Lorenz system. Lorentz attractor
Chaotic, strange attractors correspond to the unpredictable behavior of systems that do not have strictly periodic dynamics; this is a mathematical image of deterministic non-periodic processes. Strange attractors are structured and can have very complex and unusual configurations in three-dimensional space.
Rice. one.
and phase portraits (bottom row) for three different systems
(Gleick, 2001)
Although in the works of some mathematicians the possibility of the existence of strange attractors was previously established, for the first time the construction of a strange attractor (Fig. 2) as a solution to a system of differential equations was carried out in a work on computer modeling of thermoconvection and turbulence in the atmosphere by the American meteorologist E. Lorentz (E. Lorentz, 1963) . The final state of the Lorentz system is extremely sensitive to the initial state. The term "strange attractor" itself appeared later, in the work of D. Ruelle and F. Takens (D. Ruelle, F. Takens, 1971: see Ruelle, 2001) on the nature of turbulence in a fluid; the authors noted that the dimension of a strange attractor is different from the usual, or topological one. Later, B. Mandelbrot identified strange attractors, the trajectories of which, during sequential computer calculations, are infinitely stratified, split, with fractals.
Rice. 2. (Chaotic trajectories in the Lorentz system). Lorenz Attractor (Kronover, 2000)
Lorenz (1963) discovered that even a simple system of three non-linear differential equations can lead to chaotic trajectories. -second harmonics:
where s, r and b are some positive numbers, parameters of the system. Usually studies of the Lorenz system are carried out at s =10, r =28 and b =8/3 (parameter values).
Thus, systems whose behavior is determined by rules that do not include randomness show unpredictability over time due to the growth, amplification, amplification of small uncertainties, fluctuations. A visual image of a system with increasing uncertainty is the so-called billiard by Ya.G. Sinai: a sufficiently large sequence of ball collisions inevitably leads to an increase in small deviations from the calculated trajectories (due to the non-ideally spherical surface of real balls, the non-ideally uniform surface of the cloth) and the unpredictability of the system behavior.
In such systems, "randomness is created in the same way that dough is mixed or a deck of cards is shuffled" (Crutchfield et al., 1987). The so-called "baker's transformation" with successive stretching and folding, endless folding is one of the models for the emergence of the transition from order to chaos; in this case, the number of transformations can serve as a measure of chaos. There is the Aizawa Attractor, which is a special case of the Lorenz attractor.
where a = 0.95, B = 0.7, c = 0.6, d = 3.5, e = 0.25, F = 0.1. Each previous coordinate is entered into the equations, the resulting value multiplied by the time values.
Examples of other strange attractors
Attractor WangSun
Here a, b, d, e?R, c> 0 and f< 0 являются константами, cf ? 0, и x, y, z а это переменные состояния.
Rössler attractor
Where a,b,c= positive constants. With parameter values a=b=0.2 and
abstract
By discipline: Mathematics
„ Lorentz attractor“
Lorentz attractor
solution of the system atr =0,3
solution of the system atr =1,8
solution of the system atr =3,7
solution of the system atr =10
solution of the system atr =16
solution of the system atr =24,06
solution of the system atr =28 — in fact, this is the Lorentz attractor
solution of the system atr =100 - the mode of self-oscillations in the system is visible
Lorentz attractor (from English.to attract - attract) is an invariant set in a three-dimensional smooth , which has a certain complex topological structure and is asymptotically stable, it and all trajectories from some neighborhood tend to at (hence the name).
The Lorentz attractor was found in numerical experiments investigating the behavior of the trajectories of a nonlinear system:
with the following parameter values: σ=10,r =28, b =8/3. This system was first introduced as the first non-trivial one for the problem of sea water in a flat layer, which motivated the choice of the values of σ,r andb , but it also arises in other physical questions and models:
convection in a closed loop;
rotation of the water wheel;
single-mode model;
dissipative with inertial nonlinearity.
Initial hydrodynamic system of equations:
where - flow speed, - liquid temperature, - temperature of the upper limit (on the lower one, ), - density, - pressure, - gravity, - respectively, and kinematic.
In the problem of convection, the model arises when the flow velocity and temperature are decomposed into two-dimensional ones and their subsequent "cuttings" up to the first-second harmonics. In addition, the given complete system of equations is written in . The trimming of the rows is justified to a certain extent, since Soltsman in his work demonstrated the absence of any interesting features in the behavior of most harmonics.
Applicability and compliance with reality
Let us designate the physical meaning of the variables and parameters in the system of equations in relation to the mentioned problems.
Convection in a flat layer. Herex responsible for the speed of rotation of the water shafts,y andz - for the temperature distribution horizontally and vertically,r - normalized , σ - (the ratio of the kinematic coefficient to the coefficient ),b contains information about the geometry of the convective cell.
Convection in a closed loop. Herex - flow speed,y - temperature deviation from the average at a point 90 ° away from the bottom point of the loop,z - the same, but at the bottom point. Heat is supplied at the lowest point.
Rotation of the water wheel. The problem of a wheel on the rim of which baskets with holes in the bottom are fixed is considered. Top of the wheelsymmetrically a continuous stream of water flows about the axis of rotation. The task is equivalent to the previous one, turned “upside down”, with the replacement of temperature by the density of distribution of the mass of water in the baskets along the rim.
single mode laser. Herex - the amplitude of the waves in the laser,y - , z - population inversion,b and σ are the ratios of the inversion and field coefficients to the polarization relaxation coefficient,r - intensity.
It is worth pointing out that, as applied to the problem of convection, the Lorentz model is a very rough approximation, very far from reality. A more or less adequate correspondence exists in the region of regular regimes, where stable solutions qualitatively reflect the experimentally observed picture of uniformly rotating convective rolls (). The chaotic regime inherent in the model does not describe turbulent convection due to the significant trimming of the original trigonometric series.
