Geometry of strange attractors and their dimensions. consider the image of a strange attractor of the Ressler system; its geometric configuration can be visualized

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Rössler attractor- chaotic attractor, which has a system of Rössler differential equations:

$\backslash left\; \backslash (\; \backslash begin(matrix)\; \backslash frac(dx)(dt)\; =\; -y\; -\; z\; \backslash \backslash \; \backslash frac(dy)(dt)\; =\; x\; +\; ay\; \backslash \backslash \; \backslash frac(dz)(dt)\; =\; b\; +\; z\; (x-c)\; \backslash end(matrix)\; \backslash right.$ ;

Where $a,b,c$ are positive constants. For parameter values $a=b=0.2$ And $2.6\backslash le\; c\backslash le\; 4.2$ the Rössler equations have a stable limit cycle. With these values ​​of the parameters, the period and the shape of the limit cycle perform a period doubling sequence. Immediately after the dot $c\; =\; 4.2$ the phenomenon of a chaotic attractor occurs. Well-defined lines of limit cycles blur and fill the phase space with an infinite countable set of trajectories that have the properties of a fractal.

Sometimes Rössler attractors are constructed for a plane, that is, with $z\; =\; 0$.

$\backslash left\; \backslash (\; \backslash begin(matrix)\; \backslash frac(dx)(dt)\; =\; -y\; \backslash \backslash \; \backslash frac(dy)(dt)\; =\; x\; +\; ay\; \backslash end(matrix)\; \backslash right.$

Sustainable solutions for $x,\; y$ can be found by calculating the eigenvector of the Jacobi matrix of the form , for which $\backslash frac\; (a\; \backslash pm\; \backslash sqrt(a^2\; -\; 4))\; (2)$.

{2}

From this it is clear that when , the eigenvectors are complex and have positive real components, which makes the attractor unstable. Now we will consider the plane $Z$ in the same range $a$. Bye $x$ less $c$, parameter $c$ will keep the trajectory close to the plane $x,\; y$. As soon as $x$ will become more $c$, $z$-coordinate will start to increase, and a little later the parameter $-z$ will slow down growth $x$ V $\backslash frac\; (dx)\; (dt)$.

Balance points

To find the equilibrium points, the three Rössler equations are set equal to zero and $xyz$-coordinates of each equilibrium point are found by solving the resulting equations. Eventually:

$\backslash left\; \backslash (\; \backslash begin(matrix)\; x\; =\; \backslash frac(c\backslash pm\backslash sqrt(c^2-4ab))(2)\; \backslash \backslash \; y\; =\; -\backslash left(\backslash frac(c\backslash pm\backslash sqrt(c^2\; -4ab))(2a)\backslash right)\; \backslash \backslash \; z\; =\; \backslash frac(c\backslash pm\backslash sqrt(c^2-4ab))(2a)\; \backslash end(matrix)\; \backslash right.$

As shown in general equations Rössler attractor, one of these fixed points is located at the center of the attractor, while the others lie relatively far from the center.

Changing parameters a, b and c

The behavior of the Rössler attractor largely depends on the values ​​of the constant parameters. Changing each parameter gives a certain effect, as a result of which the system can converge to a periodic orbit, to a fixed point, or rush to infinity. The number of periods of the Rössler attractor is determined by the number of its turns around the central point that occur before the series of loops.

Bifurcation diagrams are a standard tool for analyzing the behavior of dynamical systems, which include the Rössler attractor. They are created by solving the equations of a system where two variables are fixed and one is changed. When constructing such a diagram, almost completely “shaded” regions are obtained; this is the realm of dynamic chaos.

Changing the parameter a

Let's fix $b\; =\; 0.2$, $c=5.7$ and we will change $a$.

As a result, empirically, we get the following table:

• $a\backslash leq\; 0$: Converging to a stable point.
• $a\; =\; 0.1$: Spinning with a period of 2.
• $a\; =\; 0.2$: Chaos (standard parameter of the Rössler equations) .
• $a\; =\; 0.3$: Chaotic attractor.
• $a\; =\; 0.35$: Similar to the previous one, but the chaos is more pronounced.
• $a\; =\; 0.38$: Similar to the previous one, but the chaos is even stronger.

Change parameter b

Let's fix $a\; =\; 0.2$, $c=5.7$ and we will now change the parameter $b$. As can be seen from the figure, at $b$ tending to zero, the attractor is unstable. When $b$ will become more $a$ And $c$, the system will be balanced and go to the stationary state.

Changing the parameter c

Let's fix $a=b=0.1$ and we will change $c$. It can be seen from the bifurcation diagram that for small $c$ the system is periodic, but rapidly becomes chaotic as it increases. The figures show exactly how the randomness of the system changes with increasing $c$. For example, when $c$= 4 the attractor will have a period equal to one, and there will be only one line on the diagram, the same will happen when $c$= 3 and so on; Bye $c$ will not become more than 12: the last periodic behavior is characterized by this value, then chaos goes everywhere.

We give illustrations of the behavior of the attractor in the specified range of values $c$, which illustrate the general behavior of such systems - frequent transitions from periodicity to dynamic chaos.

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Notes

Links

• Constructor

Literature

• Voronov V.K., Podoplelov A.V. Modern physics: Tutorial. M., KomKniga, 2005, 512 pp., ISBN 5-484-00058-0, ch. 2 Physics of open systems. pp 2.4 Rössler's chaotic attractor.

