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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…
Chaos is omnipresent in nature, and its understanding provides enormous social and economic benefits. However, the unpredictability of chaotic systems is a textbook concept due to their sensitivity to initial conditions, aperiodic behavior,…
Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime. The challenge is even greater when the physics underlying such systems is…
Deterministic chaos is phenomenon from nonlinear dynamics and it belongs to greatest advances of twentieth-century science. Chaotic behavior appears apart of mathematical equations also in wide range in observable nature, so as in there…
Chaotic systems which are due to nonlinearity have attracted a great concern in the current world and chaotic models. Systems for a wide range of operation conditions have their application in almost all branches of engineering and science.…
Chaos is an active research subject in the fields of science in recent years. it is a complex and an erratic behavior that is possible in very simple systems. in the present day, the chaotic behavior can be observed in experiments. Many…
Systems exhibiting nonlinear dynamics, including but not limited to chaos, are ubiquitous across Earth Sciences such as Meteorology, Hydrology, Climate and Ecology, as well as Biology such as neural and cardiac processes. However, System…
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…
Recently, a general data driven numerical framework has been developed for learning and modeling of unknown dynamical systems using fully- or partially-observed data. The method utilizes deep neural networks (DNNs) to construct a model for…
Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this…
The dynamics of unidirectionally coupled chaotic Lorenz systems is investigated. It is revealed that chaos is present in the response system regardless of generalized synchronization. The presence of sensitivity is theoretically proved, and…
Modern learning systems increasingly interact with data that evolve over time and depend on hidden internal state. We ask a basic question: when is such a dynamical system learnable from observations alone? This paper proposes a research…
We address the issue of how to identify the equations of a largely unknown chaotic system from knowledge about its state evolution. The technique can be applied to the estimation of parameters that drift slowly with time. To accomplish…
Given that observational and numerical climate data are being produced at ever more prodigious rates, increasingly sophisticated and automated analysis techniques have become essential. Deep learning is quickly becoming a standard approach…
In this paper a new concept, namely the critical predictable time $T_c$, is introduced to give a more precise description of computed chaotic solutions of nonlinear differential equations: it is suggested that computed chaotic solutions are…
The dynamical status of isolated quantum systems, partly due to the linearity of the Schrodinger equation is unclear: Conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation --…
Learning governing dynamics from data is a common goal across the sciences, yet it is only well-posed when the underlying mechanisms are identifiable. In practice, many data-driven methods implicitly assume identifiability; when this…
Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about…
Data Science and Machine learning have been growing strong for the past decade. We argue that to make the most of this exciting field we should resist the temptation of assuming that forecasting can be reduced to brute-force data analytics.…
Scientific discoveries are often made by finding a pattern or object that was not predicted by the known rules of science. Oftentimes, these anomalous events or objects that do not conform to the norms are an indication that the rules of…