Theoretical
Foundation for the Discrete Dynamics of Physicochemical Systems:
Chaos, Self-Organization,
Time and Space in Complex Systems
V. Gontar
International Group for Scientific
and Technological Chaos Studies, Ben-Gurion University of the Negev, P.O.B.
653, Beer-Sheva 84105, Israel
Abstract
A new theoretical foundation
for the discrete dynamics of physicochemical systems is presented. Based
on the analogy between the p-theorem
of the theory of dimensionality, the second law of thermodynamics and the
stoichiometry of complex physicochemical reactions, basic dynamic equations
and an extreme principle were formulated. The meaning of discrete time
and space in the proposed equations is discussed. Some results of numerical
calculations are presented to demonstrate the potential of the proposed
approach to the mathematical simulation of spatiotemporal physicochemical
reaction dynamics.
Introduction
Since
Lorenz's discovery of chaotic solutions of differntial equations [1],
H. Haken's analysis of self-organized systems [2],
I. Prigogine's discussion about chaos and order, time and space in complex
non-equilibrium systems [3] , and L. Chua's chaotic
electrical circuirts [4], there have been so many important
contributions to the field of complex non-linear chaotic system dynamics
that it is impossible to do them all justice. There is no other field in
science that could compete with such an interest and activity. More than
two hundred books with the word chaos in the title have been published,
seven new journals devoted to chaos studies have been launched in the last
few years....
"What is good in chaos?... Chaos has already been applied to increase the
power of lasers, synchronize the output of electronic circuits, control
oscillations in chemical reactions, stabilize the erratic beat of unhealthy
animal hearts and encode electronic messages for secure communications.
We anticipate that in the near future engineers will no longer shun chaos
but will embrace it." (Scientific American, p. 78, August 1993).
"Fractals are an effort to simulate nature’s complexity. Those computer
representations are tiny, intricate patterns from which models of virtually
anything, from snowflakes to mountains, can be created. The insights from
this exercise can be so exciting that Wall Street firms are funding extensive
research into fractals and its sister field, chaos theory, to see if they
can predict stock market behavior." (Business Week, p. 169, October 12,
1992).
"Motile cells of Escherichia coli aggregate to form stable
patterns of remarkable regularity when grown from a single point on certain
substrates. Central to this self-organization is chemotaxis, the motion
of bacteria along gradients of a chemical attractant that the cells themselves
excrete. . . . Formation of spatial patterns from a mass of initially identical
cells is one of the central problems of developmental biology." (Nature,
v. 376, no. 6535, 6 July 1995).
"The discipline of chaos has created a universal paradigm, a scientific
parlance, and a mathematical tool for grappling with nonlinear phenomena.
In every field of the applied sciences (astronomy, atmospheric sciences,
biology, chemistry, economics, geophysics, life and medical sciences, physics,
social sciences, zoology, etc.) and engineering (aerospace, chemical,
electronic, civil, computer, information, mechanical, software, telecommunication,
etc.), the local and global manifestations of Chaos and Bifurcation have
burst forth in an unprecedented universality, linking scientists heretofore
unfamiliar with one another's fields, and offering an opportunity to reshape
our grasp of reality." (Aims and Scope, International Journal of Bifurcation
and Chaos. World Scientific).
It is obvious that we are faced with a new dynamic paradigm demanding explanation
and much better understanding. How this new paradigm of chaos exist alongside
with the classical dynamics, based on the use of differential equations
for which infinitesimal changes of initial conditions and parameters result
in infinitesimal changes in solutions? Chaotic solutions, which in opposition
to the conventional solutions of differential equations, are extremely
sensitive to infinitesimal changes - bringing into question the main postulate
of the correct use of differential equations. Remembering the fact that
chaotic solutions can be obtained only by computer, we must ask whether
they reflect the dynamic laws of nature (written in the form of differential
equations before chaotic regimes were proposed), or whether they are solely
the result of an extreme sensitivity to numerical procedures and computer
errors [5]. How can the continuous time and space of
differential equations reflect catastrophic changes in behavior in chaotic
systems, and how can we combine the determinism and predictability of differential
equations and the unpredictability of chaotic solutions? What is the new
role of computers (which have become an essential part of our solutions
of differential equations through the accuracy of numerical procedures)
when chaos appears and changes in accuracy can strongly affect the results
of calculation? It is obvious that computing errors never can be made "as
small as necessary"—one of the main postulates of all existing numerical
methods and the ideology of using computers—if chaotic solutions are expected.
