\gamma$ is strictly fractional (e.g., $\gamma=1/2$ in 1D for the leading LE, corresponding to KPZ universality) (Pazó et al., 2013). Bulk exponents in the spectrum display different fractional wandering exponents ($\gamma \sim 0.80$–$0.90$), defining new universality classes.

2. Fractional Dynamics and Memory Effects

Fractional-order systems—dynamical systems where derivatives are of non-integer (Caputo) order—show continuous, monotonic tuning of the Lyapunov spectrum as a function of the fractional order parameter $q\in(0,1]$ (Danca et al., 2018, Danca, 2021). Near the chaos threshold, the largest Lyapunov exponent typically scales linearly or as a power, $\lambda(q)\sim C(q-q_c)^\alpha$ with $\alpha\approx 1$. For non-commensurate systems, the largest exponent vanishes at a critical memory parameter $q_c<1$, and the approach to chaos is smooth and governed by the memory-induced damping.

The table below summarizes scaling observations in several deterministic and fractional systems:

System	Scaling Law	Regime or Parameter
Turbulence (DNS)	$\lambda \sim Re^{0.53}$	$Re$ (large-scale)
Fractional ODE (RF system)	$\lambda_1(q) \sim (q-q_c)$	$q$ near chaos threshold
HMF model	$\lambda_i(N)\sim N^{-1/3}$	System size $N$

In all cases, the scaling exponent is fractional and typically not rational.

3. Fractional Noise and SPDEs

The introduction of noise with nontrivial temporal correlations (fractional Brownian motion with Hurst index $H\in(0,1)$) or heavy-tailed Lévy noise induces new scaling regimes for Lyapunov exponents and their moments.

For stochastic partial differential equations (SPDEs) driven by space-time Lévy noise and fractional Laplacians, moment Lyapunov exponents $\gamma(p)$ exhibit critical exponents due to the heavy tails of the corresponding heat kernel (Shiozawa et al., 28 Sep 2025):

• Upper bound: $\gammā(p)\lesssim p\beta_0$, with $\beta_0$ decreasing as the order $\alpha$ increases (i.e., faster heat kernel decay).
• Lower bound: $\gamma_(p)>0$ for $p<1+\alpha/d$, vanishing at $p=1+\alpha/d$.
• Exponential growth index: $\lambda_(p)\sim \alpha^{-1/(1-(p-1)d/\alpha)}$, manifesting heavy-tail effects and phase transitions in $p$ driven by $\alpha$.

Fractional parabolic Anderson models under colored Gaussian noise also yield non-integer moment Lyapunov exponents: $\Lambda(p)\sim p^{\alpha/(\alpha-\beta)}$ with "intermittency index" $\alpha/(\alpha-\beta)$ unbounded as $\beta\uparrow\alpha$ (Chen et al., 2016).

In SPDEs with fractional Brownian noise, the finite-time Lyapunov exponent displays explicit scaling in terms of the "distance from bifurcation" $\varepsilon$, noise strength $\sigma$, Hurst index $H$, and observation time $T$. Corrections to the deterministic Lyapunov exponent can be of order $\sigma^{2} T^{2H-1}$, amplifying or damping with time depending on whether $H>1/2$ or $H<1/2$, thus revealing critical fractional scaling laws with respect to both noise parameters and temporal windows (Blessing et al., 2023).

4. System Size and Universality Classes

Fractional (finite-size) scaling emerges in statistical, Hamiltonian, and coupled oscillator systems. In the Hamiltonian Mean Field (HMF) model, chaos indicators such as the Lyapunov spectrum scale as $\lambda_i(N)\sim N^{-1/3}$ for sufficiently large $N$, derived both from random-matrix approximations (where the exponent $1/3$ results from cubic-root dependence on fluctuation variance) and from scaling analysis of Vlasov–finite-$N$ corrections. Importantly, this scaling holds across energy regimes where the system is dominated by uniformly weak or strong chaos (Manos et al., 2010).

For weakly-coupled phase oscillators in the incoherent regime (Kuramoto model), the scaling depends on the frequency distribution:

• Regular ensembles: $\lambda(N) \sim 1/N$,
• Disordered ensembles: $\lambda(N) \sim (\ln N)/N$,

where the fractional logarithmic correction in the latter case arises from localization phenomena in the Lyapunov vector structure (Carlu et al., 2017).

5. Random Matrix Products and Special-Function Scaling

Products of random matrices near the identity and higher-dimensional random matrix products yield Lyapunov exponents exhibiting scaling functions governed by special functions (hypergeometric, Airy, Bessel, Whittaker, elliptic integrals), encapsulating a range of fractional exponents such as $1/2$ and $1/3$ depending on the multiplicity of zeros in corresponding effective diffusion polynomials (Comtet et al., 2012). For example, in the product of independent real $2\times2$ matrices, the Lyapunov exponent scales as $\gamma\propto\varepsilon^\chi$, with $\chi$ non-integer.

In products of large random matrices (Ginibre ensemble), the double-scaling limit with $N,M\to\infty$ at $N/M\to a\in(0,\infty)$ interpolates between "deterministic" ($a=0$) and "random-matrix" ($a=\infty$) regimes. At the spectral edge, the soft-edge scaling exhibits the characteristic $a^{-2/3}$ fractional exponent, coinciding with transitions to Tracy–Widom universality (Akemann et al., 2018).

6. Critical Phenomena and Order Parameters

In gravitational physics, Lyapunov exponents serve as dynamic order parameters with clear fractional critical exponents. In Reissner–Nordström–AdS black holes, the Lyapunov exponent difference $\Delta\lambda$ across the small/large black hole coexistence line vanishes as $(T-T_c)^{1/2}$ (critical exponent $\alpha=1/2$), analogous to mean-field behavior in van der Waals systems and present in both particle and string probes (Guo et al., 2022).

7. Summary and Universality

Fractional scaling of Lyapunov exponents pervades disparate domains: turbulence, random matrix theory, fractional-order dynamics, SPDEs with anomalous transport, high-dimensional chaos, and critical phenomena. The non-integer exponents and associated scaling functions expose universal structures underlying complex instability, predictability breakdown, and the statistical organization of chaos. These scaling laws derive from pathwise large deviations, heavy-tailed heat kernel estimates, memory effects, spatial or ensemble averaging in large systems, and the analytic properties of associated Fokker–Planck or variational operators. The broad reproducibility of fractional exponents such as $1/2$, $1/3$, and $2/3$, often linked to known universality classes (KPZ, Tracy–Widom, random matrix crossover), underscores the centrality of fractional scaling in the theory of dynamical instability and chaos.

---

Key References:

• "Fluctuations of Lyapunov Exponents in homogeneous and isotropic turbulence" (Ho et al., 2019)
• "Scaling with system size of the Lyapunov exponents for the Hamiltonian Mean Field model" (Manos et al., 2010)
• "Universal scaling of Lyapunov-exponent fluctuations in space-time chaos" (Pazó et al., 2013)
• "Lyapunov exponents and growth indices for fractional stochastic heat equations with space-time Lévy white noise" (Shiozawa et al., 28 Sep 2025)
• "The Lyapunov exponent of products of random $2\times2$ matrices close to the identity" (Comtet et al., 2012)
• "From Integrable to Chaotic Systems: Universal Local Statistics of Lyapunov exponents" (Akemann et al., 2018)
• "Finite-time Lyapunov exponents for SPDEs with fractional noise" (Blessing et al., 2023)
• "Probing Phase Structure of Black Holes with Lyapunov Exponents" (Guo et al., 2022)