A singularly perturbed differential equation with piecewise constant argument of generalized type

Akhmet, Marat; Dauylbayev, M. K.; Mirzakulova, A.E.

doi:10.3906/mat-1704-19

Cited by 8 publications

(3 citation statements)

References 20 publications

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“…The research of complex dynamics as well as differential equations with singularities has been of great interest in recent decades [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Strongly unpredictable solutions of difference equations

Akhmet

Tleubergenova²,

Zhamanshin³

2019

Int. j. math. phys.

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It so happens that the line of oscillations in the classical theory of dynamical systems, which is founded by H.Poincar´e and G.Birkhoff was broken at Poisson stable motions. The next oscillations were considered as actors of chaotic processes. This article discusses the new type of oscillations, unpredictable sequences, the presence of which proves the existence of Poincare chaos. The sequence is defined as an unpredictable function on the set of integers. The results continue the description of chaos which isinitiated from a single motion, an unpredictable one. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a quasilinear difference equation are proved. An example with numerical simulations is presented to illustrate the theoretical results. Since unpredictability is request for all coordinates of solutions, the concept of strong unpredictability can be useful for investigation of neural networks, brain activity, robotics, where complexity is related to optimization and effectiveness.

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“…The research of complex dynamics as well as differential equations with singularities has been of great interest in recent decades [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Strongly unpredictable solutions of difference equations

Akhmet

Tleubergenova²,

Zhamanshin³

2019

Int. j. math. phys.

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“…Note that other mathematical school of singularly perturbed equations in Kazakhstan and abroad investigate only boundary value problems, which does not have an initial jump. In our previous works in [1][2][3][4][5][6][7][8][9][10], we considered the initial and boundary value problems that are equivalent to the Cauchy problem with the initial jump for differential and integro-differential equations in the stable case.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Expansion of Solution of General BVP With Initial Jumps for Higher-Order Singularly Perturbed Integro-Differential Equation

Dauylbayev

Atakhan²,

Mirzakulova

2018

Physico-Mathematical Series

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“…Uniformly valid approximations for such functions can often be found by the modified boundary function method. In our previous works in [1][2][3][4][5][6][7][8][9][10][11][12][13] we considered the initial and boundary value problems with the initial jump for differential and integro-differential equations in the stable case. In the article [14] the constructive formula for solutions of boundary value problems for singularly perturbed higher order linear integrodifferential equations has been obtained.…”

mentioning

confidence: 99%