Of interest is the significantly higher accuracy of the model with some of its modification, which is used, in particular, to describe convection in a layer subjected to vibration in the vertical direction or to variable thermal effects. Such changes in external conditions lead to modulation of the coefficients in the equations. In this case, the high-frequency Fourier components of temperature and velocity are significantly suppressed, improving the agreement between the Lorentz model and the real system.
Noteworthy is Lorenz's luck in choosing the parameter value , since the system comes to only for values greater than 24.74, for smaller values the behavior is completely different.
System Solution Behavior
Let us consider changes in the behavior of the solution to the Lorentz system for different values of the parameter r. The illustrations for the article show the results of numerical simulation for points with initial coordinates (10,10,10) and (-10,-10,10). The simulation was carried out using the program below, written in the language, plotting on the resulting tables - due to the weak graphical capabilities of Fortran using the Compaq Array Viewer.
r <1 - the origin of coordinates is the attractor, there are no other stable points.
1< r <13,927 - the trajectories spirally approach (this corresponds to the presence of damped oscillations) to two points, the position of which is determined by the formulas:
These points determine the states of the stationary convection regime, when a structure of rotating fluid rolls is formed in the layer.
r ≈13,927 - if the trajectory leaves the origin, then, having made a complete revolution around one of the stable points, it will return back to the starting point - two homoclinic loops appear. concepthomoclinic trajectory means that it comes out and comes to the same equilibrium position.
r >13,927 - Depending on the direction, the trajectory comes to one of two stable points. Homoclinic loops are reborn into unstable limit cycles, and a family of complexly arranged trajectories also arises, which is not an attractor, but rather, on the contrary, repels trajectories from itself. Sometimes, by analogy, this structure is called the "strange repeller" (eng.to repel - repel).
r ≈24,06 - the trajectories no longer lead to stable points, but asymptotically approach unstable limit cycles - the actual Lorentz attractor appears. However, both stable points are preserved up to the valuesr ≈24,74.
For large values of the parameter, the trajectory undergoes serious changes. Shilnikov and Kaplan showed that for very larger the system goes into the self-oscillation mode, and if the parameter is reduced, a transition to chaos will be observed through a sequence of oscillation period doublings.
Model Significance
The Lorentz model is a real physical example with chaotic behavior, in contrast to various artificially constructed mappings ( , etc.).
Programs that simulate the behavior of the Lorenz system
Borland C
#include
#include
void main()
double x = 3.051522, y = 1.582542, z = 15.62388, x1, y1, z1;
double dt = 0.0001;
int a = 5, b = 15, c = 1;
int gd=DETECT, gm;
initgraph(&gd, &gm, "C:\\BORLANDC\\BGI");
do(
X1 = x + a*(-x+y)*dt;
Y1 = y + (b*x-y-z*x)*dt;
Z1 = z + (-c*z+x*y)*dt;
X=x1; y=y1; z = z1;
Putpixel((int)(19.3*(y - x*0.292893) + 320),
(int)(-11*(z + x*0.292893) + 392), 9);
) while (!kbhit());
closegraph();
Mathematica
data = table[
With[(N = 1000, dt = 0.01, a = 5, b = 1 + j, c = 1),
NestList &,
(3.051522, 1.582542, 15.62388), N
(j, 0, 5)];
[email protected][(Hue], Point[#1]) &, data]
Borland Pascal
Program Lorenz;
Uses CRT, Graph;
Const
dt = 0.0001;
a = 5;
b = 15;
c = 1;
Var
gd, gm: Integer;
x1, y1, z1, x, y, z: Real;
Begin
gd:=Detect;
InitGraph(gd, gm, "c:\bp\bgi");
x:= 3.051522;
y:= 1.582542;
z:= 15.62388;
While not KeyPressed Do Begin
x1:= x + a*(-x+y)*dt;
y1:= y + (b*x-y-z*x)*dt;
z1:= z + (-c*z+x*y)*dt;
x:= x1;
y:= y1;
z:= z1;
PutPixel(Round(19.3*(y - x*0.292893) + 320),
Round(-11*(z + x*0.292893) + 392), 9);
end;
Close Graph;
ReadKey;
end.
FORTRAN
program LorenzSystem
real,parameter::sigma=10
real,parameter::r=28
real,parameter::b=2.666666
real,parameter::dt=.01
integer,parameter::n=1000
real x,y,z
open(1,file="result.txt",form="formatted",status="replace",action="write")
x=10.;y=10.;z=10.
doi=1,n,1
x1=x+sigma*(y-x)*dt
y1=y+(r*x-x*z-y)*dt
z1=z+(x*y-b*z)*dt
x=x1
y=y1
z=z1
write(1,*)x,y,z
end do
print *,"Done"
close(1)
end program LorenzSystem
QBASIC/FreeBASIC("fbc -lang qb")
DIM x, y, z, dt, x1, y1, z1 AS SINGLE
DIM a, b, c AS INTEGER
x = 3.051522: y = 1.582542: z = 15.62388: dt = 0.0001
a=5: b=15: c=1
SCREEN 12
PRINT "Press Esc to quit"
WHILE INKEY$<>CHR$(27)
x1 = x + a * (-x + y) * dt
y1 = y + (b * x - y - z * x) * dt
z1 = z + (-c * z + x * y) * dt
x=x1
y = y1
z = z1
PSET ((19.3 * (y - x * .292893) + 300), (-11 * (z + x * .292893) + 360)), 9
WEND
END
JavaScript and HTML5
var cnv = document.getElementById("cnv");
var cx = cnv.getContext("2d");
var x = 3.051522, y = 1.582542, z = 15.62388, x1, y1, z1;
vardt = 0.0001;
var a = 5, b = 15, c = 1;
var h = parseInt(cnv.getAttribute("height"));
var w = parseInt(cnv.getAttribute("width"));
var id = cx.createImageData(w, h);
varrd = Math.round;
var idx = 0;
i=1000000; while (i--) (
x1 = x + a*(-x+y)*dt;
y1 = y + (b*x-y-z*x)*dt;
z1 = z + (-c*z+x*y)*dt;
x = x1; y=y1; z = z1;
idx=4*(rd(19.3*(y - x*0.292893) + 320) + rd(-11*(z + x*0.292893) + 392)*w);
id.data = 255;
cx.putImageData(id, 0, 0);
IDL
PRO Lorenz
n=1000000 & r=dblarr(n,3) & r= & a=5. &b=15. &c=1.