The old contradiction between continuality and discreteness is thus once
again clamoring for an answer. The traditional chain of differential equations
- numerical solution has thus been broken by the simple fact that chaotic
solutions depend not only on differential equations but strongly on numerical
algorithms and computer errors. A new link between classical mathematical
models and their solutions should be put into the chain. This link should
include the accuracy of numerical methods, computer accuracy, etc.
There are a minimum of two ways of solving these problems. The first one
is to make a careful analysis of the numerical accuracy of integration
of differential equations (which is very hard, and not always possible,
to do). The second one is to seek other methods (calculus) different from
differential equations and free of the problems connected with their use.
The question whether systems of ordinary or partial differential equations
are the single mathematical language for describing all kinds of dynamics
derived from first principles of physics is not new. Can iterational maps
or systems of difference equations be used instead of or in parallel with
differential equations? What type of postulates do we need to accept in
order to put a sound physicochemical meaning to this new calculus—calculus
of iterations? If we succeed with this program, we not only overcome many
contradictions that result from the use of differential equations, but
we also take a step towards the better understanding of the meaning of
discrete time & space, self-organization & complexity. Of course,
any activity in finding the required principles should be based on the
traditional procedure of verification: equations derived from the new principles
should not only describe existing experimental results, should not contradict
solutions of differential equation for non-chaotic regimes, and should
predict a wide range of future experimental results. This is the program
that we intend to undertake, and with this publication, which presents
one of the possible approaches, we open the discussion.
Starting from classical chemical thermodynamics for closed systems, we
have found an analogy between the second law of thermodynamics and the
p-theorem
of the theory of dimensionality, which bring us to the formulation of a
new extreme principle for chemical reaction kinetics [6,7]
and dynamics [8].
Background
The formalism of using the notations of chemical reactions is a very common
and effective application to the description of different complex multicomponent
systems. The principle of a minimum of free energy for closed multicomponent
systems and following mass action law of thermodynamics provides us with
the necessary mathematical models in a form algebraic equations for the
case of equilibrium:
(1)
and differential equations
of the kinetic mass action law for the non-equilibrium case:
(2)
where Kl
- the equilibrium constant for the l reaction, and 
, 
- rate constants for the l reaction, 
li-
matrix of stochiomeric coefficients, D - diffusion constants.
Despite the absence of a variational principle for the rate equations (2),
they are considered as fundamental equations that have verified by numerous
experiments in chemistry, physics, biology, etc. In the case that equation
(2) is applied to simulations of chaotic processes,
the same questions that were put forward in the Introduction must be raised.
To initiate the procedure of constructing a new mathematical model, we
need to start by defining the mechanism of chemical reactions, expressed
by matrix of stoichiometrical coefficients nli.
For all types of constituents Ai (atoms, molecules,
radicals, ions, clusters, cells, etc.), the mechanism of interaction can
be written in the following form:
 nli
Ai = 0 |
i = 1, 2, . . .
, N |
|
l = 1,
2, . . . , N – M
|
(3)
When nli
is given as an initial hypothesis, all the necessary equations for simulation
can be automatically written according to the equations (1)
or equations (2). By adding the equations of the
law of mass conservation:
aij
Xi = bj
j = 1,2, . . . , M
(4)
we will complete the
system of algebraic equations in the case of equilibrium (1)
and the system of differential equations (2) in
the case of chemical kinetics. Here aij
-"molecular" matrix, defining the number of components of type "j" in the
ith constituent; and Xi
- concentration of ith constituent of the system.
According to the stoichiometrical rules of chemical transformations, for
any matrixes nli
and aij the following relationship should be satisfied:
(5)
This relation can
be used in order to calculate matrix aij
from nli,
or matrix nli,
when matrix aij is given.
Now let us turn to the foundation of the theory of dimensionality. According
to the claims of this theory, all the variables used to identify the physicochemical
systems should have a dimension expressed with the aid of the so-called
main units—meters, seconds, kilograms, etc. Variables
such as velocity, energy, pressure, etc. have well-defined dimensions,
which are obviously dependent on the choice of the main units. To overcome
the dependence of arbitrary choice of the main units, we need, and recommend,
to construct invariants (i.e., dimensionless variables). Invariants are
constructed by using the p-theorem
of the theory of dimensionality and are more conveniently expressed by
the general form of matrix operation [9].