An excerpt characterizing the Rössler Attractor

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This expression apparently pleased the officer.
- The adjutant shaved off importantly, - a voice was heard from behind.
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Where is the commander in chief? Bolkonsky asked.
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At that moment he remembered his recent encounter with the doctor's wife and the Furshtat officer.
What is the Commander-in-Chief doing here? - he asked.
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Passing by Kutuzov's carriage, the tortured riding horses of the retinue, and the Cossacks, who were talking loudly among themselves, Prince Andrei entered the hallway. Kutuzov himself, as Prince Andrei was told, was in the hut with Prince Bagration and Weyrother. Weyrother was the Austrian general who replaced the slain Schmitt. In the passage little Kozlovsky was squatting in front of the clerk. The clerk, on an inverted tub, turned up the cuffs of his uniform, hastily wrote. Kozlovsky's face was exhausted - he, apparently, also did not sleep the night. He glanced at Prince Andrei and did not even nod his head at him.
- The second line ... Did you write? - he continued, dictating to the clerk, - Kiev grenadier, Podolsky ...
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At that time, Kutuzov's animatedly dissatisfied voice was heard from behind the door, interrupted by another, unfamiliar voice. By the sound of these voices, by the inattention with which Kozlovsky looked at him, by the irreverence of the exhausted clerk, by the fact that the clerk and Kozlovsky were sitting so close to the commander-in-chief on the floor near the tub, and by the fact that the Cossacks holding the horses laughed loudly under by the window of the house - for all this, Prince Andrei felt that something important and unfortunate was about to happen.
Prince Andrei urged Kozlovsky with questions.
“Now, prince,” said Kozlovsky. - Disposition to Bagration.
What about surrender?
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Prince Andrei went to the door, through which voices were heard. But just as he was about to open the door, the voices in the room fell silent, the door opened of its own accord, and Kutuzov, with his aquiline nose on his plump face, appeared on the threshold.
Prince Andrei stood directly opposite Kutuzov; but from the expression of the commander-in-chief's only sighted eye, it was clear that thought and care occupied him so much that it seemed as if his vision was obscured. He looked directly at the face of his adjutant and did not recognize him.
- Well, are you finished? he turned to Kozlovsky.
“Just a second, Your Excellency.
Bagration, not tall, with an oriental type of hard and immovable face, dry, not yet an old man, went out for the commander-in-chief.
“I have the honor to appear,” Prince Andrei repeated rather loudly, handing the envelope.
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Kutuzov went out with Bagration to the porch.
“Well, good-bye, prince,” he said to Bagration. “Christ is with you. I bless you for a great achievement.
Kutuzov's face suddenly softened, and tears appeared in his eyes. He pulled Bagration to himself with his left hand, and with his right hand, on which there was a ring, he apparently crossed him with a habitual gesture and offered him a plump cheek, instead of which Bagration kissed him on the neck.

Hi all!

This article is dedicated to the amazing features in the world of chaos. I will try to talk about how to curb such a strange and complex thing as a chaotic process and learn how to create your own simple chaos generators. Together with you, we will go from a dry theory to an excellent visualization of chaotic processes in space. In particular, using the example of well-known chaotic attractors, I will show how to create dynamic systems and use them in tasks related to field-programmable logic integrated circuits (FPGAs).

Introduction

Chaos theory is an unusual and young science that describes the behavior of nonlinear dynamic systems. In the process of its inception, chaos theory simply turned modern science upside down! It excited the minds of scientists and made them more and more immersed in the study of chaos and its properties. Unlike noise, which is a random process, chaos is deterministic. That is, for chaos, there is a law of change in the quantities included in the equations for describing a chaotic process. It would seem that with such a definition, chaos is no different from any other oscillations described as a function. But it's not. Chaotic systems are very sensitive to initial conditions, and the slightest change in them can lead to enormous differences. These differences can be so strong that it will be impossible to tell if one or more systems have been tested. From popular science sources, this property of chaos best describes a process called " butterfly Effect". Many have heard about it, and even read books and watched films that used the technique using the butterfly effect. In essence, the butterfly effect reflects the main property of chaos.

The American scientist Edward Lorenz, one of the pioneers in the field of chaos, once said:

A butterfly flapping its wings in Iowa can cause an avalanche of effects that can culminate in the rainy season in Indonesia.

So, let's plunge into chaos theory and see what improvised means can generate chaos.

Theory

Before presenting the main material, I would like to give a few definitions that will help to understand and clarify some points in the article.

dynamic system is a certain set of elements for which the functional dependence between the time coordinate and the position in the phase space of each element of the system is specified. Simply put, a dynamical system is a system whose state in space changes over time.
Many physical processes in nature are described by systems of equations, which are dynamic systems. For example, these are combustion processes, liquid and gas flows, the behavior of magnetic fields and electrical oscillations, chemical reactions, meteorological phenomena, population changes in plants and animals, turbulence in sea currents, the movement of planets and even galaxies. As you can see, many physical phenomena can be described to some extent as a chaotic process.

Phase portrait is a coordinate plane in which each point corresponds to the state of the dynamic system at a certain point in time. In other words, this is a spatial model of the system (it can be two-dimensional, three-dimensional, and even four-dimensional or more).

attractor is some set of the phase space of the dynamical system, for which all trajectories are attracted to this set over time. If at all plain language, then this is some area in which the behavior of the system in space is concentrated. Many chaotic processes are attractors, because they are concentrated in a certain region of space.

Implementation

In this article, I would like to talk about the four main attractors - Lorentz, Ressler, Rikitaka and Nose-Hoover. In addition to the theoretical description, the article reflects aspects of creating dynamic systems in the environment MATLAB Simulink and their further integration into the company's FPGA Xilinx with the help of a tool System Generator. Why not VHDL/Verilog? You can also synthesize attractors using RTL languages, but for better visualization of all processes, MATLAB is ideal. I will not touch on the complicated issues associated with the calculation of the spectrum of Lyapunov exponents or the construction of Poincaré sections. And even more so, there will be no cumbersome mathematical formulas and conclusions. So let's get started.