FOR i=0.,n-2. DO r=r + [ a*(r-r), b*r-r-r*r, r*r-c*r ]*0.0001
plot,19.3*(r[*,1]-r[*,0]*0.292893)+320.,-11*(r[*,2]+r[*,0]*0.292893)+392.
END
Literature
Kuznetsov S.P. , Lecture 3. Lorentz system; Lecture 4. Dynamics of the Lorentz system. // - M.: Fizmatlit, 2001.
Saltzman B . Finite amplitude free convection as an initial value problem. // Journal of the atmospheric science, No. 7, 1962 - p. 329-341.
Lorenz E . Deterministic non-periodic motion // Strange attractors. - M., 1981. - S. 88-116.
Usually they say that chaos is a higher form of order, but it is more correct to consider chaos as another form of order - inevitably, in any dynamic system, order in its usual sense is followed by chaos, and order follows chaos. If we define chaos as disorder, then in such disorder we will definitely be able to see our own, special form of order. For example, smoke from cigarettes at first it rises in the form of an ordered column under the influence of the external environment, it takes on more and more bizarre outlines, and its movements become chaotic. Another example of randomness in nature is leaf from any tree. It can be argued that you will find many similar leaves, such as oak, but not a single pair of identical letters. The difference is determined by temperature, wind, humidity and many other external factors besides purely internal causes (eg genetic difference).
Chaos theory
The movement from order to chaos and vice versa, apparently, is the essence of the Universe, we have not studied contributing to its manifestation. Even in the human brain, orderly and chaotic principles are present at the same time. The first corresponds to the left hemisphere of the brain, and the second to the right. The left hemisphere is responsible for the conscious behavior of a person, for the development of linear rules and strategies in human behavior, where “if ... then ...” is clearly defined. In the right hemisphere, non-linearity and chaos reign. Intuition is one of the manifestations of the right hemisphere of the brain. Chaos theory studies the order of a chaotic system that looks random, disorderly. At the same time, chaos theory helps to build a model of such a system, without setting the task of accurately predicting the behavior of a chaotic system in the future.History of chaos theory
The first elements of chaos theory appeared in the 19th century, but this theory received real scientific development in the second half of the 20th century, along with the works Edward Lorenz(Edward Lorenz) from the Massachusetts Institute of Technology and French-American mathematician Benoit B. Mandelbrot (Benoit B. Mandelbrot). Edward Lorenz at one time (early 60s of the XX century, work published in 1963) considered what is the difficulty in weather forecasting. By Lorenz's work, two opinions dominated the world of science regarding the possibility of accurate weather forecasting for an infinitely long period. First approach formulated in 1776 by a French mathematician Pierre Simon Laplace. Laplace stated that "...if we conceive of a mind that at a given moment comprehended all the connections between objects in the universe, then it would be able to ascertain the respective positions, movements and general effects of all these objects at any time in or past in the future." This approach of his was very similar to the famous words of Archimedes: "Give me a fulcrum, and I will turn the whole world upside down." Thus, Laplace and his supporters said that in order to accurately predict the weather, it is only necessary to collect more information about all the particles in the universe, their location, speed, mass, direction of movement, acceleration, etc. Laplace believed that the more a person knew, the more accurate his forecast regarding the future would be. Second approach to the possibility of weather forecasting was most clearly formulated by another French mathematician, Jules Henri Poincare. In 1903, he said: “If we knew exactly the laws of nature and the position of the universe at the initial moment, we could accurately predict the position of the same universe at a subsequent moment. But even if the laws of nature revealed all their secrets to us, even then we could know the initial position only approximately. If this allowed us to predict the subsequent position with the same approximation, that would be all we needed, and we could say that the phenomenon was predicted, that it was governed by laws. But it is not always the case that small differences in the initial conditions cause very large differences in the final phenomenon. A small error in the former will produce a huge error in the latter. Prediction becomes impossible, and we are dealing with a phenomenon that develops by chance.” In these words of Poincaré we find the postulate of chaos theory about dependence on initial conditions. The subsequent development of science, especially quantum mechanics, refuted Laplace's determinism. In 1927 a German physicist Werner Heisenberg discovered and formulated uncertainty principle. This principle explains why some random phenomena do not obey Laplacian determinism. Heisenberg demonstrated the uncertainty principle using the radioactive decay of a nucleus as an example. So, for the very small size of the nucleus, it is impossible to know all the processes occurring inside it. Therefore, no matter how much information we collect about the nucleus, it is impossible to predict exactly when this nucleus will decay.Chaos theory tools
What tools does chaos theory have? First of all, these are attractors and fractals. Attractor (from the English. To attract - attract) - a geometric structure that characterizes the behavior in the phase space at the end of a long time. I.e attractor- this is what the system strives to achieve, to which it is attracted. The simplest type of attractor is a point. Such an attractor is typical for a pendulum in the presence of friction. Regardless of the initial speed and position, such a pendulum will always come to rest, i.e. exactly. The next type of attractor is the limit cycle, which has the form of a closed curved line. An example of such an attractor is a pendulum, which is not affected by the force of friction. Another example of a limit cycle is the heartbeat. The beat frequency can decrease and increase, but it always tends to its attractor, its closed curve. The third type of attractor is a torus. In figure 1, the torus is shown in the upper right corner.
