We apply the two different definitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of 1-dimensional cellular automata. A DTDS is chaotic according to the Devaney’s definition of chaos iff it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsen’s de...

We compute the dynamics of entanglement in minimal setup producing ergodic and mixing quantum many-body dynamics, which we previously dubbed boundary chaos. This consists a free, non-interacting brickwork circuit, chaos ergodicity is induced by an impurity interaction, i.e., entangling two-qudit gate, placed at system’s boundary. both conventional bipartite entropy with respect to connected sub...

This paper examines the chaotic properties of the elementary cellular automaton rule 40. Rule 40 has been classified into Wolfram’s class I and also into class 1 by G. Braga et al. These classifications mean that the time-space patterns generated by this cellular automaton die out in a finite time and so are not interesting. As such, we may hardly realize that rule 40 has chaotic properties. In...

These are some notes related to the one-semester course Math 5535 Dynamical Systems and Chaos given at the University of Minnesota during Fall 2012 with an emphasis to the study of continuous and discrete dynamical systems of dimension one and two. An ambitious list of topics to be covered include phase portraits, fixed points, stability, bifurcations, limit sets, periodic orbit, Poincaré map a...

Our goal in these notes is to understand the long-time behavior of solutions to ODEs. For this it will be very useful to introduce the notion of ω-limit sets. A remarkable result the Poincaré–Bendixson theorem is that for planar ODEs, one can have a rather good understanding of ω-limit sets. I have been benefited a lot from the textbook Differential equations, dynamical systems and an introduct...

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulli στ -shift rule and is a member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In t...

Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are given. Nonlinear evolution equation for the considered extensive chaos model is shown to have solitary and periodical waves.

In this study, a chaotic circuit suitable for an integrated circuit is proposed. The circuit consists of two CMOS ring oscillators and a pair of diodes. By using a simplified model of the circuit, the mechanism of generating chaos is explained and the exact solutions are derived. The exact expressions of the Poincaré map and its Jacobian matrix make those possible to confirm the generation of c...