To create chaos generators, we need the following software:

• MATLAB R2014 licensed for Simulink and DSP Toolbox.
• Xilinx ISE Design Suite 14.7 with System-Generator (DSP Edition) license

These programs are quite heavy, so be patient when installing them. It is better to start the installation with MATLAB, and only then install the Xilinx software (with a different sequence, some of my friends failed to integrate one application into another). When installing the latter, a window pops up where you can link Simulink and System Generator. There is nothing complicated and unusual in the installation, so we will skip this process.

Lorentz attractor

Lorentz attractor- this is perhaps the most famous dynamical system in chaos theory. For several decades now, it has attracted great attention of many researchers for the description of certain physical processes. The first mention of the attractor is given in 1963 in the works of E. Lorenz, who was engaged in modeling atmospheric phenomena. The Lorentz attractor is a three-dimensional dynamical system of nonlinear autonomous differential equations of the first order. It has a complex topological structure, is asymptotically stable, and is stable in the sense of Lyapunov. The Lorentz attractor is described by the following system of differential equations:

In the formula, the dot above the parameter means taking the derivative, which reflects the rate of change of the value with respect to the parameter ( physical meaning derivative).

For parameter values σ = 10, r= 28 and b= 8/3 this simple dynamical system was obtained by E. Lorenz. For a long time he could not understand what was happening with his computer, until he finally realized that the system was exhibiting chaotic properties! It was obtained in the course of experiments for the problem of modeling fluid convection. In addition, this dynamical system describes the behavior of the following physical processes:

• is a single-mode laser model,
• – convection in a closed loop and a flat layer,
• – rotation of the water wheel,
• – harmonic oscillator with inertial nonlinearity,
• – swirls of cloud masses, etc.

The following figure shows the Lorentz attractor system in the MATLAB environment:

The figure uses a number of the following symbols:

• subtractors: SUB0-3;
• constant multipliers: SIGMA, B, R;
• multipliers: MULT0-1;
• integrators with a cell for specifying the initial condition: INTEGRATOR X,Y,Z;
• output ports OUT: DATA X,Y,Z for signals XSIG, YSIG, ZSIG;

In addition, the diagram presents auxiliary analysis tools, these are:

• saving calculation results to a file: To Workspace X,Y,Z;
• construction of spatial graphs: Graph XY, YZ, XZ;
• building time charts: Scope XYZ;
• tools for estimating the occupied resources of the crystal and generating HDL code from the model " Resource Estimator" And " System Generator».

Inside each node of mathematical operations, you must specify the bit depth of intermediate data and their type. Unfortunately, it is not so easy to work with floating point in FPGAs and in most cases all operations are performed in fixed point format. Setting parameters incorrectly can lead to incorrect results and frustrate you when building your systems. I experimented with different values, but settled on the following data type: a 32-bit vector of signed numbers in fixed-point format. 12 bits are allocated for the integer part, 20 bits for the fractional part.

By setting the X, Y, Z integrators in the trigger block to the initial value of the system, for example, {10, 0, 0} , I ran the model. In the time base, the following three signals can be observed:


Even if the simulation time tends to infinity, then the implementation in time will never repeat. Chaotic processes are non-periodic.

In three dimensions, the Lorentz attractor looks like this:

It can be seen that the attractor has two points of attraction, around which the whole process takes place. With a slight change in the initial conditions, the process will also be concentrated around these points, but its trajectories will differ significantly from the previous version.

Rössler attractor

The second attractor by the number of references in scientific articles and publications. For Rössler attractor characteristic is the presence of a boundary point for the manifestation of chaotic or periodic properties. At certain parameters of the dynamic system, oscillations cease to be periodic, and chaotic oscillations arise. One of the remarkable properties of the Rössler attractor is the fractal structure in the phase plane, that is, the phenomenon of self-similarity. It can be seen that other attractors, as a rule, have this property.

The Rössler attractor is observed in many systems. For example, it is used to describe fluid flows, as well as to describe the behavior of various chemical reactions and molecular processes. The Rössler system is described by the following differential equations:

In the MATLAB environment, the attractor is built as follows:

Temporal implementation of spatial quantities:

Three-dimensional model of the Rössler attractor:

Bang! The values ​​have changed a little:

Attractor under slightly changed initial conditions (the trajectories are different!)

Attractor with other coefficients in the system of equations (a chaotic process has turned into a periodic one!)

Compare pictures of 3D attractors with different initial conditions and coefficients in the system of equations. Do you see how the trajectories of movement changed dramatically in the first case? But one way or another, they are concentrated near a single area of ​​attraction. In the second case, the attractor generally ceased to show signs of chaos, turning into a closed periodic loop (limit cycle).

Attractor Rikitake

Dynamo Rikitake is one of the well-known third-order dynamical systems with chaotic behavior. It is a model of a two-disk dynamo and was first proposed in problems of chaotic inversion of the Earth's geomagnetic field. The scientist Rikitake investigated a dynamo system with two interconnected disks constructed in such a way that the current from one coil of the disk flowed into the other and excited the second disk, and vice versa. At some point, the system began to fail and show unpredictable things. Active studies of the attractor made it possible to project the Rikitake dynamo onto a model of the connection of large magnetic field vortices in the Earth's core.

Dynamo Rikitake is described by the following system of equations:

Rikitake dynamo model in MATLAB:

Temporary implementation:

Attractor (first version):

Dynamo (second version)

You can see that the Rikitake dynamo is somewhat similar to the Lorenz attractor, but these are completely different systems and describe different physical processes!