Figure 1 - Main types of attractors Shown at the top are three predictable, simple attractors. Below are three chaotic attractors. Despite the complexity of the behavior of chaotic attractors, sometimes called strange attractors, knowledge of the phase space allows one to represent the behavior of the system in a geometric form and, accordingly, to predict it. And although the stay of the system at a particular point in time at a particular point in the phase space is practically impossible, the area where the object is located and its tendency to the attractor are predictable. Lorenz attractor
The first chaotic attractor was the Lorenz attractor.
Figure 2 - Chaotic Lorenz attractor Lorentz attractor calculated on the basis of only three degrees of freedom - three ordinary differential equations, three constants and three initial conditions. However, despite its simplicity, the Lorenz system behaves in a pseudo-random (chaotic) manner. Having simulated his system on a computer, Lorentz identified the reason for its chaotic behavior - the difference in the initial conditions. Even a microscopic deviation of two systems at the very beginning in the process of evolution led to an exponential accumulation of errors and, accordingly, their stochastic disagreement. At the same time, any attractor has boundary dimensions, so the exponential divergence of two trajectories of different systems cannot continue indefinitely. Sooner or later, the orbits will converge again and pass next to each other or even coincide, although the latter is very unlikely. By the way, the coincidence of trajectories is the rule for the behavior of simple predictable attractors. convergence-divergence(also called compounding and stretching, respectively) of a chaotic attractor systematically removes the initial information and replaces it with new information. When ascending, the trajectories approach each other and the effect of myopia begins to appear - the uncertainty of large-scale information increases. When the trajectories diverge, on the contrary, they diverge and the farsightedness effect appears when the uncertainty of small-scale information increases. As a result of the constant convergence-divergence of the chaotic attractor, the uncertainty is growing rapidly, which makes it impossible for us to make accurate predictions with each moment of time. What science is so proud of - the ability to establish connections between causes and effects - is impossible in chaotic systems. There is no causal relationship between the past and the future in chaos. It should also be noted here that the rate of convergence-divergence is a measure of chaos, i.e. a numerical expression of how chaotic the system is. Another statistical measure of chaos is the dimension of the attractor. Thus, it can be noted that the main property of chaotic attractors is the convergence-divergence of the trajectories of different systems, which randomly gradually and infinitely mix. Izv. universities "PND", v. 15, No. 1, 2007 UDC 517.9
LORENTZ ATTRACTOR IN SHEAR FLOWS
A.M. Mukhamedov
Within the framework of the previously proposed model of chaotic dynamics of a continuous medium, a realization of the three-dimensional regime of flow velocity fluctuations corresponding to a Lorentz-type attractor is obtained. The solution is a set of structures that determine the geometry of the layered manifold reduced to the three-dimensional case, formed by pulsations of the medium flow velocities. The dynamics of the Lorentz attractor itself manifests itself in the form of a time dependence of velocity fluctuations along the streamlines of the mean flow.
As is known, one of the classic examples of deterministic chaos, the Lorentz attractor, discovered as a result of applied hydrodynamic research, has not yet been adequately reproduced in the formalism of the existing turbulent mechanics. In the works of the author, a hypothesis was expressed that the classical hydrodynamic solution of this problem cannot be obtained in principle, and a justification for such a conclusion was proposed. It was based on the understanding that attractor models of chaotic dynamics affect the mesoscopic level of motion of a continuous medium, and that this level is not represented in the classical Navier-Stokes equations. This led to the proposal to expand the options for solving the problem of the Lorentz attractor by explicitly including additional mesostructures in the mathematical formalism of hydrodynamics, which take the apparatus of this theory beyond the framework of classical operations with the Navier-Stokes equations.
At present, the attractor regimes of the dynamics of continuous media are constructed within the framework of models that are far-reaching abstractions of the motion of a continuous medium, almost without using the concept of mechanical interactions of the particles of the medium with each other. In some cases, these abstractions reflect the properties of evolutionary type operators acting in a hierarchy of nested Hilbert spaces. In other cases, they reflect the dynamics of finite-dimensional systems that reproduce changes in the states of the environment, but in this case, each of the states is actually represented by just a point of the corresponding phase manifold. Such modeling does not correspond to the applied purpose of hydromechanics, which requires the reproduction of all essential structures directly, that is, in the space occupied by a continuous medium. If we take into account the arguments of theoretical and experimental data in favor of
existence of such a representation, then the reproduction of attractors in the context of the dynamics of the space-time characteristics of the environment seems to be an urgent need.
In this paper, the Lorentz attractor is constructed within the framework of the turbulent dynamics proposed in the model. According to this model, the phase spaces of turbulent regimes are stratifications of jets of fluctuations of hydrodynamic quantities. The geometry of fluctuating bundles is assumed to be a priori arbitrary, determined by the modeled features of the corresponding chaotic regimes. The main object of modeling is a chaotic structure, which is a complex of unstable trajectories of movement of points in the medium. It is assumed that each established turbulent regime corresponds to a well-defined chaotic structure. In the trajectory of a chaotic structure, they were identified with the set of integral curves of a non-integrable (non-holonomic) Pfaff-type distribution defined on a bundle of fluctuations of dynamic variables.
A characteristic feature of the proposed model is the Lagrange method of describing the motion of a medium, which, in the general case, does not reduce to describing the motion in terms of Euler's variables. At the same time, it turned out that Lagrange's description is admirably adapted to reflect the dynamics of systems with strange attractors. Instead of the strict restrictions of the Euler paradigm, Lagrange's description imposes much softer conditions that serve to determine the geometric objects of the corresponding nonholonomic distributions. Such a change in the emphasis of modeling makes it possible to reproduce various attractors in the dynamics of particle beams in continuum media.
