Real-world systems, from climate models to power grids, often fluctuate due to sensor noise, drift, or environmental variability, yet standard chaos diagnostics assume fixed parameters and asymptotic horizons. We introduce a finite-time framework for two-dimensional maps under independent, identically distributed parameter noise. First, we prove that the maximal finite-time Lyapunov exponent converges, after centering and scaling, to a Gaussian law whose mean and variance depend explicitly on the map's Jacobian statistics. Second, we develop an attractor separation algorithm that uses FTLE histograms and a geometry-based classifier to partition phase space into chaotic and periodic regions under noise. Third, we validate our theory numerically on the noisy Domenicali map, demonstrating Gaussian FTLE distributions, predictable shifts in Kaplan-Yorke dimension, and a sharp noise threshold for basin escape. Finally, we estimate the critical noise level $\sigma_c$ at which attractor coalescence occurs, using three complementary numerical methods to bound it above and to pinpoint an estimate.
Empirical entropy rates are widely used to quantify unpredictability from symbolic or time-series data, yet their interpretation is subtle in weakly chaotic dynamics, where ordinary Lyapunov exponents vanish and invariant measures are infinite. We address this issue by studying the empirical one-step conditional entropy for the fixed finite partitions considered below in one-dimensional intermittent maps with infinite invariant measures. For the modified Bernoulli map and the Boole transformation in the infinite-measure weak-chaos regime, we prove that this per-step empirical entropy converges to zero. Thus, the usual entropy-rate normalization becomes asymptotically blind to subexponential instability. The finite-time information sum, however, remains informative. Rare transitions between long laminar phases occur on the return-sequence scale, and their empirical self-information contributes an additional logarithmic factor. Under the stated regularity and moment assumptions, this mechanism yields a two-term estimate for the ensemble mean decay, supported by numerical simulations. Although the raw entropy rate vanishes, self-normalized fluctuations remain nontrivial and are numerically consistent with normalized Mittag-Leffler laws. A comparison with generalized Lyapunov sums shows that the corresponding information sum is not a Krengel entropy estimator, but a computable, partition-dependent finite-time measure of sparse symbolic transitions. These results clarify what empirical Markov entropy can, and cannot, measure in infinite-measure weak chaos.
We study the long-time, small-noise behavior of a class of dissipative stochastic evolution equations in a separable Hilbert space, driven by a cylindrical Wiener process whose covariance degenerates to a limiting operator in the strong operator topology. A prototypical example is a stochastic reaction-diffusion equation on a bounded domain with spatially homogeneous but spectrally regularized noise that becomes spatially rough in the limit. We establish a large deviation principle for occupation measures as the time horizon becomes large, the noise intensity tends to zero, and the noise becomes increasingly rough. The result covers a broad class of dissipative equations in infinite dimensions, including those driven by asymptotically rough cylindrical noise whose covariance need not be trace class. Extending the finite-dimensional work of Budhiraja and Zoubouloglou, the infinite-dimensional setting introduces substantial new difficulties: the bound arguments require careful handling of the unbounded evolution and inverse covariance operators, and the increasing roughness of the noise must be balanced against its vanishing amplitude through uniform estimates. Proofs combine analytic semigroup techniques, fractional domain space estimates and a careful treatment of stochastic convolutions in weighted spaces. The rate function is given by a simple explicit formula, the average over the measure of the squared Cameron-Martin cost of canceling the deterministic drift at each point. The proof follows the weak convergence approach based on the Bou\`{e}-Dupuis variational formula and constructs near-optimal controls by alternating travel phases that steer the process between prescribed target states and hold phases that stabilize it near a target while shaping the occupation measure.
Randomly connected neural networks undergo a transition from a stable fixed point to chaos as the coupling strength increases. In the thermodynamic limit, this transition has been shown theoretically to occur abruptly at a critical point. In finite-size systems, however, a variety of bifurcation cascades appear between the stable fixed point and chaos. In this study, we systematically characterize routes to chaos in finite-size random neural networks. By analyzing individual realizations, we identify multiple scenarios, including the Ruelle-Takens-Newhouse route, torus doubling, and fractalization, as well as chaotic dynamics with persistent toroidal geometry. We also study these behaviors at the ensemble level by quantifying the fraction of chaotic trajectories as a function of coupling strength and system size. The resulting finite-size crossover sharpens with increasing system size and exhibits empirical scaling trends, providing a statistical characterization of the broad onset of chaos in finite random networks.
