Related papers: Extension of Lorenz Unpredictability
In this work, the dynamics of a system of mutually coupled Generalized Lorenz systems (GLS) is investigated. The state variables of two Lorenz oscillators are coupled mutually via non-linear controls and synchronization is achieved between…
The evolution of the spectrum of cosmological fluctuations from one cycle to the next is studied. It is pointed out that each cycle leads to a reddening of the spectrum. This opens up new ways to generate a scale-invariant spectrum of…
The study of chaos in relativistic systems has been hampered by the observer dependence of Lyapunov exponents (LEs) and of conditions, such as orbit boundedness, invoked in the interpretation of LEs as indicators of chaos. Here we establish…
In the framework of causal perturbation theory renormalization consists of the extension of distributions. We give the explicit form of a Lorentz invariant extension of a scalar distribution, depending on one difference of space time…
We look afresh at the deduction of the "Lorentz contraction" of a "rod" from the Lorentz transformation equations of the special theory of relativity. We show that under special conditions, which include acceleration of the "rod", length…
A generalized fluctuation-response relation is found for thermal systems driven out of equilibrium. Its derivation is independent of many details of the dynamics, which is only required to be first-order. The result gives a correction to…
We treat $n$-dimensional piecewise-linear continuous maps with two pieces, each of which has exactly one unstable direction, and identify an explicit set of sufficient conditions for the existence of a chaotic attractor. The conditions…
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…
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…
We generalize the exact solution to the Bernoulli shift map. Under certain conditions, the generalized functions can produce unpredictable dynamics. We use the properties of the generalized functions to show that certain dynamical systems…
Randomly-assembled dynamical systems are theoretically predicted to be unstable upon crossing a critical threshold of complexity, as first shown by May. Yet, empirical complex systems exhibit remarkable stability, indicating the presence of…
The relationship between chaos and quantum mechanics has been somewhat uneasy -- even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our…
Cascades are self-reinforcing processes underlying the systemic risk of many complex systems. Understanding the universal aspects of these phenomena is of fundamental interest, yet typically bound to numerical observations in ad-hoc models…
We prove the holding of chaos in the sense of Li-Yorke for a family of four-dimensional discrete dynamical systems that are naturally associated to ODE systems describing coupled oscillators subject to an external non-conservative force,…
We consider transformations between uniformly accelerated systems, assuming that the Clock Hypothesis is false. We use the proper velocity-time description of events rather than the usual space-time description in order to obtain linear…
We describe a computational method for constructing a coarse combinatorial model of some dynamical system in which the macroscopic states are given by elementary cycling motions of the system. Our method is in particular applicable to time…
We demonstrate the deterministic coherence and anti-coherence resonance phenomena in two coupled identical chaotic Lorenz oscillators. Both effects are found to occur simultaneously when varying the coupling strength. In particular, the…
The butterfly effect is today commonly identified with the sensitive dependence of deterministic chaotic systems upon initial conditions. However, this is only one facet of the notion of unpredictability pioneered by Lorenz, who actually…
The idea of controlling the dynamics of spatially extended systems using a small number of localized perturbations is very appealing - such a setup is easy to implement in practice. However, when the distance between controllers generating…
Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here,…