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Computation plays a key role in predicting and analyzing natural phenomena. There are two fundamental barriers to our ability to computationally understand the long-term behavior of a dynamical system that describes a natural process. The…
We introduce a new analytical method, which allows to find out chaotic dynamics in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered as an example. The corresponding…
It has been generally recognized that stochasticity can play an important role in the information processing accomplished by reaction networks in biological cells. Most treatments of that stochasticity employ Gaussian noise even though it…
We develop a statistical theory of waveform shaping of incident waves that aim to efficiently deliver energy at weakly lossy targets which are embedded inside chaotic enclosures. Our approach utilizes the universal features of chaotic…
The defining feature of chaos is its hypersensitivity to small perturbations. However, we report a stability of branched flow against large perturbations where the classical trajectories are chaotic, showing that strong perturbations are…
Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed…
A new theoretical approach to non-equilibrium statistical systems has recently been proposed by the author, a co-author and others. It is based on a variational principle which is associated with the discrepancy of a path through…
Classical shadows are a computationally efficient approach to storing quantum states on a classical computer for the purposes of estimating expectation values of local observables, obtained by performing repeated random measurements. In…
We investigate functions that are exact solutions to chaotic dynamical systems. A generalization of these functions can produce truly random numbers. For the first time, we present solutions to random maps. This allows us to check,…
The problem of determining the mathematical model of the dynamics of multi-dimensional control systems in the presence of noise under the condition that the correlation functions cannot be found. Known statistical dynamics of linear systems…
Some intriging connections between the properties of nonlinear noise driven systems and the nonlinear dynamics of a particular set of Hamilton's equation are discussed. A large class of Fokker-Planck Equations, like the Schr\"odinger…
Turbulence, namely, irregular fluctuations in space and time characterize fluid flows in general and atmospheric flows in particular.The irregular,i.e., nonlinear space-time fluctuations on all scales contribute to the unpredictable nature…
Predictability horizon properties of chaotic dynamical systems can be related to their spectral properties. It is shown, using this relationship, that the spectral properties of the leading large-scale climate daily indices indicate a…
Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. This fact is used to find a unique semiclassical representation without the…
In the chaotic quantization approach one replaces the Gaussian white noise of the Parisi-Wu approach of stochastic quantization by a deterministic chaotic process on a very small scale. We consider suitable coupled chaotic noise processes…
Here we define natural chaotic systems, like the earths weather and climate system, as chaotic systems which are open to the world so have constantly changing boundary conditions, and measurements of their states are subject to errors. In…
It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependance on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error)…
We analyze a class of linear shell models subject to stochastic forcing in finitely many degrees of freedom. The unforced systems considered formally conserve energy. Despite being formally conservative, we show that these dynamical systems…
The property of shadowing has been shown to be fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise…
We introduce a general statistical learning theory for processes that take as input a classical random variable and output a quantum state. Our setting is motivated by the practical situation in which one desires to learn a quantum process…