Suppose we have N variables qi with the dimensions
defined by use of matrix Qij, where
j = 1, 2,. . . , and M - number of main units; for example, q1
= s - distance, q2
= t - time, q3 =
v - velocity, q4 = a - acceleration, variables
defining the kinematics of an arbitrary mechanical system with main units:
u1- sec (s), and u2 - meter (m).
In this case, matrix Q can be written in the form:
|
|
m
|
s
|
|
| s |
|
|
1 |
0 |
| |
| t |
|
|
0 |
1 |
| |
| v |
|
|
1 |
-1 |
| |
| a |
|
|
1 |
-2 |
| |
(6)
Now to define the
invariants according to the p-theorem,
we need to present matrix Qij in
the following form:
|
| |
R
|
| |
|
Q =
|
| |
– – – –
|
| |
|
| |
P
|
|
|
(7)
where R is
nondegenerate matrix of dimensions M ´
M, and P is a matrix
of dimensions (N-M) ´ M.
Then, the determinant matrix zli
for L=N-M invariants pl
take the following form:
zli
= |-P
R-1 
I
|
(8)
where I is the unit matrix (N-M) ´
(N-M).
For our example we
will get matrix:
(9)
and two invariants:
p1
= s-1 t1
v1 a0 =
vt/s
p2
= s-1 t2
v0 a1 =
at2 /s.
(10)
The general expression
for N-M invariants pl
is given by the p-theorem:
(10a)
Now let us transport
this formalism to the stoichiometry of chemical reactions.
Again, to make things
clear, let us use a simple example of a chemical system with four constituents
X1 = A, X2
= B, X3 = AB, X4
= AB2 and as the components of the system,
let us choose A and B. In this case, the molecular matrix aij
takes the following form:
|
|
A
|
B
|
|
|
A
|
|
|
1
|
0
|
| |
|
B
|
|
|
0
|
1
|
| |
|
AB
|
|
|
1
|
1
|
| |
|
AB2
|
|
|
1
|
2
|
| |
(11)
and the chemical "invariants"
are easy to define in the same way used in equation (8). As a result we
will get the following expressions for the two "invariants":
(12)
Now, the analogy between the equations of the thermodynamic mass action
law (1) and the p-theorem
(10a) is obvious: the matrixes aij
& Qij and nli
& zli
are mathematically equivalent, as well as equations (1)
and (10a). Of course, there is a vast difference
in the meaning of the dimensionless invariants pl
and the "chemical invariants" Pl,
which have a dimension in the sense of the theory of dimensionality, but
nonetheless we intend to use some similarities and the analogies.
As is known, equations (1) were derived in classical
thermodynamics from the principle of a minimum of free energy (or maximum
entropy) for a mixture of ideal gases. This assumption serves as a strong
constraint for the use these equations for a condensed systems. If we view
this from the position of the theory of dimensionality and suppose that
the dimensionless character of the "chemical invariants" reflects the independence
of the thermodynamic constants Kl (mathematical
analog to pl)
on the concentrations of M components of the system, we can apply equations
(1) to any systems with chemical reactions, without
constraints, based on ideal gases assumptions. If the initial system of
chemical reactions does not give an adequate description, according to
the theory of dimensionality, we need to change our initial hypothesis
about the mechanism and add some more components and constituents, or change
the type of chemical reaction.
Another recommendation that can be taken from the theory of dimensionality
is the claim for an invariant to be constant (not dependent on time, initial
concentrations, or concentrations of constituents) for similar systems.
If a new situation arises (for example, from simulation of a kinematic
system, we are going to analyze the dynamics), we need to add a new component
and variables in the matrix zli
(kg as a main unit in this particular case, and force, energy, as new variables
etc.). Following this rule for moving to the description of the dynamics
of the physiochemical systems, we need to add a new "time component" in
the molecular matrix aij, when going to
dynamic simulations from thermodynamic equilibrium. The details of such
an approach can be found in ref. [10]. If we do this
for a closed system, we will get, after simple operations of the matrixes,
the time-dependent functions Pl(tq)
= Kl exp[-Wl/tq]
on the right side of equations (1), where q = 1,2.
. . :
=
Kl exp[-Wl/tq]
(13)
which together with
equations (4) represent a complete system of N
non-linear algebraic equations for N variables Xi(tq).
