Replication of chaos

2013-10-01
We propose a rigorous method for replication of chaos from a prior one to systems with large dimensions. Extension of the formal properties and features of a complex motion can be observed such that ingredients of chaos united as known types of chaos, Devaney's, Li-Yorke and obtained through period-doubling cascade. This is true for other appearances of chaos: intermittency, structure of the chaotic attractor, its fractal dimension, form of the bifurcation diagram, the spectra of Lyapunov exponents, etc. That is why we identify the extension of chaos through the replication as morphogenesis.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION

Suggestions

Non-autonomous equations with unpredictable solutions
Akhmet, Marat (Elsevier BV, 2018-06-01)
To make research of chaos more amenable to investigating differential and discrete equations, we introduce the concepts of an unpredictable function and sequence. The topology of uniform convergence on compact sets is applied to define unpredictable functions [1,2]. The unpredictable sequence is defined as a specific unpredictable function on the set of integers. The definitions are convenient to be verified as solutions of differential and discrete equations. The topology is metrizable and easy for applica...
Domain-Structured Chaos in a Hopfield Neural Network
Akhmet, Marat (World Scientific Pub Co Pte Lt, 2019-12-30)
In this paper, we provide a new method for constructing chaotic Hopfield neural networks. Our approach is based on structuring the domain to form a special set through the discrete evolution of the network state variables. In the chaotic regime, the formed set is invariant under the system governing the dynamics of the neural network. The approach can be viewed as an extension of the unimodality technique for one-dimensional map, thereby generating chaos from higher-dimensional systems. We show that the dis...
Modeling and analyzing finite state automata in the finite field F 2
Reger, J.; Schmidt, Klaus Verner (Elsevier BV, 2004-06-29)
A method for determining multilinear state space models for general finite state automata is presented. The obtained model resides on F-2, the finite field of characteristic 2 with the operations addition and multiplication, both carried out modulo 2. It is functionally complete in the sense that it is capable of describing all finite state automata, including non-deterministic and partially defined automata. For those cases in which the model over F-2 is linear, means for a complete analysis of the cyclic ...
Nonlocal hydrodynamic type of equations
Gürses, Metin; Pekcan, Asli; Zheltukhın, Kostyantyn (Elsevier BV, 2020-06-01)
We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and integrable. They admit Lax representations and hence possess infinitely many conserved quantities.
LOCAL OPERATOR ALGEBRAS FRACTIONAL POSITIVITY AND THE QUANTUM MOMENT PROBLEM
Dosi, Anar (American Mathematical Society (AMS), 2011-02-01)
In the present paper we introduce quantum measures as a concept of quantum functional analysis and develop the fractional space technique in the quantum (or local operator) space framework. We prove that each local operator algebra (or quantum *-algebra) has a fractional space realization. This approach allows us to formulate and prove a noncommutative Albrecht-Vasilescu extension theorem, which in turn solves the quantum moment problem.
Citation Formats
M. Akhmet, “Replication of chaos,” COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, pp. 2626–2666, 2013, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/40295.