Nose-Hoover attractor

A less famous but no less important three-dimensional dynamical system is Nose-Hoover thermostat. It is used in molecular theory as a time-reversible thermostatic system. Unfortunately, I don’t know as much about this attractor as about the others, but it seemed interesting to me and I included it in the review.

The Nose-Hoover thermostat is described by the following system of equations:

Nose-Hoover Model in MATLAB:

Temporary implementation:

In this book, we have taken an empirical approach to chaotic oscillations and have outlined a whole series of different physical phenomena in which chaotic dynamics play an important role. Of course, not all readers have access to a laboratory or have a penchant for experimentation, although most of them can use digital computers. With this in mind, we present in this appendix a series of numerical experiments that can be carried out either on a personal computer or on a microcomputer, in the hope that they will help the reader to investigate the dynamics of the now classical chaos models.

B.1. LOGISTIC EQUATION: DOUBLING THE PERIOD

One of the simplest problems to start with in new dynamics must be the population growth model, or logistic equation

Period doubling phenomena have been observed by various investigators (see, for example, May's work) and, of course, by Feigenbaum, who discovered the famous parameter similarity laws (see Chaps. 1 and 5). A personal computer makes it extremely easy to reproduce two numerical experiments.

In the first experiment, we have a graph of dependence on in the range . Period doubling mode is observed at values ​​below. Starting with you can see a trajectory with a period of 1. To see longer trajectories, mark the first 30-50 iterations with dots, and subsequent iterations with a different symbol.

Of course, by plotting the dependence on , you will be able to observe the transitional and stationary regimes. Chaotic trajectories can be detected at . In the vicinity, one can detect a trajectory with a period of 3 .

The following numerical experiment is related to the construction of a bifurcation diagram. To do this, it is necessary to plot the dependence at large on the control parameter. Choose some initial condition (for example, and do 100 iterations of the display. Then plot the values ​​obtained from the next 50 iterations on the vertical axis, and the corresponding value on the horizontal axis (or vice versa). Step by choose about 0.01 and go through the range On the diagram at the period doubling points should produce classical pitchfork-type bifurcations.Can you determine the Feigenbaum number from the data of a numerical experiment?

May also gives a list of numerical experiments with other one-dimensional mappings, for example, with the mapping

He describes this mapping as a model of population growth of a single species, regulated by an epidemic disease. Explore the area. The period doubling accumulation point and the beginning of chaos correspond to . May's article also contains data on some other numerical experiments.

B.2. LORENTZ EQUATIONS

A remarkable numerical experiment, undoubtedly worthy of repetition, is contained in the original work of Lorentz. Lorentz simplified the equations derived by Saltzman from the equations of thermal convection in a liquid (see Chap. 3). The priority in the discovery of non-periodic solutions of the convection equations, according to Lorentz, belongs to Salzman. To study chaotic motions, Lorentz chose the now classical values ​​of the parameters in the equations

The data shown in fig. 1 and 2 of the Lorentz article, can be reproduced by choosing the initial conditions and the time step and projecting the solution either on a plane or on a plane

To get the one-dimensional mapping induced by this flow, Lorentz considered the successive maxima of the variable z, which he labeled The plot of dependence on showed that in this case the mapping is given by a curve resembling the shape of the roof of a house. Lorentz then explored a simplified version of this map, called the "house type map", a bilinear version of the logistic equation

B.3. Intermittency and the Lorentz Equations

An illustrative example of intermittency can be found by numerically integrating the Lorentz equations using a computer:

with parameters according to the Runge-Kutta method. At , you will get a periodic trajectory, but at and more, "bursts", or chaotic noises will appear (see the work of Manneville and Pomo). By measuring the average number N of periodic cycles between bursts (laminar phase), you should get the scaling law

B.4. OENON ATTRACTOR

A generalization of the quadratic mapping on the line for the two-dimensional case (on the plane) was proposed by the French astronomer Hénon:

When , the Hénon mapping reduces to the logistic mapping explored by May and Feigenbaum. The values ​​of a and b at which a strange attractor arises include, in particular, . Build a graph of this mapping on the plane, bounding it with a rectangle. Having received an attractor, focus your attention on some small part of it and increase this part using a similarity transformation. Follow a significantly larger number of iterations of the mappings and try to reveal the fine-scale fractal structure. If you have the patience or you have a fast computer at hand, then perform another similarity transformation and repeat all over again for an even smaller area of ​​the attractor (see Fig. 1.20, 1.22).

If you have a program for calculating the Lyapunov exponents, then it is useful to keep in mind that the value of the Lyapunov exponent is given in the literature, and the fractal dimension of the attractor in the Hénon map is . By varying the parameters a and b, one can try to determine the area of ​​those values ​​for which the attractor exists, and find the period doubling area on the plane (a, b).

B.5. DUFFING EQUATION: UEDA ATTRACTOR

This model of an electric circuit with a nonlinear inductance was considered in Ch. 3. The equations of this model, written as a system of first-order equations, have the form

The chaotic oscillations in this model have been studied in great detail by Ueda. Use some standard numerical integration algorithm, such as a fourth-order Runge-Kutta scheme, and consider the case . At , you should get a periodic trajectory with a period of 3. (Place the Poincaré section at ) In the vicinity of the value, a trajectory with a period of 3 should, after a bifurcation, turn into a chaotic motion.

At , the periodicity is restored again with a transient chaotic regime (see Fig. 3.13).

Compare the fractal nature of the attractor as the damping decreases, assuming and 0.05. Note that at , only a small part of the attractor remains, and at , the motion becomes periodic.

