On Mathematical Modeling of Dynamics of Complex Systems. Synergetics
As in all other areas of knowledge, progress in the study of the nature of systems occurs through the accumulation of experimental information and the development of models of the object of knowledge in the direction of increasing their completeness and accuracy. In the limit, this leads to the desire to build mathematical models of systems (see Section 3.4, Part I). It is not surprising that in mathematical modeling of systems physicists advanced the farthest and farthest away: the highest form of knowledge of physical systems are models of theoretical (mathematical) physics, generalizing and explaining the results obtained by experimental physics.
The finding by physicists the main system-wide property of nature,—emergence, occurred when considering the dynamics of physical systems, that is, in the description of natural changes over time, the state of a particular physical system. The manifestation of emergence in dynamics is called synergy (from the Greek “joint action”), emphasizing the nonlinearity of the common action, that is, not just the sum of particular actions, but the effect of emergence of their interactions.
In physics, it is customary to describe the state of a physical system by determining its observable and quantitatively measured characteristics: the values of constant (during the experiment) parameters, and the values of variable parameters in the course of observation.
For example, a free-swinging pendulum is characterized by constant parameters: the initial amplitude, weight (mass) of the pendulum, the distance between the suspension point and the center of mass, the coefficient of friction on the suspension axis, the coefficient of air resistance to the movement of the pendulum; and variable parameters: the position of the pendulum at each time (deviation from the equilibrium point), and vectors of speed and acceleration of motion at a given time.
At each moment of time, the state of the system is characterized by a specific set of variable values, that is, coordinates of a point in the state space: over time, this point moves, forming a certain trajectory. The specificity of the nature of this system is manifested in certain stable features of its trajectories in the state space.
In physics, the state space is called phase space because for some physical systems different areas of this space correspond to qualitatively different phase states of the system (e.g., for molecular matter in the space “temperature—pressure” is the phase of solid, liquid, gaseous, and plasma states).
It is easy to see that the quantitative modeling of dynamics occurs according to the general scheme with qualitative modeling of static systems. The set of parameters is a model of the system composition. The model of the system structure (i.e., the description of the interactions between the parts) is an equation that relates the parameters to each other. The problem is a description of the inputs (setting the initial state of the system), and its solution is a description of the outputs of the system (together it is a black box model). The solution of the system equation is a quantitative description of the synergetic behavior of the system (determination of the trajectory of the emergent state of the system in phase space; the “phase portrait” of the system): each structure corresponds to its inherent archetype of behavior (in synergetics it is an attractor).
Already from the qualitative model we know that the archetype is determined by the specific structure of the feedback (positive and negative) links and time delays of resource flows in the links. These features of the system in the mathematical model are displayed by a function that quantitatively describes the relationship between the parameters. For example, the simplest equations of motion describe the relationships between the parameters x = [x{, x2,..., xn) and their velocities dxjdf.
Positive (amplifying) feedbacks are displayed by nonlinear terms in polynomials Vt(x); negative (weakening) by members of attenuation, energy dissipation; and delays by parameters of inertia.
Naturally, different structural schemes of the system, that is, different functions V(x) = ={Vj(x)} correspond to different emergent (synergetic) effects. Convergence of dynamic processes to certain specific configurations is interpreted as a process of self-organization of the system, which gave reason to talk about synergetics as a theory of self-organization.
In world science, there are scientific schools that develop different specific branches of synergetics: the I. Prigogine’s Brussels school of nonequilibrium thermodynamics of macro systems, the Lorenz-Haken school of irreversible microsystems, the Gorky Mandelstam-Andronov school of autowave oscillations, the Zeeman school of the theory of catastrophes, the theory of fractals of Mandelbrot, etc. The specificity of these branches of synergetics is based primarily on the study of different types of differential equations (ordinary and partial derivatives of different orders) and different types of function V(x) (which, incidentally, required a serious development of a number of sections of the mathematics).
However, along with the results specific to each branch, some surprising and fascinating system-wide patterns have emerged. Three related topics attracted the most attention.
1. Order and chaos in nature are related: in particular, this is manifested in the fact that in systems described by equations containing only nonrandom parameters, at certain values of these parameters, the trajectories become random; and in natural chaotic processes, there are internal regularities.
- 2. All (both random and chaotic) trajectories of systems in phase space form stable configurations called attractors.
- 3. The structure of natural systems is hierarchical, with uniform rules of organization on all floors of the hierarchy; this feature of nature is called fractality.
- (All these features are manifested already on the example of the simplest model of system dynamics, Equation (8.1).)
These three aspects of the dynamics of systems together have allowed some progress in understanding the mechanisms of the observed amazing process of selforganization of nature, the development of our cosmic universe; the formation of a discrete spectrum of material entities; the appearance of living organisms; the observed diversity of trends in their changes, one of which led to the emergence of human beings and social systems; the development of diverse interactions between all existing systems (of which we are primarily interested in human interactions with each other and the world around us). Let us briefly discuss the detected features of the dynamics of systems.
