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We consider a family of singular maps as an example of a simple model of dynamical systems exhibiting the property of robust chaos on a well defined range of parameters. Critical boundaries separating the region of robust chaos from the…
The dynamics of decisions in complex networks is studied within a Markov process framework using numerical simulations combined with mathematical insight into the process mechanisms. A mathematical discrete-time model is derived based on a…
To make predictions or design control, information on local sensitivity of initial conditions and state-space contraction is both central, and often instrumental. However, it is not always simple to reliably determine instability fields or…
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 propose a theory of deterministic chaos for discrete systems, based on their representations in binary state spaces $ \Omega $, homeomorphic to the space of symbolic dynamics. This formalism is applied to neural networks and cellular…
A dynamical network, a graph whose nodes are dynamical systems, is usually characterized by a large dimensional space which is not always accesible due to the impossibility of measuring all the variables spanning the state space. Therefore,…
Stability and analysis of multi-agent network systems with state-dependent switching typologies have been a fundamental and longstanding challenge in control, social sciences, and many other related fields. These already complex systems…
In this paper, we introduce two deterministic models aimed at capturing the dynamics of congested Internet connections. The first model is a continuous-time model that combines a system of differential equations with a sudden change in one…
Ever since the link between nonlinear science and cryptography became apparent, the problem of applying chaotic dynamics to the construction of cryptographic systems has gained a broad audience and has been the subject of thousands of…
For spatiotemporal chaos described by partial differential equations, there are generally locations where the dynamical variable achieves its local extremum or where the time partial derivative of the variable vanishes instantaneously. To a…
The full family of discrete logistic maps has been widely studied both as a canonical example of the period-doubling route to chaos, and as a model of natural processes. In this paper we present a study of the stochastic process described…
We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory…
This work redefines the framework of chaos in dynamical systems by extending Devaney's definition to multiple mappings, emphasizing the pivotal role of nonlinearity. We propose a novel theorem demonstrating how nonlinear dynamics within a…
Simple dynamical systems -- with a small number of degrees of freedom -- can behave in a complex manner due to the presence of chaos. Such systems are most often (idealized) limiting cases of more realistic situations. Isolating a small…
We study nonlinear dynamics on complex networks. Each vertex $i$ has a state $x_i$ which evolves according to a networked dynamics to a steady-state $x_i^*$. We develop fundamental tools to learn the true steady-state of a small part of the…
A measure describing the chaos of a dynamics was introduced by two complexities in information dynamics, and it is called the chaos degree. In particular, the entropic chaos degree has been used to characterized several dynamical maps such…
The paper examines the discrete-time dynamics of neuron models (of excitatory and inhibitory types) with piecewise linear activation functions, which are connected in a network. The properties of a pair of neurons (one excitatory and the…
We consider the dynamics of continuously measured many-body chaotic quantum systems. Focusing on the observable of state purification, we analytically describe the limits of strong and weak measurement rate, where in the latter case…
We show that under certain simple assumptions on the topology (structure) of networks of strongly interacting chaotic elements a phenomenon of long range action takes place, namely that the asymptotic (as time goes to infinity) dynamics of…
Autonomous randomly coupled neural networks display a transition to chaos at a critical coupling strength. We here investigate the effect of a time-varying input on the onset of chaos and the resulting consequences for information…