arXiv subjects

M. U. Akhmet

Publications and source records attributed to M. U. Akhmet.

8 recordsLinked to original sources

arXiv2026

Integral of an alpha unpredictable function

The present paper is devoted to the study of integral properties of alpha unpredictable functions. The problem of invariance of recurrence under integration is one of the most challenging and interesting in the theory of functions. Let us start with periodicity. Next, the problem was solved for quasiperiodic, almost periodic, and Poisson stable functions. All of the problems were subjected to the condition of bounded integrals. The alpha-unpredictable functions are the endpoint in the row of recurrent functions. The class is the cross-border point from regularity to chaos in the dynamics presented through the elements of the row. In the present research, we finalized the proof that any theoretical functional recurrence is invariant with respect to integration, provided the result is bounded.

nlin.CD↗

arXiv2011

Neural networks with non-smooth and impact activations

In this paper, we consider a model of impulsive recurrent neural networks with piecewise constant delay. The dynamics are presented by differential equations with discontinuities such as impulses at fixed moments and piecewise constant argument of generalized type. Sufficient conditions ensuring the existence, uniqueness and global asymptotic stability of the equilibrium point are obtained. By employing Green's function we derive new result of existence of the periodic solution. The global asymptotic stability of the solution is investigated. Examples with numerical simulations are given to validate the theoretical results.

math.DS↗

arXiv2010

On the principles of B-smooth discontinuous flows

In this paper, we define B-smooth discontinuous dynamical systems which can be used as models of various processes in mechanics, electronics, biology and medicine. We find sufficient conditions to guarantee the existence of such systems. These conditions are easy to verify. Appropriate examples are constructed.

math.DS↗

arXiv2007

The discontinuous dynamics and non-autonomous chaos

A multidimensional chaos is generated by a special initial value problem for the non-autonomous impulsive differential equation. The existence of a chaotic attractor is shown, where density of periodic solutions, sensitivity of solutions and existence of a trajectory dense in the set of all orbits are observed. The chaotic properties of all solutions are discussed. An appropriate example is constructed, where the intermittency phenomenon is indicated. The results of the paper are illustrating that impulsive differential equations may play a special role in the investigation of the complex behavior of dynamical systems, different from that played by continuous dynamics.

nlin.CD↗

arXiv2007

A ptototype compartmental model of blood pressure distribution

We consider a system of differential equations the behavior of which solutions possesses several properties characteristic of the blood pressure distribution. The system can be used for a compartmental modeling of the cardiovascular system. It admits a unique bounded solution such that all coordinates of the solution are separated from zero by positive numbers, and which is periodic, eventually periodic or almost periodic depending on the moments of heart contraction. Appropriate numerical simulations are provided.

q-bio.TO↗

arXiv2006

On the reduction principle for differential equations with piecewise constant argument of generalized type

In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA. The Reduction Principle is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so called critical case and investigate the stability of the zero solution.

math.GM↗

arXiv2006

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

We consider existence and stability of an almost periodic solution of the quasilinear system of differential equations with piecewise constant argument of generalized type. The associated linear homogeneous system satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. The problem of positive (almost periodic) solutions of the logistic equation is discussed as an example. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.

math.DS↗

arXiv2005

Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

In this paper we establish asymptotic (biasymptotic) equivalence between spaces of solutions of a given linear homogeneous system and a perturbed system. The perturbations are of either linear or weakly linear characters. Existence of a homeomorphism between subspaces of almost periodic and asymptotically (biasymptotically) almost periodic solutions is also obtained.

math.DS↗