Predictability: a way to characterize Complexity

Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kind of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports. Related information at this http://axtnt2.phys.uniroma1.i

Testing the SOC hypothesis for the magnetosphere

As noted by Chang, the hypothesis of Self-Organised Criticality provides a theoretical framework in which the low dimensionality seen in magnetospheric indices can be combined with the scaling seen in their power spectra and the recently-observed plasma bursty bulk flows. As such, it has considerable appeal, describing the aspects of the magnetospheric fuelling:storage:release cycle which are generic to slowly-driven, interaction-dominated, thresholded systems rather than unique to the magnetosphere. In consequence, several recent numerical "sandpile" algorithms have been used with a view to comparison with magnetospheric observables. However, demonstration of SOC in the magnetosphere will require further work in the definition of a set of observable properties which are the unique "fingerprint" of SOC. This is because, for example, a scale-free power spectrum admits several possible explanations other than SOC. A more subtle problem is important for both simulations and data analysis when dealing with multiscale and hence broadband phenomena such as SOC. This is that finite length systems such as the magnetosphere or magnetotail will by definition give information over a small range of orders of magnitude, and so scaling will tend to be narrowband. Here we develop a simple framework in which previous descriptions of magnetospheric dynamics can be described and contrasted. We then review existing observations which are indicative of SOC, and ask if they are sufficient to demonstrate it unambiguously, and if not, what new observations need to be made?Comment: 29 pages, 0 figures. Based on invited talk at Spring American Geophysical Union Meeting, 1999. Journal of Atmospheric and Solar Terrestrial Physics, in pres

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.Comment: 21 pages, 5 figures, Final revision to appear in Bulletin of Mathematical Biolog

Numerical methods for stochastic sensitivity analysis of 2D chaotic attractors

The paper presents constructive algorithms for finding the outer boundaries of chaotic attractors, based on a geometric selection of points of critical lines belonging only to the outer boundary. In the theory of dynamical discrete-time systems, critical lines play a key role. These lines facilitate the study of the dynamic properties of noninvertible maps and to describe the boundaries of a chaotic attractor. The previously constructed stochastic sensitivity function for chaotic attractors is based on critical lines and lets us estimate the dispersion of random states around the chaotic attractor. However, the technical problem is complicated by the fact that the critical lines describe not only the external boundaries, but also structures inside the chaotic attractor. Our algorithms are tested for complex non-convex forms of chaotic attractors. Based on the algorithms, we solve the problem of finding confidence domains around chaotic attractors of stochastic systems. © 2022 Author(s).Russian Science Foundation, RSF, (N 21-11-00062)The work was supported by Russian Science Foundation (N 21-11-00062)

Dispersal and noise: Various modes of synchrony in\ud ecological oscillators

We use the theory of noise-induced phase synchronization to analyze the effects of dispersal on the synchronization of a pair of predator-prey systems within a fluctuating environment (Moran effect). Assuming that each isolated local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive a Fokker–Planck equation describing the evolution of the probability density for pairwise phase differences between the oscillators. In the case of common environmental noise, the oscillators ultimately synchronize. However the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of correlated (shared) and uncorrelated (unshared) noise with dispersal can lead to a multistable\ud steady-state probability density

Predicting catastrophes: the role of criticality

Is prediction feasible in systems at criticality? While conventional scale-invariant arguments suggest a negative answer, evidence from simulation of driven-dissipative systems and real systems such as ruptures in material and crashes in the financial market have suggested otherwise. In this dissertation, I address the question of predictability at criticality by investigating two non-equilibrium systems: a driven-dissipative system called the OFC model which is used to describe earthquakes and damage spreading in the Ising model. Both systems display a phase transition at the critical point. By using machine learning, I show that in the OFC model, scaling events are indistinguishable from one another and only the large, non-scaling events are distinguishable from the small, scaling events. I also show that as the critical point is approached, predictability falls. For damage spreading in the Ising model, the opposite behavior is seen: the accuracy of predicting whether damage will spread or heal increases as the critical point is approached. I will also use machine learning to understand what are the useful precursors to the prediction problem

Стохастическая чувствительность квазипериодических и хаотических аттракторов дискретной модели Лотки-Вольтерры

Целью исследования, представленного в данной статье, является анализ возможных динамических режимов детерминированной и стохастической модели Лотки-Вольтерры. В зависимости от двух параметров системы строится карта режимов. Изучаются параметрические зоны существования устойчивых равновесий, циклов, замкнутых инвариантных кривых, а также хаотических аттракторов. Описываются бифуркации удвоения периода, Неймарка-Саккера и кризиса. Демонстрируется сложная форма бассейнов притяжения нерегулярных аттракторов (замкнутой инвариантной кривой и хаоса). Помимо детерминированной системы подробно изучается стохастическая, описывающая влияние внешнего случайного воздействия. Здесь ключевым является нахождение чувствительности таких сложных аттракторов, как замкнутая инвариантная кривая и хаос. В случае хаоса дан алгоритм нахождения критических линий, описывающих границу хаотического аттрактора. Опираясь на найденную функцию стохастической чувствительности, строятся доверительные полосы, позволяющие описать разброс случайных состояний вокруг детерминированного аттрактора

Stochastic transformations of multi-rhythmic dynamics and order-chaos transitions in a discrete 2D model

A problem of the analysis of stochastic effects in multirhythmic nonlinear systems is investigated on the basis of the conceptual neuron map-based model proposed by Rulkov. A parameter zone with diverse scenarios of the coexistence of oscillatory regimes, both spiking and bursting, was revealed and studied. Noise-induced transitions between basins of periodic attractors are analyzed parametrically by statistics extracted from numerical simulations and by a theoretical approach using the stochastic sensitivity technique. Chaos-order transformations of dynamics caused by random forcing are discussed. © 2021 Author(s).This work was supported by the Russian Science Foundation (No. 21-11-00062)