1. Let us set the equations for the dynamics of pulsations of the three-mode regime
(yi + 4 (x, y!) (xk = Ar(x, y^)(U (1,3,k = 1,2,3), (1)
where xk and yz form the sets of spatial and dynamic coordinates of the stratification of pulsations, and the objects mkk(x, yt)(xk and Ar(x, yt)M determine the nature of the intermode interactions of the regime. These objects and equation (1) itself can be considered as the rules for the formation of derivatives of dynamic coordinates with respect to spatial coordinates and time, determined by real turbulent evolution.The invariant geometric meaning of these objects is that in the bundle of pulsations they determine the object of internal connection and the vertical vector field, respectively.
Let us assume that the dynamic coordinates introduced above have the meaning of fluctuations in the flow velocity of the medium, that is, the actual velocity of the medium can be expanded into the velocity field of the mean flow and fluctuations according to the formula
u (x, y) = u0 (x) + y. (2)
We will take the mass and momentum balance equations in the form of the standard continuity equation and the Navier-Stokes equation
Chr + udi. (4)
This system of equations is not yet complete, since equation (4) includes pressure, which is a thermodynamic variable, the dynamics of which, in the general case, goes beyond the scope of kinematics. To describe pressure fluctuations, new dynamic coordinates are required, which increases the number of required degrees of freedom to describe the corresponding turbulent motion regime. We introduce a new dynamic variable that has the meaning of pressure fluctuations, that is, we take
p(x,y)= po(x) + y4. (5)
Thus, the initial set of required dynamic coordinates for displaying the motion of a continuous medium is four-dimensional.
The possibility of reduction to a three-dimensional system with dynamics similar to the dynamics of the Lorentz system lies in the fact that pressure enters Eq. (4) in the form of a gradient. Hence it follows that the reduction to the three-dimensional dynamics of velocity fluctuations can be performed if the pressure gradient entering Eq. (4) contains only the first three dynamic coordinates. To do this, it suffices to require that in the equations of dynamics for the fourth coordinate
dy4 + wj (x, y)dxk = A4 (x, y)dt (6)
the coefficients of the connection forms w4(x,yj)dxk depended only on the first three dynamical coordinates. Note that the three-dimensional regime may turn out to be unstable from the point of view of a more complete description, which includes consideration of all excited degrees of freedom. However, we will restrict ourselves to modeling precisely this a priori possible dynamics.
Let us consider the conditions imposed by the balance equations (3), (4) on the expressions for the unknown quantities wk(x,yj)dxk and Ai(x,yj)dt included in the dynamic equation (1). To do this, we substitute (2) and (5) into (3) and (4), and use equations (1) and (6). To simplify the resulting expressions, we assume that the spatial coordinates xk are Cartesian. In this case, you can not distinguish between superscripts and subscripts, raising and lowering them as necessary to write covariant expressions. Then we obtain the following equations for the coefficients of equation (1)
dkuk - wj = 0, (7)
Ai + (uk + yk)(djuk - wj) = -(dipo - w4i) - vDjwik. (eight)
where the notation Dj = dj - wk^y is introduced.
For what follows, we concretize the formulation of the problem. We will consider a regime whose average velocity field describes the flow of a simple shear
uk = Ax3à\. (nine)
In addition, we make assumptions about the geometry of the fibred pulsation space. We will assume that the bundle is connected as a linear function in dynamic coordinates, that is, w^ = waj (x)yj (a = 1,..., 4). In this case, it immediately follows from Eq. (8) that the second object acquires a structure that is polynomial in dynamic coordinates. Namely, the vertical vector field becomes a second-order polynomial in dynamic coordinates, i.e.
Ai = Ak (x) + Aj (x)yk + j (x)yj yk.
Thus, the unknown functions that determine the equation for the dynamics of pulsations of the three-mode regime under consideration are the coefficients waak(x), Ar0(x), Ark(x), and A3k(x), for which we have equations (3) and (4). Note that equation (4) essentially reduces to determining the coefficients of the vertical vector field, while the choice of connection coefficients limits only the continuity equation (3). This equation leaves considerable arbitrariness in determining the connectivity coefficients, thus leaving the breadth of modeling the spatial structure of the fluctuation dynamics consistent with the chosen average flow.
2. Consider the possibility of obtaining an attractor of the Lorentz type in this problem. For this purpose, first of all, we will discuss the expansion of the actual values of the speed into the average speed and fluctuations around the average.
According to the meaning of pulsations, their time average should be equal to zero, i.e.
(y)t - 0. (10)
At the same time, pulsations are defined as deviations of the actual speed values from the average value. If the mean flow is assumed to be given, then the noted circumstance does not allow us to choose an arbitrary system of equations with chaotic dynamics as a model chaos equation. In order for the variables of the model system of equations to be considered as pulsations of real hydromechanical quantities, conditions (10) must be satisfied. If (10) is not satisfied, then this means the existence of an unaccounted drift in the pulsation dynamics. Accordingly, the adopted model system turns out to be inconsistent either with the acting factors taken into account, or with the structure of the allowed mean flow.
Further, equation (1) is, in the general case, a not completely integrable Pfaff-type system. The property of non-integrability of this equation is fundamentally important, corresponding to a feature characteristic of turbulent motion. Namely, in the process of movement, any macroscopically small turbulent formations, particles, moths, globules, lose their individuality. This feature is taken into account by the non-integrability of Eq. (1). In essence, (1) describes an ensemble of possible trajectories of motion of the points of a continuum formed by a continuous medium. These trajectories are defined in the bundle of fluctuations. Their projections onto the space occupied by a continuous medium determine the dynamics of the development of fluctuations along the corresponding spatial curves. Note that the latter can be chosen arbitrarily, determining the possibility of considering the dynamics of fluctuations along any spatial curve.