B.6. DUFFING EQUATION WITH TWO POTENTIAL WELLS: HOLMES ATTRACTOR

This example was considered in our book. Several numerical experiments deserve to be repeated. The dimensionless equations in this case have the form

(Assuming and introducing an additional equation z = w, they can be written as autonomous system third order.) The factor 1/2 makes the natural frequency of small oscillations in each potential well equal to one. The chaos criterion for a fixed attenuation coefficient and variables was considered by us in Chap. 5. The area of ​​interest for research is . In this region, there should be a transition from the periodic regime to the chaotic one, periodic windows in the chaotic regime, and exit from the chaotic regime at . There is another area of ​​interest: In all studies, we strongly recommend the reader to use the Poincaré mapping. When using a personal computer, a high speed of information processing can be achieved through special tricks when compiling a program (see Fig. 5.3).

Another interesting numerical experiment is to fix the parameters, for example, to set and vary the phase of the Poincaré mapping, i.e. plot the points at by changing from 0 to Note the reversal of the mapping at Is this related to the symmetry of the equation? (See Figure 4.8.)

B.7. CUBIC MAPPING (HOLMES)

We have illustrated many concepts of the theory of chaotic oscillations by the example of an attractor in a model with two potential wells. The dynamics of such a model is described by an ordinary nonlinear differential equation second order (see Chap.

2 and 3), but an explicit formula for the Poincaré map of such an attractor is unknown. Holmes proposed a two-dimensional cubic mapping that has some of the properties of a Duffing oscillator with negative stiffness:

The chaotic attractor can be found near the parameter values

B.8. Bouncing BALL DISPLAY (STANDARD DISPLAY)

(See the article by Holmes and the book by Lichtenberg and Lieberman.) As noted in Chap. 3, the Poincaré map for a ball bouncing on a vibrating table can be written exactly in terms of the dimensionless velocity of the ball hitting the table and the phase of the table movement.

where is the energy loss during the collision.

Case (conservative chaos). This case is investigated in the book by Lichtenberg and Lieberman as a model of electron acceleration in electromagnetic fields. After iterating the display, apply the obtained points to the plane. To calculate, use the expression

in an improved version of BASIC. To get a good picture, you have to vary the initial conditions. For example, select and follow a few hundred mapping iterations at various v from the interval -

You will find interesting cases with. At , one can observe quasi-periodic closed trajectories around periodic fixed points of the map. At , regions of conservative chaos should appear near the points of the separatrices (see Fig. 5.21).

case. This case corresponds to a dissipative mapping, where energy is lost in every collision between the ball and the table. Start with . Note that although the first iterations look chaotic, as in case 1, the movement becomes periodic. To get fractal-like chaos, the values ​​of K must be increased to . A strange attractor, even more reminiscent of a fractal, you get by setting .

B.9. MAPPING THE CIRCLE ON ITSELF: SYNCHRONIZATION OF THE NUMBER OF ROTATIONS AND FAIRY TREES

A point moving along the surface of a torus can serve as an abstract mathematical model of the dynamics of two coupled oscillators. The oscillator motion amplitudes serve as small and large torus radii and are often assumed to be fixed. The phases of the oscillators correspond to two angles that define the position of a point along a small circle (meridian) and a large circle (parallel) on the surface of the torus. The Poincaré section along the small circles of the torus generates a one-dimensional difference equation called the self-mapping of the circle:

where is a periodic function.

Each iteration of this mapping corresponds to the trajectory of one oscillator along the great circle of the torus. A popular object of study is the so-called standard circle mapping (normalized to )

The possible movements observed in this mapping are: periodic, quasi-periodic and chaotic modes. To see periodic cycles, plot points on a circle with rectangular coordinates

When the parameter is 0, there is nothing but the number of rotations - the ratio of two frequencies of unrelated oscillators.

When the mapping can be periodic and when is an irrational number. In this case, the oscillators are said to be locked or that mode pulling has occurred. At , one can observe synchronized or periodic motions in regions of finite width along the O axis, which, of course, contain irrational values ​​of the parameter . For example, when a cycle with period 2 can be found in the interval and a cycle with period 3 can be found in the interval To find these intervals at calculate the number of rotations W as a function of the parameter at 0 01.

In practice, to get the number of rotations with sufficient accuracy, you need to take N > 500. Plotting W against , you will see a series of plateaus corresponding to synchronization areas. To see more synchronization regions, one should choose a small AP region and construct W for a large number of points in this small region.

Each synchronization plateau on the graph ) corresponds to a rational number - the ratio of cycles of one oscillator to q cycles of another oscillator. The relationships are ordered in a sequence known as the Fairy tree. If two regions of mode synchronization are specified for the values ​​of the parameters , then between them in the interval there will certainly be one more region of synchronization with the number of rotations

Starting with 0/1 at and 1/1 at , one can construct the entire infinite sequence of synchronization regions. Most of them are very narrow.

Note that the width of these regions tends to zero at and becomes larger at Synchronization regions in the () plane are in the form of long protrusions, and are sometimes called Arnold tongues.

B.10. Rössler attractor: chemical reactions, one-dimensional approximation of multidimensional systems

Each of the main areas of classical physics has created its own model of chaotic dynamics: hydromechanics - the Lorentz equations, structural mechanics - the Duffing-Holmes attractor with two potential wells, electrical engineering - the Duffing-Ueda attractor. Another simple model arose in the dynamics of chemical reactions occurring in a certain vessel with stirring. Rbssler suggested it.

where is the sum of diagonal first-order minors of the matrix A

is the sum of second-order diagonal minors of the matrix A

is the sum of third-order diagonal minors of the matrix A

Leta= - ,b= , then the XY of the 3rd order has the form:

Condition:

F(a,b,c)<0 – все собст.знач.-я ХП вещественные

F(a,b,c)>

Two characteristic Ressler equations.

