Projection dynamics of highly dissipative systems

Lubashevskii, I. A.; Gafiychuk, V. V.

doi:10.1103/physreve.50.171

Cited by 11 publications

(7 citation statements)

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“…I should be noted that solution of the system using Grünwald-Letnikov derivative approximation (16) instead of Caputo one leads practically to the same attractor. The only difference is contained in transition dynamics due to the influence of the last term in Eq.…”

Section: Computer Simulation Of Pattern Formationmentioning

confidence: 97%

“…(see, for example, [23,3,21,13,4,5,16,9,30,22]). The main result obtained from these systems is that nonlinear phenomena include diversity of stationary and spatio-temporary dissipative patterns, oscillations, different types of waves, excitability, bistability, etc.…”

Section: Introductionmentioning

confidence: 99%

“…Our particular interest is the analysis of the specific nonlinear system of FRD equations. We consider a very well-known example of the RDS with cubical nonlinearity [21,13,16] which probably is the simplest one used in RD systems modeling…”

Section: Introductionmentioning

confidence: 99%

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Mathematical modeling of time fractional reaction–diffusion systems

Gafiychuk

Datsko

Meleshko

2008

Journal of Computational and Applied Mathematics

201

101

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We study a fractional reaction-diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction-diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.

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Section: Computer Simulation Of Pattern Formationmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Mathematical modeling of time fractional reaction–diffusion systems

Gafiychuk

Datsko

Meleshko

2008

Journal of Computational and Applied Mathematics

201

101

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“…We remark that the possibility of applying very simple variational principles, for example, the principle of Gauss, depends on the particular properties of the dissipative system (the geometry of the attractor or the geometry of the manifold f~, the Frrchet derivative, and so forth) [3]. However, when the order of the small parameter e is low, such variational principles make it possible to obtain the evolution equations that describe the attractor.…”

Section: Udc 5179:541128mentioning

confidence: 99%

Analysis of dissipative structures in the Gierer-Mainhardt model

Gafiichuk¹,

Demchuk²

1998

J Math Sci

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To analyze a manifold that approximates an attractor of a dissipative system we propose a Gaussian variational principle.For a Gierer-Mainhardt system we obtain closed-form expressions for the approximating stationary solution and study its stability.The evolution of distributed dissipative systems can be described by the following nonlinear equations:Ot "'where i = l, N, N eN, and {gti(r, t)} arc fields that describe the state of the system and are regarded as realvalued functions of the time t and the spatial coordinates r; (F~} are the components of the nonlinear evolution operator F, which depends on the fields {tg~} and the external parameters A I ..... A~t, M e N. Because of the dissipation, as t -~ oo the system tends to a ccrtain state in the space T of functions {~i}. In what follows we shall denote this state by f~" and call it an attractor of the dissipative system. The problem is to study the geometry of the attractor of the corresponding system and the behavior of the system in a neighborhood of it. For isolated dissipative systems numerical analysis can bc used to determine a priori the general form of an attractor, that is the form of the fields {gti(r, t)} that are approximate solutions of Eqs. (I) as t--~ co. Using such a method we can construct a certain manifold f2 in the space W that characterizes these solutions and is an approximating manifold of the attractor ~'. Such a manifold may have finite dimension p, and in that case it can bc described as follows: f~ = {v,(r) = (I),(r, u, ..... u,)},where ( The main idea of the approximation in the present work is to reduce the system of equations (1) to evolution equations that contain only the variables (u~, ..., up), and whose time dependence characterizes the motion of the system along the attractor.To analyze dissipative systems we consider the well-known variational principle of Gauss, which is connected with the following functional, called the minimal constraint functional:We regard the fields {~, ~u, } as pairwise independent variables, and the conditionleads to Eq. (1). To obtain the evolution equations in the variables (u~ ..... up), we substitute the functions %(r, t) = dO~ ( r, u~ (t) ..... up(t) ) into the functional (3) and minimize it with respect to ti~, zi 2 .... , zip. It should be remarked that this procedure requires justification. Indeed, these variational principles come from classical mechanics, where reducing the dimension of a system requires the existence of additional ideal (holonomic) constraints on the variables of the system. These constraints cause additional forces to arise and act on the

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“…For dissipative systems some variation principles can be constructed as well. Some approaches in this direction were made in [7,8]. In this case using model functions for an approximate attractor present in the system appears to be very productive and makes it possible to find an approximate form of the governing solutions.…”

Section: Introduction Complex Dynamics -Immanent Property Of Nonlinementioning

confidence: 99%

Oscillatory pattern formation in a neural dynamical system governed by a mutual Hamiltonian and gradient vector field structure

Gafiychuk¹,

Prykarpatsky²

2004

Condens. Matter Phys.

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We analyze dynamical systems of general form possessing gradient (symmetric) and Hamiltonian (antisymmetric) flow parts. The relevance of such systems to self-organizing processes is discussed. Coherent structure formation and related gradient flows on matrix Grassmann type manifolds are considered. The corresponding graph model associated with the partition swap neighborhood problem is studied. The criterion for emerging gradient and Hamiltonian flows is established. As an example we consider nonlinear dynamics in a neuron network system described by a simulative vector field. A simple criterion was written in order to establish conditions for the formation of an oscillatory pattern in a model neural system under consideration.

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Projection dynamics of highly dissipative systems

Cited by 11 publications

References 1 publication

Mathematical modeling of time fractional reaction–diffusion systems

Mathematical modeling of time fractional reaction–diffusion systems

Analysis of dissipative structures in the Gierer-Mainhardt model

Oscillatory pattern formation in a neural dynamical system governed by a mutual Hamiltonian and gradient vector field structure

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