The reasons for our choice of an exponential function can be found in the
asymptotic behavior of chemical systems [10]. The
equations (4,13) ha unit
solution for all Xi(tq)
> 0. This is the way in which we can construct a system of algebraic equations
for mathematical simulation of the dynamics of N concentrations Xi(tq)
without using of differential equations (2) for
any mechanisms of chemical reactions given by matrix nli.
Numerous numerical calculations have been made to compare the solutions
that have come out of the ordinary differential equations (without diffusion)
of the kinetic mass action law and the proposed equations for the same
mechanisms of chemical reactions, and good qualitative agreement was obtained
[6]. System (13,4)
is far more simple from a computational point of view (no problems with
the stiffness of differential equations) and can serve as good approximation
to the solutions of differential equations.
Continuing to the mathematical simulations of open catalytic systems, we
intend to include in our molecular matrix aij a new
component reflecting time delay interaction, which after simple transformations
[10], will bring us to a system of difference equations. In this case,
Pl
becomes a function of all concentrations Xi,
calculated at previous moments of time tq-s,
s = 1, 2, . . . For an open system, when tq changes
from t0 to tc - a constant (characteristic
time for any specific system, when the steady state is reached), system
(13,4) transforms from
algebraic equations to a set of non-linear difference equations. In the
same conceptional way, we insert space coordinate r through dependence
Xi(rÄ)
in our matrix aij, and the final function
for Pl
can be written in the form:
(14)
where ali,
bli
are empirical parameters, characterizing the intensity of feedbacks in
time - and the intensity influences of the space-distributed concentrations
of the neighbors Xi(rÄ)
- on the lth chemical reaction.
We have now constructed a full system of basic equations for simulations
of all types of complex chemical reaction dynamics, starting with the initial
hypotheses of the mechanism given by matrix nli.
Using the fact that
equations (1) express the minimization of free
energy, we would like to generalize this principle and formulate a new
extreme principle for chemical reaction dynamics: Reactions in multicomponent
physicochemical systems proceed in such a way that at each instant of time
tq (q = 1,2. . . ) and at every point of
considered space r(r1, r2,
r3) the function:
F(tq,r)
= 
{lnXi(tq)
– fi – 1} i = 1,2, . . . ,N
(15)
reaches its minimum
in the concentration space Xi, subject
to the to the law of mass conservation (4).
Here:
where
Pl
is defined by equation (14).
Results of Numerical
Calculations
Some
results of numerical calculations can now be presented to demonstrate possible
applications of our approach to the mathematical simulation of the spatial
dynamics of pattern formation and time series. Let us consider the following
mechanism of chemical transformations:
(16)
where A = X1,
B = X2, C = X3
and X3 affects reaction A®B
and X1 affects reaction B®C.
We constrain ourselves with the case, in which only the previous moment
of time is included in the consideration s=1. According to the mechanism
(15), we have the following stoichiometric and
molecular matrixes:
|
|
| |
-1
|
1 |
0 |
| |
|
|
|
|
|
|
|
1
|
| |
|
nli
|
= |
| |
|
|
|
| |
|
|
|
aij
|
=
|
|
|
1
|
| |
|
|
| |
-1
|
0
|
1
|
| |
|
|
|
|
|
|
|
1
|
| |
|
X2(tq,r)
|
|
|
|
--------
|
=
|
K1{[a13X3(tq–1)
+ b11X1(tq–1,rÅ)
+ b12X2(tq–1,rÅ)
+ b13X3(tq–1,rÅ)]/tc}
|
|
X1(tq,r)
|
|
|
|
|
|
|
X3(tq,r)
|
|
|
|
--------
|
=
|
K2{[a11X1(tq–1)
+ b21X1(tq–1,rÅ)
+ b22X2(tq–1,rÅ)
+ b23X3(tq–1,rÅ)]/tc}
|
|
X1(tq,r)
|
|
|
X1(tq,r)
+ X2 (tq,r) + X3
(tq,r) = b
(17)
Concerning spatiotemporal dynamics, we constrain ourselves in this publication
to the plane r = r(r1, r2)
represented by a square lattice with Â*Â
elements with coordinates r(r1,
r2). In each cell, we calculate Xi(tq,r)
according to equations (17), which are reflect
the influence of the concentrations of the neighbors Xi(tq-1,rÅ)
on the reaction rate in each cell with coordinates r(r1,r2).
The initial conditions are: Xi(tq,r)
= b, X2(tq,r)
= X3(tq,r)
= 0. The boundary conditions reflect the fact of the absence of neighbors
for all cells with coordinates |r|= Â.
