Given a scalar observable of an ergodic dynamical system with a low-dimensional attractor, two families of methods reconstruct and predict the underlying state: recurrence-based methods (the method of analogues and its descendants), which wait for the trajectory to return to an $\varepsilon$-neighborhood of a previously observed state, and observer-based methods, which fit a converging state estimator on the delay reconstruction. We formalize and empirically verify an exponential separation between the two: the expected cost of recurrence scales as $\varepsilon^{-d}$, where $d$ is the pointwise dimension of the invariant measure (a consequence of the Kac lemma and quantitative Poincare recurrence), whereas a detectable linear observer converges in $\Theta(\log(1/\varepsilon)/(1-\rho(A_{cl})^2))$ steps, where $\rho(A_{cl})$ is the closed-loop spectral radius of the Riccati fixed point. Both laws are verified numerically (return-time exponent $-1.8$ on the Lorenz attractor against the theoretical $-2.05$; observer cost linear in $\log(1/\varepsilon)$ with $R^2=1.000$ and in $(1-\rho^2)^{-1}$ with $R^2=0.985$), yielding a measured cost gap of $\sim 10^{9}$ at $\varepsilon=10^{-6}$ for $d\approx 2$. We complement the theorem with an admission protocol (the Kac-Riccati gate) deciding whether a signal lies inside the theorem's class, via surrogate-data prediction gating; it also explains the folklore of"universal"fractal dimensions as a dataset-size artifact bounded by $2\log_{10}N$. On real data the gate admits the Santa Fe laser benchmark ($\hat D_2=2.0$) and refuses the monthly sunspot series, reproducing the settled resolution of historical low-dimensionality claims. All results reproduce from a single verification script (17/17 checks).
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and R\"ossler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the R\"ossler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and R\"ossler systems.
Symbolic discovery of governing equations from data is limited not only by algorithm design and data volume, but by the geometry of the attractor: what the long-run dynamics allow to be recovered. Using a within-system design on Lorenz-84, where one forcing parameter drives fixed-point, limit-cycle, and chaotic regimes while the governing equations and library stay fixed, we show that a single number, $\lambda_{\min}(M)$, the smallest eigenvalue of the invariant-measure moment matrix, sets the identifiability ceiling for both sparse regression (SINDy) and evolutionary symbolic regression (PySR). Derived from the Birkhoff ergodic theorem and obtained from a short reference trajectory before any run, $\lambda_{\min}(M)$ measures how fully the attractor covers function space: where it vanishes, recovery is impossible for any algorithm, sparse or combinatorial alike; as it grows, both algorithms improve. Chaos raises $\lambda_{\min}(M)$ by spreading the attractor, but also enlarges it and amplifies noise; because noise enters SINDy's regression bottleneck linearly and PySR's discrimination channel superlinearly, the same transition can push the two methods in opposite directions, so deeper chaos is not uniformly better. Parameter-free mechanistic scores from this framework transfer without refitting to a held-out Lorenz-96 system, confirming mechanism rather than curve-fitting; a criterion read from the equations predicts when added chaos will not improve conditioning. We also introduce Soft F1, a coefficient-weighted structural metric that resolves performance differences invisible to binary-success and predictive scores. The first question of discovery is then not which algorithm, but what the attractor permits.
Matteo Gallo, Fabio Anselmi, Paolo Lazzari· arXiv.org· 2 citations
Intensity of attraction quantifies the largest amplitude of a persistent bounded disturbance that an attractor can withstand without loss of controlled confinement in its basin. Although intensity has been formulated separately for flows and maps, its behavior under temporal sampling has remained unresolved. We establish an explicit correspondence between the intensity $\mu(A)$ of a continuous-time attractor and the intensity $\mu_h(A)$ of its exact time-$h$ map. For an $L$-Lipschitz vector field, \[ \frac{\mu(A)}{1+Lh} \leq \frac{\mu_h(A)}{h} \leq \mu(A)\frac{e^{Lh}-1}{Lh}, \] and hence $\mu_h(A)/h\to\mu(A)$. The resulting first-order rate is sharp in general, while smooth scalar escape geometries can exhibit second-order convergence. We extend the framework to one-step numerical methods through a stability theory for block intensity and to attracting invariant graphs over compact invertible nonautonomous bases, obtaining uniform sampling convergence over the forcing phase. For bounded-support random perturbations, normalized discrete intensity is identified with the pathwise safety threshold; above it, finite escape follows under an explicit finite-exit condition, while escape probabilities require additional assumptions on the noise law. We also show that the discrete state--normal boundary map converges to the normalized Pontryagin boundary system governing extremal reachable-set boundaries. Exact scalar benchmarks, a grazing resilience model, planar Duffing escape, anisotropic disturbances, periodic and quasiperiodic forcing, and transfer-operator computations illustrate the theory. These results give intensity estimated from discrete observations or simulations a sampling-independent continuous-time meaning.
Persistence changes character as an autoregressive coefficient approaches one: for each fixed $0<a<1$, survival above zero decays exponentially, whereas at the unit root symmetric random-walk survival is of order $n^{-1/2}$. We study this transition for AR($1$) sequences driven by symmetric $\alpha$-stable innovations and write $\Lambda(a,\alpha)$ for their exponential persistence rate. The entire chain admits an exact representation through a single stable L\'{e}vy process observed on a geometrically expanding time grid. Comparison with continuous half-line survival gives $\Lambda(a,\alpha) \leq \frac{\alpha}{2}\log{(1/a)}$. For $0<\alpha<2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel for regularly varying innovation tails. \rev{Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order.} This proves $\Lambda(a,\alpha) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that the ratio $\Lambda(a,\alpha)/\log{(1/a)}$ converges to a limit in $(0,\alpha/2]$, equal to its supremum over $0<a<1$. Finally, a Lamperti transformation reduces identification of this constant to a dense-sampling persistence problem for a stationary stable Ornstein--Uhlenbeck process. Existing Gaussian theory determines the sharp value at $\alpha=2$. For $0<\alpha<2$, identifying the value requires controlling paths that cross below zero and return above zero between consecutive observations.
We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter $(\varepsilon_t)_{t \geq 0}$ satisfying $\varepsilon_t \to 0$ as $t \to \infty$. In contrast to approaches based on noise-induced smoothing or elliptic regularity, our approach relies on dissipativity of the frozen fast dynamics and therefore permits degenerate diffusion coefficients. We prove a maximal $L^p$-estimate between the slow variable and the averaged ODE at late times, with the classical strong convergence rate of order $1/2$. Under an additional decay condition on $(\varepsilon_t)_{t \ge 0}$, this estimate implies that the slow variable is almost surely an asymptotic pseudo-trajectory of the averaged ODE. As a consequence, we obtain criteria for the identification of possible limit points and for convergence toward asymptotically stable equilibria for the slow variable by analyzing the dynamical behavior of the averaged equation.
We develop a structural framework for irregularity in linear dynamics centered on the space $\Upsilon$ of observation schemes. This approach separates the underlying dynamical behavior from the observation mechanism and provides a unified setting in which classical notions such as Li--Yorke chaos, mean Li--Yorke chaos, and distributional chaos arise as particular cases corresponding to Dirac measures and Ces\`aro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(\mu_m)$-irregular and distributionally $(\mu_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $\Upsilon$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.
Manuel Saavedra· 0 citations
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