Let us consider, for definiteness, the dynamics of fluctuations along the streamlines of the mean flow. Then we have the following dynamic equations:
xr = u0, (11)
y + w) k y3 4 \u003d Ar. (12)
Before considering this system, we transform it to dimensionless variables. To do this, in the original equation (4), instead of the viscosity coefficient, we introduce
Reynolds number. Then remove the explicit dependence on this number by replacing
<сг = 1_<юг, ю4 = со4, х = х^Иё, у = у^Кё, и0 = и0^Иё, рг = Иер0. (13)
Omitting the underscore over the variables, from (12) we obtain
y \u003d DiO - and! kdkiO - dgro + y3 (-dziO +<г - дкюЗ^ + ю\кю*к) + у3ук<3к. (14)
Let us analyze (13). Note that the model used assumes developed turbulence, that is, the Reynolds number should be considered sufficiently large. Then, if the dimensionless quantities have values of the order of unity, then the real dimensional quantities in accordance with (13) will indicate the scale of the manifestation of the dynamics. In particular, it follows from (13) that the spatial scales turn out to be small. Thus, the model used should be considered, first of all, as a model of turbulent mixing processes at the mesoscopic level of resolution of a continuous medium.
Let us now turn to the analysis of (11) and (12). It is easy to see that for the chosen mean flow, equation (11) has simple integrals. The corresponding mean flow streamline equations are straight lines parallel to the x1 coordinate axis. Eliminating spatial coordinates, from (12) we obtain in the general case a system of non-autonomous differential equations. In this case, if the connectivity coefficients and the pressure gradient do not depend on the x1 coordinate, then system (14) becomes autonomous, containing the remaining spatial coordinates x2 and x3 as parameters. In this case, a real way opens up for direct modeling of the spatially inhomogeneous quasi-stationary pulsation dynamics. An example of such a simulation will be given below.
In conclusion of this paragraph, we note that the appearance of a nonholonomic distribution given by the Pfaff system (1), (6) is a consequence of the assumption that in the state of steady strong turbulence, the class of possible trajectories of motion of the particles of the medium is a stable formation. A necessary condition for this new stability is the requirement for the instability of the trajectories of motion of points, which, in turn, implies large values of the Reynolds number. An attempt to extend the approach to small values of the number Re is unfounded.
3. Let us turn to the construction of an example in which velocity fluctuations along the trajectories of the mean flow are described by a Lorentz-type canonical system. For simplicity, we will assume that all connection coefficients are constant. In this case, we obtain dynamics that is spatially homogeneous along the streamlines of the mean flow, but, nevertheless, along arbitrary lines is not spatially homogeneous. We will call this assumption the quasi-homogeneous approximation.
Our task is to give equation (14) the form of the canonical Lorentz system. The first visible obstacle to this is the uncertainty of identifying the dynamic coordinates and the corresponding variables
from the canonical system. Assuming that various types of mechanisms of intermode interactions will make it possible to simulate any of these identifications, we will choose the following option. Let the structure of equation (14) have the following form:
y1 = a(-y1 + y2), (15)
y2 = (r - (r))y1 - y2 - y1y3, (16)
y3 = -y(y3 + (r)) + y1y2, (17)
where the regular term is explicitly singled out, which, in accordance with what was said in Section 2, must be excluded from the expression for pulsations.
x \u003d o (-x + y), y \u003d rx - y - xr, r \u003d -y r + xy. (eighteen)
For this, we assume that time averages for the variables of system (18) exist. Based on the invariance of this system with respect to transformations
x ^ -x, y ^ -y, z ^ z (19)
it is natural to expect that the means for the first two variables should be zero. Then the substitution
x ⩽ x, y ⩽ y, z ⩽ z + (z) (20)
in (18) gives the system of equations (15) - (17).
In this regard, we note that for various values of the parameters of the Lorentz system, solutions are possible with both zero and non-zero average values of the first two variables . With this in mind, we restrict our subsequent consideration to the first of these possibilities. In addition, we note that substitution (20) can also be performed in the case when the term in the third expression (20) does not have the meaning of the time average. In this case, a new definition of the averaging procedure may be required for subsequent interpretation. In the general case, a suitable definition will require a refinement of the time scales of the phenomena under consideration. It is clear that such redefinitions will require more detailed consideration of both the initial data and variations in the system parameters. The well-known effect of the interaction of chaotic attractors shows how ambiguities can arise in the determination of the averages for small variations in the motion parameters .
Let's return to our consideration. Comparing the coefficients of system (15) -(17) and (14), we obtain
(DiO - u£dki0 - c/ro) =
(-3]u0 + - dkyu] + u^) =
V -U (g)) (-o
g - (g) -1 0 V 0 0 -y
In addition, from (7) we have
dk u0 = 0, 0.
Consider (21) and (24). Substituting expression (9), it is easy to see that (24) is fulfilled identically, and (21) reduces only to the determination of the average pressure gradient. In this case, the gradient turns out to be perpendicular to the average flow velocity, which is a consequence of the chosen identification of the variables of the Lorentz canonical system and the velocity fluctuation components.
Let us turn to equations (23) and (25). From (23) we obtain single-valued expressions for the subscript-symmetrized components of the connection object. The antisymmetric part is determined from (25) with some arbitrariness. The general solution of these equations is given by the following expression:
/ ae,x2 - bxx - aix1 + sd,x3 bx1 - cx2 \
eix2 - /dix3 -eix1 + bix3 (/ - 1)dix1 - bix2 V ra1x2 - eix3 (-p + 1)dix1 + aix3 eix1 - aix2)
Let us turn to the remaining equation (22). This matrix equation is a system of 9 quadratic algebraic equations
b2 - c(p + /) +
ae - bp + Yur \u003d r - (r),
eb - a/ + o43 = 0,
ae - bp + b + 1021 = o,
C/ + e2 + b2 - (1 - /) (1 - p) + o42 \u003d -1,
Ec + ab + u43 = 0,
A/ + eb + a - A + u31 = 0,
Ec + ab + u42 = 0,
Cp - (1 - /) (1 - p) + e2 + a2 + u33 \u003d -y.
