65 research outputs found
Nonlinear oscillations and stability domains in fractional reaction-diffusion systems
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 of periodic solutions for different parameters of the
problem are obtained. A computer simulation of the corresponding nonlinear
fractional ordinary differential equations is presented. It is established by
computer simulation that there exists a set of stable spatio-temporal tructures
of the one-dimensional system under the Neumann and periodic boundary
condition. The features of these solutions consist in 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.Comment: 15 pages, 5 figure
Analysis of instabilities and pattern formation in time fractional reaction-diffusion systems
We analyzed conditions for Hopf and Turing instabilities to occur in
two-component fractional reaction-diffusion systems. We showed that the
eigenvalue spectrum and fractional derivative order mainly determine the type
of instability and the dynamics of the system. The results of the linear
stability analysis are confirmed by computer simulation of the model with cubic
nonlinearity for activator variable and linear dependance for the inhibitor
one. It is shown that pattern formation conditions of instability and transient
dynamics are different than for a standard system. As a result, more
complicated pattern formation dynamics takes place in fractional
reaction-diffusion systems.Comment: 7 pages, 6 figure
Replicator dynamical systems and their gradient and Hamiltonian properties
We consider the general properties of the replicator dynamical system from
the standpoint of its evolution and stability. Vector field analysis as well as
spectral properties of such system has been studied. Lyaponuv function for
investigation of system evolution has been proposed. The generalization of the
replicator dynamics for the case of multi-agent systems has been introduced. We
propose a new mathematical model to describe the multi-agent interaction in
complex system
Mathematical description of the heat transfer in living tissue (Part II)
In the present monograph we formulate a simple model for heat transfer in
living tissue with self - regulation. The initial point of the model is the
governing equations describing heat transfer in living tissue at the mesoscopic
level, i.e. considering different vessels individually. Then, basing on the
well known equivalence of the diffusion type process and random walks, we
develop a certain regular procedure that enables us to average these mesoscopic
equations practically over all scales of the hierarchical vascular network. The
microscopic governing equations obtained in this way describe living tissue in
terms of an active medium with continuously distributed self - regulation. One
of the interesting results obtained in the present monograph is that there can
be the phenomena of ideal self - regulation in large active hierarchical
systems. Large hierarchical systems are characterized by such a great
information flow that none of its elements can possess whole information
required of governing the system behavior. Nevertheless, there exists a
cooperative mechanism of regulation which involves individual response of each
element to the corresponding hierarchical piece of information and leads to
ideal system response due to self - processing of information. The particular
results are obtained for bioheat transfer. However, self - regulation in other
natural hierarchical systems seems to be organized in a similar way. The
characteristics of large hierarchical systems occurring in nature are discussed
from the stand point of regulation problems. By way of example, some ecological
and economic systems are considered. An cooperative mechanism of
self-regulation which enables the system to function ideally is proposed
Analysis of the optimality principles responsible for vascular network architectonics
The equivalence of two optimality principles leading to Murray's law has been
discussed. The first approach is based on minimization of biological work
needed for maintaining the blood flow through the vessels at required level.
The second one is the principle of minimal drag and lumen volume.
Characteristic features of these principles are considered. An alternative
approach leading to Murray's law has been proposed. For that we model the
microcirculatory bed in terms of delivering vascular network with symmetrical
bifurcation nodes, embedded uniformly into the cellular tissue. It was shown
that Murray's law can be regarded as a direct consequence of the organism
capacity for controlling the blood flow redistribution over the
microcirculatory beds
The properties of quasispecies dynamics in molecular evolution
We consider the general properties of the quasispecies dynamical system from
the standpoint of its evolution and stability. Vector field analysis as well as
spectral properties of such system has been studied. Mathematical modelling of
the system under consideration has been performed.Comment: 11 pages, 1 figur
Hamiltonian and gradient properties of certain type of dynamical systems
From the sandpoint of neural network dynamics we consider dynamical system of
special type pesesses gradient (symmetric) and Hamiltonian (antisymmetric)
flows. The conditions when Hamiltonian flow properties are dominant in the
system are considered. A simple Hamiltonian has been studied for establishing
oscillatory patern conditions in system under consideration
Anomalous properties of heat diffusion in living tissue caused by branching artery network. Qualitative description
We analyze the effect of blood flow through large arteries of peripheral
circulation on heat transfer in living tissue. Blood flow in such arteries
gives rise to fast heat propagation over large scales, which is described in
terms of heat superdiffusion. The corresponding bioheat heat equation is
derived. In particular, we show that under local strong heating of a small
tissue domain the temperature distribution inside the surrounding tissue is
affected substantially by heat superdiffusion.Comment: 4 pages, 3 figures, RevTeX
Remarks on scaling properties inherent to the systems with hierarchically organized supplying network
We study the emergence of a power law distribution in the systems which can
be characterized by a hierarchically organized supplying network. It is shown
that conservation laws on the branches of the network can, at some
approximation, impose power law properties on the systems. Some simple examples
taken from economics, biophysics etc. are considered
Quasi-stationary Stefan problem and computer simulation of interface dynamics
The computer simulation of quasistationary Stefan problem has been realized.
Different representations of the Laplacian growth model are considered. The
main attention has been paid for the interface dynamics represented by integro
differential equations. Numerical approach has been realized by use of
interpolating polynomials and exact quadrature formulae. As a result system of
ordinary differential equations has been simulated.Comment: 13 pages, 9 figure
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