Non-autonomous equations with unpredictable solutions

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 applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation are proved as well as for quasilinear discrete systems. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. The results are demonstrated numerically, and an application to Hopfield neural networks is provided. In particular, Poincare chaos near periodic orbits is observed. The completed research contributes to the theory of chaos as well as to the theory of differential and discrete equations, considering unpredictable solutions.


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...
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Akhmet, Marat (Springer Science and Business Media LLC, 2020-01-01)
In this paper, we study existence of the bounded solutions and asymptotic behavior of an impulsive Bernoulli equations. Nonautonomous pitchfork and transcritical bifurcation scenarios are investigated. An examples with numerical simulations are given to illustrate our results.
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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.
Integral manifolds of differential equations with piecewise constant argument of generalized type
Akhmet, Marat (Elsevier BV, 2007-01-15)
In this paper we introduce a general type of differential equations with piecewise constant argument (EPCAG). The existence of global integral manifolds of the quasilinear EPCAG is established when the associated linear homogeneous system has an exponential dichotomy. The smoothness of the manifolds is investigated. The existence of bounded and periodic solutions is considered. A new technique of investigation of equations with piecewise argument, based on an integral representation formula, is proposed. Ap...
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Akhmet, Marat (Elsevier BV, 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. Th...
Citation Formats
M. Akhmet, “Non-autonomous equations with unpredictable solutions,” COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, pp. 657–670, 2018, Accessed: 00, 2020. [Online]. Available: