Quasilinear systems with unpredictable relay perturbations

It is rigorously proven under certain assumptions that a quasilinear system with discontinuous right-hand side possesses a unique unpredictable solution. The discontinuous perturbation function on the right-hand side is defined by means of an unpredictable sequence. A Gronwall–Coppel type inequality is utilized to achieve the main result, and the stability of the unpredictable solution is discussed. Examples with exponentially asymptotically stable and unstable unpredictable solutions are provided.

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[1] Akhmet M, Fen MO. Unpredictable points and chaos. Communications in Nonlinear Science and Numerical Simulation 2016; 40: 1-5.
[2] Akhmet M, Fen MO. Existence of unpredictable solutions and chaos. Turkish Journal of Mathematics 2017; 41: 254-266.
[3] Akhmet M, Fen MO. Poincaré chaos and unpredictable functions. Communications in Nonlinear Science and Numerical Simulation 2017; 48: 85-94.
[4] Akhmet M, Fen MO. Non-autonomous equations with unpredictable solutions. Communications in Nonlinear Science and Numerical Simulation 2018; 59: 657-670.
[5] Akhmet M, Fečkan M, Fen MO, Kashkynbayev A. Perturbed Li-Yorke homoclinic chaos. Electronic Journal of Qualitative Theory of Differential Equations 2018; no. 75: 1-18.
[6] Akhmet M, Fen MO, Tleubergenova M, Zhamanshin A. Poincaré chaos for a hyperbolic quasilinear system. Miskolc Mathematical Notes 2019; 20: 33-44.
[7] Akhmet M, Tleubergenova M, Zhamanshin A. Quasilinear differential equations with strongly unpredictable solutions. Carpathian Journal of Mathematics 2020; 36: 341-349, 2020.
[8] Akhmet M, Tleubergenova M, Zhamanshin A. Inertial neural networks with unpredictable oscillations. Mathematics 2020; 8: 1797.
[9] Akhmet M, Aruğaslan Çinçin D, Tleubergenova M, Nugayeva Z. Unpredictable oscillations for Hopfield-type neural networks with delayed and advanced arguments, Mathematics, 2021, 9(5), 571.
[10] Akhmet M, Tleubergenova M, Zhamanshin A. Modulo periodic Poisson stable solutions of quasilinear differential equations, Entropy, 2021, 23(11),1535.
[11] Akhmet M, Tleubergenova M, Nugayeva Z. Unpredictable oscillations of impulsive neural networks with Hopfield structure. In: Hemanth J, Yigit T, Patrut B, Angelopoulou A (editors). Trends in Data Engineering Methods for Intelligent Systems - Proceedings of the International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2020), Switzerland: Springer, 2021, pp. 625-642.
[12] Akhmet M. Domain Structured Dynamics: Unpredictability, Chaos, Randomness, Fractals, Differential Equations and Neural Networks. Institute of Physics Publishing, 2021.
[13] Akhmet M, Tleubergenova M, Nugayeva Z. Unpredictable solutions of impulsive quasi-linear systems. Discontinuity, Nonlinearity, and Complexity 2022; 11: 73-89.
[14] Bainov D, Simeonov P. Integral Inequalities and Applications. Dordrecht: Kluwer Academic Publishers, 1992.
[15] Cartwright ML, Littlewood JE. On non-linear differential equations of the second order: I. the equation y − k(1 − $y^2$) ˙y + y = bλk cos(λt + α), k large. Journal of the London Mathematical Society 1945; 1: 180-189.
[16] Corduneanu C. Principles of Differential and Integral Equations. The Bronx, New York, USA: Chelsea Publishing Company, 1977.
[17] Fen MO, Tokmak Fen F. Unpredictable oscillations of SICNNs with delay. Neurocomputing 2021; 464: 119-129.
[18] Filippov AF. Differential Equations with Discontinuous Righthand Sides: Control Systems. Berlin, Germany: Springer, 1988.
[19] Hale JK. Ordinary Differential Equations. Malabar, Florida, USA: Krieger Publishing Company, 1980.
[20] Hale J, Koçak H. Dynamics and Bifurcations. New York, USA: Springer-Verlag, 1991.
[21] Miller A. Unpredictable points and stronger versions of Ruelle-Takens and Auslander-Yorke chaos. Topology and its Applications 2019; 253: 7-16.
[22] Sell GR. Topological Dynamics and Ordinary Differential Equations. London, UK: Van Nostrand Reinhold Company, 1971.
[23] Thakur R, Das R. Strongly Ruelle-Takens, strongly Auslander-Yorke and Poincaré chaos on semiflows. Communications in Nonlinear Science and Numerical Simulation 2020; 81: 105018.
[24] Thakur R, Das R. Sensitivity and chaos on product and on hyperspatial semiflows. Journal of Difference Equations and Applications 2021; 27: 1-15.
[25] Ueda Y. Survey of regular and chaotic phenomena in the forced Duffing oscillator. Chaos, Solitons & Fractals 1991; 1: 199-231.
[26] Van der Pol B, Van der Mark J. Frequency demultiplication. Nature 1927; 120: 363-364.