The unknowns in it are 6 connectivity coefficients (26), 9 components of the pressure tensor, 1 coefficient that determines the value of the average velocity, and 3 parameters of the Lorentz system. Hence it follows that the solution of this system is determined with considerable parametric arbitrariness. In the three-dimensional regime under consideration, the pressure gradient tensor ω > 4r is arbitrary, and due to its concretization, it is possible to simulate the desired dynamics for any, pre-fixed, choice of connectivity coefficients. For multidimensional regimes, the components of the pressure tensor are included in a more complete system of equations that take into account the dynamics of all excited degrees of freedom. In this case, the pressure tensor can no longer be arbitrary. In this regard, it is interesting to consider various particular options for determining the pressure tensor, assuming that physically reasonable assumptions should find their representations in more complete equations that take into account multidimensional dynamics. We will assume that the pressure gradient tensor is diagonal with a zero component corresponding to the y2 coordinate. In this case, (22) has the following exact analytical solution:
o!1 = .1 - a, o43 = .1 - y + 1, .1 = (K - a) a - A2, K = r - (r), (27)
K - a t Ka, K - a AK
a \u003d A, b \u003d a - K, c \u003d - -.1, p \u003d -, f \u003d - K, e \u003d - - -. (28)
Consider the resulting solution (27), (28). The quantities A, r, a, y, which determine the magnitude of the mean current velocity gradient, and three parameters of the Lorentz model system remained arbitrary in it. All other motion characteristics are expressed as functions of the above set of quantities. Due to the choice of certain values of these quantities, it is possible to vary the dynamics of fluctuations, and using formulas (26), (27) to find the corresponding values of the components of the connectivity object. If we take into account that each object determines the nature of the interactions of pulsations, then it becomes possible to vary the different types of interactions themselves. In particular, to vary the magnitude of the pressure tensor components. It should be noted that in some cases these components can be turned identically to zero. A feature of solutions (27), (28) is that it turns out to be impossible to turn the pressure tensor components to zero, while remaining in the range of those values of the system parameters for which the Lorentz dynamics arises. (However, this is quite possible in the region of those parameter values for which the pulsation dynamics is regular.)
Let's make some estimates. Let the parameters of the model system correspond to the Lorentz attractor with parameters a = 10, r = 28, y = 8/3. In this case, calculations show that the pulsations have a characteristic time t ~ 0.7. Within the calculated time interval b = 0 + 50, the pulsation values belong to the intervals y1 = -17.3 + 19.8, y2 = -22.8 + 27.2, and y3 = -23.2 + 23.7.
Let us compare the absolute values of the velocity fluctuations and the average velocity gradient. From (13) it follows that the pulsations are obtained by dividing the relative values by the number l/d, while the average velocity gradient remains unchanged. Let us take for the velocity gradient a value equal to unity in order of magnitude, then
is A ~ 1. Then, at the value of Re = 2000, that is, at the lower critical value of , for pulsations we obtain an order of magnitude equal to 50% of the gradient value. For the case of Re=40000, the velocity fluctuations reach only 10%% of the accepted value of the average velocity gradient. This shows that reasonable proportions between the average velocity and pulsations can be ensured only in a certain range of Re numbers.
4. New data are revealed when considering the motion of points in the medium. For the Lorentz dynamics in the quasi-homogeneous approximation, the equations of motion of points have the form
r -(z) -l 0 0 0 -Y
Aox3 -A(r - (z))x3
This system turns out to be linear with constant coefficients. Its general solution can be easily obtained by elementary integration. Therefore, we note only the qualitative features of the trajectories of movement of points. From the characteristic equation for the speeds of motion, we find that there are two negative and one positive roots. Thus, at each point in space, two compressive and one tensile directions are distinguished. These features of the dynamics are invariant characteristics that can be used to classify attractors corresponding to flows with the same average velocity.
As it follows from the general solution of system (29) and (30), the possible displacements of medium points in directions transversal to the mean flow streamlines are not limited. Namely, a regular drift occurs in the projection onto the x3 axis. In this case, the points, moving perpendicular to the streamlines of the mean current, fall into the region of high velocities. In this case, the number Re increases, which leads to a decrease in the relative magnitude of the fluctuations. In the framework of the quasi-homogeneous approximation made, this effect leads to a relative decrease in fluctuations and, ultimately, to their degeneration into fluctuations.
Bibliographic list
1. Mukhamedov A.M. Turbulent models: problems and solutions //17 IMACS Congress, Paper T4-1-103-0846, http://imacs2005.ec-lille.fr.
2. Mukhamedov A.M. Towards a gauge theory of turbulence // Chaos, Solitons & Fractals. 2006 Vol. 29. P. 253.
3. Ruelle D., Takens F. On the nature of turbulence // Commun. Math. Phys. 1971 Vol. 20. P. 167.
4. Babin A.V., Vishik M.I. Attractors of evolution equations. M.: Nauka, 1989. 296 p.
5. Mandelbrot B. The fractal geometry of nature. freeman. San Francisco, 1982.
6. Benzi RPaladin G., Parisi G., Vulpiani A. On the multifractal nature of fully developed turbulence and chaotic systems // J. Phys. A. 1984. Vol.17. P.3521.
7 Elnaschie M.S. The Feynman path integrals and E-Infinity theory from the two-slit Gedanken experiment // International Journal of Nonlinear sciences and Numerical Simulations. 2005 Vol. 6(4). P. 335.
8. Mukhamedov A.M. Ensemble regimes of turbulence in shear flows // Bulletin of KSTU im. A.N. Tupolev. 2003, No. 3. S. 36.