Our intention with this mathematical model (17)
was to demonstrate the ability of the proposed theoretical approach to
generate different types of complex chemical oscillations behavior and
to simulate pattern formation dynamics.
In Picture
1, we present different time series obtained from equation (17).
Pictures 2
and 3 present a process of creation of a patterns that
are growing and moving in space. Picture 4 demonstrates the creation of
two patterns and their interaction. Picture 5 demonstrates the evolution
of chemical spiral waves. In Picture 6 we present some symmetrical patterns.
All patterns were obtained in a square lattice with 160´160
cells, except picture 4: 70x70 cells. The color of each cell reflects the
value of the concentration of Xi(tq,r).
Calculations were performed on a PC-486 computer.
Discussion and
Conclusions
We have presented a brief description of a new theoretical approach to
physicochemical reaction dynamics. We consider this approach to be the
basis for constructing a solid theoretical foundation for the application
of a system of algebraic and difference equations, instead of, or in parallel
to, the commonly used system of differential equations. The advantages
of difference equations from the computational point of view are obvious
(the dynamics of pattern formation presented here takes about 10 min for
300 iterations on a PC-486).
Now let us clarify the meaning of time and space that we face in our approach
and equations. We think that there is a general problem of defining the
meaning of so-called "discrete time and space" appearing when iterational
maps or difference equations are used. The questions arise when we try
to make a link between the continuous time and space of differential equations
and discrete nature of our numerical calculations. As a precondition for
sampling, we suppose the existence of a "small-as-necessary" time interval
Dt.
This concept fails when chaotic solutions can be obtained from differential
equations. When problems of computer accuracy and error diffusion problems
are also considered, an analysis of the real meaning and quality of the
obtained solutions becomes extremely complicated.
Our approach from the very beginning was directed towards constructing
algebraic equations—and then difference equations—which are, by definition,
free of the above-mentioned problem. For a closed system, equations (1)
and (13) demonstrate a new time tq.
Trajectories Xi(tq)
are defined by the solutions of the system of non-linear algebraic equations
(1,4,13).
Despite of the close similarity of these solutions to the solutions of
differential equations, the meaning of time tq
and that of continuous time t in (2) is different
from the calculation point of view, and tq can
be considered as discrete time by virtue of the way that this time appears.
The continuality of time t in differential equations is connected to the
existence of lim dX/dt and dt ®
0, in contrast to tq, which runs in arbitrary
way from t0 in any order we need.
The next step in the proposed theory is the description of chemical reactions
with feedbacks, when the parameters of the initial mathematical model get
the dependence of the concentration Xi
calculated at the previous "moment of time tq-s".
In this case, the algebraic equations transform into a system of difference
equations, and for an open system our discrete time tq
becomes a constant or in another words disappears as a dynamic variable.
Difference equations, being dynamic or evolutionary equations by their
very origin, have no time in the sense of time of the systems of differential
equations (2) (through derivatives dX/dt) or even
in the sense of the discrete time tq in
equations (13,1,4).
Astronomic continuous time t commonly used for interpretation of all types
of dynamic behavior has disappeared from our difference equations, being
transformed in the set of internal times resulting from our calculations.
The initial scale of time is not important for difference equations and
can be easily attached to the astronomic time in our experiments by normalization
of parameters of the model when trajectory X is calculated. This means,
in another words, that the system is "self-organizing in our physical time
and space," and only the character of the interactions is responsible for
any time intervals appearing in the system's dynamics. The smallest time
interval ?t, as well as all the other intervals, can be calculated by spectrum
analysis of trajectories X. The fact that the smallest interval no longer
depends on the limitations of the numerical procedure of integrating differential
equations is very important for practical computer calculations.
The situation with simulation of spatial dynamics by our system of difference
equations is also different from use of partial differential equations
(2). Spatial coordinates are not included in the
basic equations, and are present only through dependence of parameters 
l
from concentrations Xi(tq,r).
The absence of derivatives in our basic equations frees us from the
necessity of existence of continuous curves, as happened in the case of
solutions of partial differential equations (2).
If we look at Picture 2, at the early stages of distribution
of the concentration we see that concentrations are not organized in any
specific form. But later, due solely to chemical transformations going
in each cell of the square lattice we can see some process of concentration
organization in a form that reminds us of spiral and ring curves. In fact,
these are not curves in the sense of continuous curves that can be obtained
from partial differential equations. What we obtain from the new basic
equations is the direct concentration distribution of particular constituent
X in space according to our extreme principle and initial hypothesis of
the chemical reaction mechanism. This principle is self-sufficient for
the simulation of different types of spatiotemporal behavior of chemical
waves and other pattern formations, which was previously the only partial
differential equations [11] or "cellular automata"
method with very little physicochemical background [12].