We investigate the stochastic stability of master--slave synchronization for a class of nonlinear dissipative PDEs with Burgers-type nonlinearity and polynomial linear operator, including the Burgers, Kuramoto--Sivashinsky, Kawahara, Benney--Lin, and Nikolaevskiy equations. Under periodic boundary conditions, each equation is represented by a finite-dimensional Fourier truncation coupled to a slave driven by observed master data. We establish local exponential stability of the deterministic zero-error synchronization manifold under a simple coupling condition. Introducing observational noise in the coupling transforms the slave into an It\^o diffusion, preventing exact synchronization. We analyze the deviation $\Delta(t)$ between the stochastic error and the exponentially stable deterministic reference error, proving an $\mathcal{O}(\sigma^2)$ finite-time mean-square bound localized near the synchronization manifold, with a tail-probability estimate. Under global one-sided dissipativity, the localization is removed and a time-uniform $\mathcal{O}(\sigma^2)$ bound is obtained. Bounds are derived in Fourier space and transferred to the physical domain via Parseval's identity. Under uniform Sobolev and Galerkin stability assumptions, we also bound the error with respect to the infinite-dimensional master uniformly in truncation order. Results are illustrated by numerical simulations of Kawahara and Nikolaevskiy master--slave systems.
We analyze inverse stochastic resonance (ISR) in a bistable FitzHugh--Nagumo neuron driven by additive noise in the voltage variable, focusing on how neural excitability and timescale separation regulate the noise-induced modulation of spiking activity. A codimension-two bifurcation analysis identifies a narrow bistable region in which a stable fixed point and a stable limit cycle coexist, separated by an unstable periodic orbit. Finite-time Monte Carlo simulations show that the occupation of the limit-cycle basin may appear to depend on the initial basin when rare transitions are not fully resolved. We prove that this dependence is not asymptotic: the stochastic system admits a unique invariant probability measure, so long-time firing statistics are independent of the initial basin of attraction. The parameter dependence of ISR is characterized by quasi-potential barriers computed with a geometric minimum action method for the degenerate noise. The difference between the limit-cycle and fixed-point quasi-potentials partitions the bistable wedge into two escape-dominated regimes. A reduced metastable two-state Markov approximation yields a weak-noise formula for the limit-cycle basin occupation probability, a sign criterion for genuine ISR, and a semiquantitative prediction of the ISR-minimizing noise amplitude. In the fitted regime with a negative effective exponent, a genuine ISR minimum occurs only when the limit-cycle quasi-potential exceeds that of the fixed point. These results provide an escape-balance mechanism linking intrinsic neuronal parameters to asymptotic noise-induced spike suppression.
Marius E. Yamakou, Torben Krüger, H. Schulz-Baldes· 0 citations
Intermittency represents a fundamental route to chaos in nonlinear dynamical systems. In this work we introduce an adaptive control strategy in which the control parameter of an intermittent system is promoted to a dynamical variable that evolves autonomously under an auxiliary nonlinear map drawn from the same functional hierarchy as the system itself. The construction eliminates the need for orbit identification, local linearization, and trajectory-triggered perturbations, which are central ingredients of conventional feedback schemes. The theoretical framework is developed within a class of one-dimensional nonlinear ergodic maps with exactly known invariant (Sinai--Ruelle--Bowen) measures, for which we derive in closed form (i) the dynamics and invariant measure of the evolving control parameter, (ii) the invariant measure of the coupled system, and (iii) the $q$-generalized Lyapunov exponents before and after control. The generalized Lyapunov spectrum serves as an analytical order parameter for the control process: the collapse of its positive regions provides a quantitative and initial-condition-independent signature of chaos suppression, and yields the sensitivity to initial conditions in explicit form. To establish the physical relevance of the approach beyond low-dimensional maps, we apply the same construction to a cluster of three interacting ultrasound-driven microbubbles described by the Keller--Herring model, promoting the experimentally accessible acoustic driving frequency to a dynamical variable. Systematic bifurcation and Lyapunov analyses, performed over wide ranges of driving pressure, frequency, and equilibrium radii, demonstrate that intermittent chaotic radial oscillations are progressively suppressed and replaced by stable periodic motion.
M. Yahyavi, Sina Gholizadeh, S. Behnia et al.· Physical Review Research· 7 citations
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