When solving a system of differential equations, there are 2 singular points P10(0,0,0) and P20==(c-ab,b-c/a,c/a-b), if you do all the operations with finding the Jacobian and the sums of diagonal elements, then 2 equations will be learned Ressler:

3.3 Condition for determining the type of eigenvalues ​​of the characteristic equation of the third order.

Condition:

F (a, b, c) \u003d (9c-ab) 2 - (6b-2a 2) (6ac-2b 2)

F(a,b,c)<0 – все собст.знач.-я ХП вещественные

Ф(a,b,c)=0 – two (three) multiple substances. root

Ф(a,b,c)>0 – two complex conjugate roots

Roots of the characteristic equation with parameters: 0.38; 0.30; 4.82 (unstable focus saddle).

Integral curves must be built with respect to each singular point.

All “conditions” are considered + condition (s-av)> 0i (s-av)<0 рассматирваием для Ро1=(0,0,0)

If we consider equations with parameters 0.38 ..., then an interesting trajectory is obtained, the trajectory repels from Po1 (0,0,0) along R 2 (x1, x2) in the phase space R 3, and attracts along a one-dimensional curve, forming a fixed point of the saddle type -focus. The representing point leaves the region of the unstable equilibrium point of type Po1 in the plane of variables (x1,x3), and then returns to this point again.

Homoclinic trajectory in the phase space of the system.

The phase portrait makes it possible to depict a qualitative characteristic of the entire set of free motions (processes) for a selected region of the NU of the root space.

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 arise (The concept of a homoclinic trajectory means that it goes out and comes to the same equilibrium position).

Homoclinic trajectory– does not occur if the parameters do not satisfy some strict constraint.

Structural instability of a homoclinic trajectory.

At large values ​​of the parameter, the trajectory undergoes serious changes. Shilnikov and Kaplan showed that at very large r, the system goes into self-oscillation mode, and if the parameter is reduced, a transition to chaos will be observed through a sequence of oscillation period doublings.

Homoclinic trajectories- structurally unstable.

strange attractor

strange attractor: unstable equilibrium position is the main feature of chaotic behavior. Trajectories are very sensitive to changes in initial conditions - this quality is inherent in strange attractors.

A strange attractor is an attractor that has two significant differences from an ordinary attractor: the trajectory of such an attractor is non-periodic (it does not close) and the mode of operation is unstable (small deviations from the mode increase). The main criterion for the randomness of an attractor is the exponential growth of small perturbations in time. The consequence of this is "mixing" in the system, non-periodicity in time of any of the coordinates of the system, a continuous power spectrum and an autocorrelation function decreasing in time.

The dynamics on strange attractors is often chaotic: predicting a trajectory that has fallen into an attractor is difficult, since a small inaccuracy in the initial data after a while can lead to a strong discrepancy between the prediction and the real trajectory. The unpredictability of the trajectory in deterministic dynamic systems is called dynamic chaos, distinguishing it from the stochastic chaos that occurs in stochastic dynamic systems. This phenomenon is also called the butterfly effect, implying the possibility of transforming weak turbulent air currents caused by the flapping of a butterfly's wings at one point on the planet into a powerful tornado on the other side of it due to their multiple amplification in the atmosphere over some time.

Is stochastic and regular behavior possible at the same time? Or is it always either regular or stochastic?

Both regular and chaotic behavior of dynamic dissipative systems with many variables (n>2) are possible, and not only separately (either-or), but also simultaneously.

It cannot be said that the system goes into chaos immediately after the first bifurcation (since it left in one place, it came in another)

Why third order? Is it possible for strange attractors to appear in second-order systems? And in systems above the third order?

More precise mathematical conditions for the emergence of chaos look like this:

The system must have non-linear characteristics, be globally stable, but have at least one unstable oscillatory-type equilibrium point, while the dimension of the system must be at least 1.5 (i.e., the order of the differential equation must be at least 3rd).

Linear systems are never chaotic. For a dynamical system to be chaotic, it must be non-linear. According to the Poincaré-Bendixson theorem, a continuous dynamical system on a plane cannot be chaotic. Among continuous systems, only non-planar spatial systems have chaotic behavior (at least three dimensions or non-Euclidean geometry are required). However, a discrete dynamical system at some stage can exhibit chaotic behavior even in one-dimensional or two-dimensional space.

Lecture 3. Integrable and non-integrable systems. conservative systems

Integrated systems

1. Reducibility to free (unperturbed) motion of systems. What happens in case of irreducibility?

For integrable systems, interactions can be excluded and the problem can be reduced to the problem of free movement. For free movement it is not difficult to find expressions for coordinates and velocities in the form of explicit functions of time. For non-integrable systems, it is necessary to abandon the description in terms of trajectories and go over to a probabilistic description (if irreducible).

Is it possible to describe a non-integrable system in terms of trajectories?

no impossible. We are talking about a fundamentally probabilistic description that cannot be reduced to a description in terms of individual trajectories.

Can a system given by a deterministic equation have stochastic dynamics?

D. s. opposed to a probabilistic system, the outputs of which only randomly, and not uniquely depend on the inputs. (in the ds, it uniquely depends on the inputs). But any system, even if it is deterministic, will contain some randomness.