9. Yudovich V.I. Asymptotics of the limit cycles of the Lorentz system for large Rayleigh numbers // VINITI. 07/31/78. No. 2611-78.
10. Anishchenko V.S. Complex oscillations in simple systems. M.: Nauka, 1990. 312 p.
11. Loitsyansky L.G. Mechanics of liquid and gas. M.: Nauka, 1987. 840 p.
Kazan State University Received January 23, 2006
Technical University Revised 08/15/2006
LORENZ ATTRACTOR IN FLOWS OF SIMPLE SHIFT
In the frame of a model given before for simulation of chaotic dynamics of continuum medium the Lorenz attractor is represented. The simulation is given with the help of the structures that define the geometry of a fiber bundle associated with 3-dimensional regime of velocity pulsations. Lorenz dynamics appears as time dependence of pulsations along the lines of average flow.
Mukhamedov Alfarid Mavievich - was born in Kazan (1953). Graduated from the Faculty of Physics of Kazan State University in the Department of Gravity and Theory of Relativity (1976). Doctoral student of the Department of Theoretical and Applied Mechanics, Kazan State Technical University named after V.I. A.N. Tupolev. Author of 12 papers on this topic, as well as the monograph "Scientific search and methodology of mathematics" (Kazan: KSTU Publishing House, 2005, co-authored with G.D. Tarzimanova). Area of scientific interests - mathematical models of chaotic dynamics, geometry of fibred manifolds, methodology of modern mathematics.
LORENTZ SYSTEM
LORENTZ SYSTEM
The system of three non-linear differentials. ur-tions of the first order:
solutions to-swarm in a wide range of parameters are irregular functions of time and many others. their characteristics are indistinguishable from random. L. s. was obtained by E. Lorenz from the equations of hydrodynamics as a model for describing thermal convection in a horizontal layer of liquid heated from below ( R r - Prandtl number,
-
reduced R e -ley number, b- is determined by the choice in the Fourier expansion of the field of velocity and temperature).
Rice. 1. Illustration of successive bifurcations in the Lorentz system with increasing parameter r: a) ; b) ; c) d) e) f)
L. s. is one of the examples dynamic system, having a simple physical meaning; it demonstrates stochastic. system behavior. AT phase space this system in the range of parameters shown in Fig. 1 exists strange attractor, the movement of the representative point on the krom corresponds to a "random" - turbulent flow of fluid during thermal convection.
Rice. 2. Convective loop - a physical model for which the Lorentz equations are derived.
L. s. (at b=l) describes, in particular, the motion of a fluid in a convective loop located in a vertical plane in a homogeneous gravity toroidal cavity filled with fluid (Fig. 2). A time-independent (but angle-dependent) temperature is maintained on the cavity walls. T(); lower part of the loop is warmer than the top. The equations for the motion of a fluid in a convective loop are reduced to L. s., where x(t] - fluid velocity, y(t) - temp-pa at the point N, a z(t) - temp-pa at the point M at large t. With growth G the nature of the fluid motion changes: first (at r<1) неподвижна, далее (при ) устанавливается циркуляция с пост. скоростью (либо по часовой стрелке, либо против); при ещё больших r the entire flow becomes sensitive to small changes in the beginning. conditions, the rate of circulation of the liquid changes already irregularly: the liquid rotates sometimes clockwise, sometimes counterclockwise.
At commonly used values Pr=10, b= 8/3 HP has . properties: ur-tion L. s. transformation invariants , the phase volume is reduced from post. speed
per unit of time, the volume is reduced by 10 6 times. With the growth of g in L. s. following occur. main bifurcations. 1) When
the only equilibrium state is the stable node at the origin O(O, O, 0). 2) At , where r 1 \u003d 13.92, L. s. except for the mentioned trivial ( O) has two more equilibria , . Balance state O is a saddle with two-dimensional stable and one-dimensional unstable, consisting of O and two separatrices and tending to and (Fig. 1, a). 3) When r=r 1 each of the separatrices becomes doubly asymptotic to the saddle O(Fig. 1, b). During the transition r through r 1 from closed loops of separatrices are born unstable (saddle) periodic. movements - limit cycles L 1
and L 2 .
Along with these unstable cycles, a very complexly organized limit is born; it, however, is not attractive (attractor), and at (Fig. 1, in), where r 2 = 24.06, all trajectories still tend to . This situation differs from the previous one in that now the separatrices _ and go to "not their own" equilibrium states and respectively. 4) At, where = 24.74, in L. s. along with stable equilibrium states, there is also an attracting set characterized by a complex behavior of trajectories, the Lorentz attractor (Fig. 1 , d iris. 3). 5) When saddle cycles L 1 and L 2 contract to equilibrium states and , which lose their stability at
the gesture of L. s. is the Lorentz attractor. Thus, if we strive for k from the side of smaller values, then the stochasticity in L. s. arises immediately, abruptly, i.e., there is a hard onset of stochasticity.
Rice. 3. Trajectory reproducing the Lorentz attractor (leaving the origin); the horizontal plane corresponds r = = 27, r=28.
To L. s. reduced not only ur-tion, describing the convective motion of the fluid, but also other physical. models (three-level, disk dynamo, etc.).
Lit.: Lorenz E., Deterministic nonperiodic flow, "J. Atmos. Sci.", 1963, v. 20, p. 130; in Russian trans., in the book: Strange attractors, M., 1981, p. 88; Gaponov - Grekhov A. V., Rabinovich M. I., Chaotic simple systems, "Nature", 1981, No. 2, p. 54; Afraimovich V. S., Bykov V. V., Shilnikov L. P., On attracting non-rough limit sets of Lorentz attractor type, Proceedings of the Moscow Mathematical Society, 1982, v. 44, p. 150; Rabinovich M. I., Trubetskov D. I., Introduction to the theory of oscillations and waves, M., 1984. V. G. Shekhov.
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .
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