We should also relate to the stochastic process of mathematical simulation.
Because chaotic regimes are present
in the proposed basic difference equations, there is no special need to
add a "random number generator", as should be done with the differential
equations. The random number generator and chaotic regimes are the same
in their mathematical nature.
In summary, we have demonstrated the existence of other theoretical and
mathematical foundations to chemical reaction dynamics than the differential
equations of the kinetic mass action law.
We are perched at the beginning of the development of this new paradigm
for discrete dynamics and see as our goal for future studies the development
of a calculus of difference equations in the same manner as it was done
with the calculus of the infinitesimal for differential equations. The
results already coming out from proposed approach prove the tremendous
potential of the proposed mathematical model that can be effectively used
in real-time control systems, neural networks, and in signal processing
as practically unlimited source of different types of oscillatory and spatiotemporal
behavior simulations. These approaches can be effectively used for mathematical
simulations of living systems. We are sure that the application of the
concept of proposed discrete dynamics and the new basic equations will
find application in economics, medicine, computer science and many other
theoretical and practical areas for which extensive computing time is required,
without losing the physicochemical sense and theoretical background of
mathematical models.
Pictures:
Pic.1
Pic.2 Dynamics of pattern formation, obtained by
new basic equations for the chemical reactions mechanism: A-->B-->C. 1-
50th iteration, 2-100th, etc.
Pic.3 Dynamics of pattern formation for the same
mechanism of chemical reactions, but different parameters of the mathematical
model.
Pic.4. Dynamics of patterns
formation and their interaction.
Pic.5. Evolution of spiral
chemical waves , obtained by new basic equations for B-Z reaction: 1- 300th
iteration, 2- 400th, ..., 6- 800th iteration.
Pic.6. Symmetrical patterns obtained by new basic
equations.
Acknowledgment
I would like to express
my gratitude to the Ben-Gurion University of the Negev, the Lady Davis
Trust, and the Doron Foundation for Education and Welfare, for their support.
References
1.
E.N. Lorenz, "Deterministic non periodic flow", J. Am. Sci. 1963, 20, 130-141
2.
H. Haken, Synergetic: An Introduction, Springer Ser. Synergetics, Vol.1,
1978.
3.
I. Prigogine, “From Being to Becoming: Time and Complexity in the Physical
Sciences”, W.H. Freeman, San-Francisco, 1980.
4.
L. O. Chua, C. A. Desoer, and E. S. Kuh, "Linear and Nonlinear Circuirts"
New York, NY: McGraw-Hill, 1987.
5.
J.H.E. Cartwright and O. Piro, “The Dynamics of Runge-Kutta Methods”, Int.
Journal of Bifurcation and Chaos, V. 2, N. 3, 1992, pp. 427-450.
6.
V. Gontar, “Theory of Dimensionality and Chemical Reactions Kinetics. A
New Model of Chemical Kinetics”, Vest. Mosc. University, Chem. V. 17, N.
1, 1976, pp. 108-110.
7.
V. Gontar, “New Principle and Mathematical Model for Chemical Reactions
Kinetics”, R. Journal Phys. Chem. V. 55, N. 9, 1981, pp. 2301-2304.
8.
V. Gontar, I. Ilin. “New Mathematical Model of Physicochemical Dynamics”.
Contrib. Plasma Phys. 31, N. 6, 1991, pp. 681-690
9.
L. Brand, “p-Theorem
of the Theory of Dimensionality”, Arch. Rational Mech. Anal., V. 1, 1957,
pp.35-43.
10
V. Gontar, “New Theoretical Approach for Physicochemical Reactions Dynamics
with Chaotic Behavior”, In: “Chaos in Chemistry and Biochemistry”, World
Scientific, London, 1993, pp. 225-247.
11.
I. R. Epstein, K. Showalter, "Non-linear Chemical Dynamics: Oscillations,
Patterns, and Chaos", J. Phys. Chem. 1996, 100, pp. 13132-13147.
12.
T. Toffoli, N. Margolus: Cellular Automata Mashines - A New Environment
For Modelling, MIT Press Cambridge/Mass, London/ Engl. !987.
{
– 
} 