Order and Chaos in Nature
When performing any of his purposeful actions, a person often faces the fact that what is really happening is not quite or not at all what he expected. Such events are called random, and the need to achieve the goal even in the presence of randomness demanded to understand its nature, that is, to build models that allow to operate successfully in conditions of incomplete predictability of events.
The randomness is most evident in gambling; with them began the knowledge of the nature of chance: theory of probability was built (the mathematical axiomatic of which was completed by A.N. Kolmogorov in the middle of the last century) and the game theory based on it. It turned out that the randomness of the events in this case is associated with incomplete knowledge of the conditions for their coming true. The probability (measure of possibility) of an event is determined by a certain set of conditions, and if we plan our actions on the basis of inaccurate knowledge or ignorance of any of the conditions, the event (naturally!) occurs in unexpected ways.
A typical example is the dice game. Under the condition of “correctness” (uniformity) of the bone and “honesty” of its throwing, the probability of any points is the same and is equal to 1:6. There is a “probabilistic” anecdote. Players in the market place bet on different points, and the dice thrower on the six. When he won six times in a row, the players began to beat the thrower, accusing him of fraud. So it was: the bone was shifted center of gravity. But why did they beat him only after the sixth time?
The concept of random events was soon extended to their time sequence, first, a discrete (statistical series), and then a continuous (random process). It was necessary to introduce the account of variability of conditions (in particular, different variants of dependence of events spaced in time, the concept of ergodicity and stationarity of processes).
In general, this direction of research into the nature of randomness has focused on the cause of the chance, which is connected with what the person interacting with reality knows (and does not know) about it. In this case, random (for a person!) events arise because of his ignorance of some objective conditions of their origin.
Another important result of this branch of knowledge of randomness is that even random phenomena are subject to certain laws. All information about stable, nonrandom properties of random events is concentrated in the probability distribution function for such events. For example, in the one-dimensional case, we can talk about the parameters of the position and scale of the distribution function, in the multidimensional case, about the measures of dependence of variables, etc. On the basis of the theory of probability there were developed various applied disciplines — game theory, mathematical statistics, operations research, information theory, theory of economic risks, actuarial mathematics, etc.
But even more important progress in the knowledge of the nature of chance and the relation of chaos and order was the discovery that randomness is inherent in nature itself, regardless of whether the person is involved in the events with his knowledge.
Actually, this fact was discovered at the beginning of the century in quantum mechanics with its uncertainty principle and wave functions, but at first it looked more like a feature of the mathematical model of the microcosm. But later it turned out that even in strictly determined systems described by exact equations that do not contain any random parameters or functions can occur typically random events. To describe this phenomenon, more general concepts of chaos and stochastic events, reversibility and stability of systems were introduced; various quantitative characteristics of stochastic uncertainty were introduced: entropy, moments of distributions, measures of the amount and value of information. (Detailed critical reviews of this section of dynamic systems theory are contained in books [8-10].) There were found out two (for now?) sources of objective stochastic events in nature.
The first was discovered at the beginning of the century by Poincare in the form of a class of unstable systems whose phase trajectories diverge sharply at the slightest differences in the initial conditions. It turned out to be impossible to obtain analytically the exact solution of the equations of such (“nonintegral”) systems, but it was possible to carry out a qualitative analysis of their dynamics in computer experiments.
In particular, the sensational popularity of this phenomenon was obtained after the publication of the case that occurred in weather forecasting by solving a multidimensional system of equations on the computer linking previously observed in the area of weather data with possible upcoming data. Having received the forecast with very accurate initial data (up to six decimal places), and leaving for lunch, the meteorologist E. Lorentz decided just in case to recalculate the forecast to speed up the calculations by setting the same data with an accuracy of only three characters. Imagine his surprise when he returned from lunch and saw the second forecast drastically different from that of the first! The investigation revealed that the decision of some nonrandom (!) equations is extremely sensitive to initial conditions: a very small change in initial conditions can lead to very different results.
This systemic effect was found not only in mathematical models but also in real nature, and even got the figurative name “butterfly effect”. (Scientific folklore connects this name with two sources. The first is a joke: “If a butterfly flaps its wings at the mouth of the Amazon river, a typhoon will break off the coast of Japan.” The second is the story of the science fiction writer Ray Bradbury about how a tourist traveling by a time machine in the era of dinosaurs, and warned about the categorical ban to change there anything, accidentally crushed a butterfly on the path. When he returned home, he saw America as a completely different society.)
The high sensitivity of a complex system to small changes in the initial conditions became the subject of the study of its stability factors, giving rise to the corresponding branch of the mathematical theory of differential equations.
Along with this, another manifestation of objective randomness in the behavior of systems described by completely deterministic equations was discovered: the presence of stochastic “attractors” in their phase space.
