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We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional…
This chapter offers a principled approach to the prediction of chaotic systems from data. First, we introduce some concepts from dynamical systems' theory and chaos theory. Second, we introduce machine learning approaches for…
The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The…
The main purpose of these lectures is to discuss briefly recent methods of calculation of statistical properties of quantum eigenvalues for chaotic systems based on semi-classical trace formulas. Under the assumption that periodic orbit…
Classically integrable approximants are here constructed for a family of predominantly chaotic periodic systems by means of the Baker-Hausdorff-Campbell formula. We compare the evolving wave density for the corresponding exact quantum…
Using the Lorenz equations, we have investigated whether unstable periodic orbits (UPOs) associated with a strange attractor may predict the occurrence of the robust sharp peaks in histograms of some experimental chaotic time series.…
The striking fractal geometry of strange attractors underscores the generative nature of chaos: like probability distributions, chaotic systems can be repeatedly measured to produce arbitrarily-detailed information about the underlying…
We show that in the semiclassical limit, classically chaotic systems have universal spectral statistics. Concentrating on short-time statistics, we identify the pairs of classical periodic orbits determining the small-$\tau$ behavior of the…
We compare different analytical and numerical methods for studying the partitions of a finite system into fragments. We propose a new numerical method of exploring the partition space by generating the Markov chains of partitions based on…
Due to the dynamic nature, chaotic time series are difficult predict. In conventional signal processing approaches signals are treated either in time or in space domain only. Spatio-temporal analysis of signal provides more advantages over…
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behavior of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics…
The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions…
Chaos indicators, like the Lyapunov exponent lambda, are widely used in celestial mechanics to characterize the dynamical behavior of bodies. The stability of their orbit can be determined by the calculation of the local exponential…
A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the physical processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have…
The literature is rich with studies, analyses, and examples on parameter estimation for describing the evolution of chaotic dynamical systems based on measurements, even when only partial information is available through observations.…
We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation…
Quantized, compact graphs were shown to be excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity we show that they display all the features which characterize scattering systems with an underlying…
Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step…
We prove that transport in the phase space of the "most strongly chaotic" dynamical systems has three different stages. Consider a finite Markov partition (coarse graining) $\xi$ of the phase space of such a system. In the first short times…
For a simple model of chaotic dynamical systems with a large number of degrees of freedom, we find that there is an ensemble of unstable periodic orbits (UPOs) with the special property that the expectation values of macroscopic quantities…