Domain-Structured Chaos in a Hopfield Neural Network

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 Poincare. Mathematical analysis and numerical simulation are provided to confirm the presence of chaos in the network.


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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...
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In this paper, we present a new method for chaos generation in nonautonomous impulsive systems. We prove the presence of chaos in the sense of Li-Yorke by implementing chaotic perturbations. An impulsive Duffing oscillator is used to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Moreover, a procedure to stabilize the unstable periodic solutions is proposed.
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Tiryaki, Aydin; Zafer, Ağacık (Elsevier BV, 2013-09-01)
In this paper we obtain new conditions for the global existence and boundedness of solutions for nonlinear second-order equations of the form
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, “Domain-Structured Chaos in a Hopfield Neural Network,” INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, pp. 0–0, 2019, Accessed: 00, 2020. [Online]. Available: