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 discrete Hopfield neural network considered is chaotic in the sense of Devaney, Li–Yorke, and Poincaré. Mathematical analysis and numerical simulation are provided to confirm the presence of chaos in the network.
A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.
In this article, we show that a chaotic behavior can be found on a cube with arbitrary finite dimension. That is, the cube is a quasi-minimal set with Poincarè chaos. Moreover, the dynamics is shown to be Devaney and Li-Yorke chaotic. It can be characterized as a domain-structured chaos for an associated map. Previously, this was known only for unit section and for Devaney and Li-Yorke chaos.Chaos has become a very important concept that is deeply integrated into many, if not most, fields of science such as physics, biology, medicine, engineering, culture, and human activities [1,2]. The chaotic behavior of some physical and biological properties was formerly attributed to random or stochastic processes or uncontrolled forces [3,4]. Appearance of chaos in deterministic systems drew the borderline between (deterministic) chaos and stochastic noise. The idea is manifested in the chaotic behavior of simple dynamical systems. However, the randomness theory of KolmogorovMartin-Lfwhich still can provide a deeper understanding of the origins of deterministic chaos [1]. The fundamental theoretical framework of chaos was developed in last quarter of the twentieth century. During that period, different types and definitions of chaos where formulated. In general, chaos can be defined as aperiodic longterm behavior in a deterministic system that exhibits sensitive dependence on initial conditions [5]. Devaney [6] and Li-Yorke [7] chaos are the most frequently used types, which are characterized by transitivity, sensitivity, frequent separation and proximality. Another common type occurs through period-doubling cascade which is a sort of route to chaos through local bifurcations [8-10]. In the papers [11,12], Poincar chaos was introduced through the unpredictable point concepts. Further, it was developed to unpredictable functions and sequences.Whoever searches in this field can discern from the literature that there is a scientific conception that chaos is everywhere. Realizing such an ideation needs to developed our mathematical tools to conceptualize all manifestation of the phenomenon. Strictly speaking, we should develop simple chaotic mechanisms that has the ability to emulate complex behaviors. Investigating the fundamental aspects of high-dimensional chaotic states is necessary in this direction. Indeed, mathematical modeling of real-world problems show that real life is very often a highdimensional chaos and even chaotic activities in our everyday lives are difficult to described via low-dimensional systems [13].Recently, in papers [14, 15], we have developed a new method of chaos formation which depends rather on the way of partition of the domain, than on a map. That is, the map is a natural consequence of the structure of the domain to be chaotic. In the present study, we extend the approach of consideration of the methodological problem on generosity of chaos as dynamical phenomenon in real world, science and industry. This question has not been discussed in our previous researches and the present study is a co...
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