Consider the image of a strange attractor of the Ressler system. Its geometric configuration can be visualized as follows. Take a paper tape that expands towards one end (a). At the wide end, fold the tape in half and then glue it into a ring as shown in Fig. (b-d). Such a paper model gives a good idea of ​​the Ressler attractor and the spatial arrangement of its trajectories. It is, however, inaccurate in one essential detail. The Ressler SDE solution can be constructed both forward and backward in time, and the uniqueness theorem holds. Therefore, no two different phase trajectories can converge into one, which means that the gluing procedure is illegal.

The resolution of the contradiction lies in the fact that the "tape" from which the Ressler attractor is "glued" is actually a layered formation, a set of sheets. The gluing procedure is equivalent to establishing a one-to-one correspondence between the set of sheets of the original tape and the set of sheets of the tape folded in half. Such a correspondence can take place only if both sets are infinite. Thus, the Ressler attractor must have an infinite number of layers in cross section and, therefore, represent a fractal object with a complex structure, as they say.

The same nature of the structure is also characteristic of other strange attractors. The figure shows a diagram from Heno's article illustrating the structure of the attractor in the mapping (2). It can be noted that the main point in the motivation of this work was precisely the intention to present a more illustrative example of the fractal structure of the attractor for consideration than the Lorentz model, known by that time, demonstrated. Reproduction of the fractal structure of the Henault attractor at different resolution scales

Fractals Fractals are understood as sets that demonstrate similarity properties (or scale invariance) in a strict or approximate sense at different resolution scales of their geometric structure, as well as objects in nature that have this property, at least approximately, in a fairly wide range of scales. The concept of a fractal came into use thanks to the mathematician Benoit Mandelbrot to refer to non-trivial geometric objects. He drew attention to the fact that fractal objects can be considered not only as "mathematical monsters", but as models of the geometric properties of quite real formations in nature (coastline, clouds, mountain ranges, trees, eddies in a turbulent fluid, etc.) . Classification of fractals 1. Constructive (built using certain recursive geometric or algebraic procedures). 2. Dynamic (generated by dynamic systems). 3. Natural (observed in nature). 4. Stochastic (the trajectory of a Brownian particle or an arbitrary trajectory of a diffusion random process).

The simplest constructive fractal is associated with the construction proposed back in 1883 by the founder of set theory, Georg Cantor. Having a unit segment, we divide it into three equal parts and discard the interval that occupies the middle third. We again divide each of the remaining segments into three parts and throw out the middle third, and so on ad infinitum. What remains in the end is the Cantor set or "Cantor dust". The Cantor set satisfies the definition of a fractal: each of its fragments, obtained from some segment at a certain level of construction, is similar to the entire set and passes into it with an appropriate recalculation of the scale. We note two properties of the Cantor set. 1) This set has zero measure (zero length), i.e. the total length of all discarded intervals is equal to 1, the length of the original interval. At the 1st step, an interval of length 1/3 is discarded, at the 2nd step - two intervals of length 1/9, at the nth step - 2 n intervals of length 3 -n+1. Calculating the sum, we get

2) The Cantor set has the cardinality of the continuum, i.e. allows the establishment of a one-to-one correspondence with the set of all points of the unit interval due to the algorithm for its construction. By changing the rule for dividing a single segment and introducing a division into three unequal parts, you can get a more complex two-scale Cantor set (multifractal). The Koch snowflake is an example of an area with a fractal boundary. We start construction with an equilateral triangle. Then on each side we replace the middle third with a broken line of two segments of the same length. Repeating the procedure many times to infinity, we eventually arrive at a fractal object. The first 4 iterations of the 7 steps of building a Koch snowflake

To construct a Sierpinski napkin (triangle), we take an equilateral triangle, which can be thought of as being made up of four smaller triangles. Discard the middle triangle. Further, we perform the same actions with each of the remaining triangles ad infinitum. The Sierpinski carpet is built on the basis of a square, which is divided by vertical and horizontal lines into 9 equal parts, and the middle square is discarded. With each remaining square, the same procedure, and so on ad infinitum.

Fractals generated by the deterministic dynamics of nonlinear systems are called dynamic. Dynamic fractals can be attractors or other limit sets in the phase space, the dimension of which N for flows should be N > 2, and for systems with discrete time N 2. When talking about irregular attractors, they separate the concepts of "strange" and "chaotic". It is the property of "strangeness" that refers to its non-trivial (fractal) geometry. Fractal properties are possessed by the boundaries of basins of attraction of several coexisting attractors, and this is a characteristic feature of nonlinear DS. 2, "> 2, and for systems with discrete time N 2. When talking about irregular attractors, they separate the concepts of "strange" and "chaotic". It is the property of "strangeness" that refers to its non-trivial (fractal) geometry. Fractal properties have boundaries basins of attraction of several coexisting attractors, and this is a characteristic feature of nonlinear DS."> 2, " title=" Fractals generated by the deterministic dynamics of nonlinear systems are called dynamic. Dynamic fractals can be attractors or other limit sets in the phase space, dimension which N for streams must be N > 2,"> title="Fractals generated by the deterministic dynamics of nonlinear systems are called dynamic. Dynamic fractals can be attractors or other limit sets in the phase space, the dimension of which N for flows must be N > 2,"> !}

Many dynamic fractals known for their beauty are associated with the following simple Julia mapping: where Z is a complex variable and C is a complex parameter. The Julia set is an example of a fractal boundary between the pools of attraction of an attractor at infinity (maroon area) and periodic motion (colored area). The tone (color) is determined by the number of iterations required to reach the attractor.

Mandelbrot set This fractal structure is obtained by repeatedly applying an algebraic transformation (recurrent relation) using a function of a complex variable. The black color in the middle shows that at these points the function tends to zero - this is the Mandelbrot set. Beyond this set, the function tends to infinity. The most interesting thing is the boundaries of the set. They are also fractal. At the boundaries of this set, the function behaves unpredictably - chaotically.

Dimensions of attractors A distinctive feature of strange attractors is the presence of the property of scale invariance (scaling), which is expressed in the repeatability of their structure on ever smaller scales. A consequence of the laws of similarity is the universality in the geometry of chaotic sets of Poincare sections, in the distribution of vibration energy over frequencies and amplitudes in the spectrum, etc. The notion of dimension is introduced to characterize strange attractors. The dimension determines the amount of information required to set the coordinates of a point belonging to the attractor within the specified accuracy. For regular attractors that are manifolds, the dimension is an integer: a fixed point has dimension 0, a limit cycle has dimension 1, and a two-dimensional torus has dimension 2. Due to the complexity of the geometric structure, strange attractors are not manifolds and have a fractional dimension. Dimension definitions are generally divided into two types: those that depend only on the metric properties of the attractor and, apart from the metric, those that depend on the statistical properties of the flow due to the dynamics. In typical cases, the metric dimensions take the same value, which is usually called the fractal dimension of the attractor D. The dimension, determined taking into account the probability of visiting the trajectory various areas attractor in the phase space is called the information or dimension of the natural measure.

(29)

Applying definition (29) to calculate the dimension of a point, line, and surface, one can verify the usual values ​​of 0, 1, and 2, respectively. For non-trivial sets, the fractal dimension is always fractional. This property is used as feature"strangeness" of the attractor. The fractal dimension defined by covering the set with cells of a fixed shape and size is called the capacity of the set. If elements of an arbitrary shape and size are used as a cover of a set, then the dimension calculated in this way is called the Hausdorff dimension. For fractals, this dimension and capacity coincide and simply speak of the fractal dimension of the object.

Information dimension Along with the fractal dimension, a number of others are introduced and used, including information, correlation, and generalized Rényi dimensions. Why is one metric dimension not enough? Let's imagine that the attractor is non-uniform - some areas (coverage elements) are visited more often, others less often. This circumstance is not reflected in the definition of capacitance. Let an invariant measure be defined for the attractor, and we have constructed a cover of this attractor, while each cell of the cover will have its own specific measure value. In other words, each i-th cell of the coverage will correspond to some probability p i of being in it. Assuming that the cells completely cover the attractor and do not overlap, we now consider the sum (30) This value can be interpreted as the amount of information in the statement that the representative point is found in one specific cell of the coverage.

It is clear that as the size of the coverage cells decreases, the value of the sum (30) will increase: the smaller the cells, the more information in the statement that the point fell into a given specific cell. This increase follows the law (31) or, equivalently, there is a limit (32) The quantity D I is called the information dimension.

Correlation dimension and the Grassberger-Procaccia algorithm Consider again covering the attractor with cells of the same size and suppose that two points belonging to the attractor, x 1 and x 2, are chosen at random. What is the probability that both of them will be in the i-th cell? The probability that one point falls into i-th element coverage is equal to p i. If the hit of both points in a given cell can be considered independent events, then the probability will be p i 2. Consider the sum (33) As the size of the cells decreases, the sum will decrease and this will happen according to the power law (34) or, equivalently, there is a limit (35) The value of D C is called the correlation dimension.
Generalized dimension It is possible to generalize the dimensions D F, D I, D C and introduce a dimension of order q, using the generalized entropy of order q (Rényi entropy) (37) where P i is the probability of finding a point of the set in i-th element coatings. Then the dimension of order q is (38) It can be shown that D 0 = D F, D 1 = D I, D 2 = D C.

Lyapunov dimension The fractal dimension of the DS attractor in the phase space RN can be estimated using the spectrum of Lyapunov characteristic exponents (LXI). Such an estimate is called the Lyapunov dimension D L and is given by a certain relation called the Kaplan-Yorke formula. Let the LHP spectrum of a strange attractor of an N-dimensional system be known, the dimension of which must be estimated: 1 2 … N. The sum of all spectrum indices is negative due to the dissipativity of the system. Let us consider the first k indicators of the LHP spectrum, where k is largest number, which satisfies the condition All positive, all zero and some negative indicators are included in the specified number of indicators so that the sum remains non-negative. Since the sum of the exponents determines the nature of the local change in the element of the phase volume in the attractor, then the phase volume of dimension k

Thus, we can assume that the dimension of the attractor lies in the interval k D L k + 1. It is reasonable to require that the motion on the attractor obeys the condition that corresponds to the physical concepts of the stationarity of the process, where d is the fractional part of the dimension. The full Lyapunov dimension of the attractor will be the sum of the integer k and fractional d parts: (39) Differences in the signature of the LHP spectra and the dimension D L can be a sign of the classification of regular and strange attractors. From the Kaplan-York formula (39) for regular attractors, we obtain the following values ​​of the Lyapunov dimension, coinciding with the fractal dimension of the corresponding set and equal to the number of zero exponents in the LHP spectrum: equilibrium state (-, -, -, …) – D L = 0; limit cycle (0, -, -, -, ...) – D L = 1; two-dimensional torus (0, 0, -, -, …) – D L = 2; N-dimensional torus (0, 0, 0, …,0, -, …) – D L = N.

For regular attractors, the following are in full agreement: the Lyapunov dimension, the fractal dimension, and the signature of the LHP spectrum of the attractor. With regard to strange attractors, such an interaction can only be spoken of in relation to three-dimensional differential systems and two-dimensional reversible mappings with constant expansion and contraction. It was proved that for attractors in such systems, the fractal dimension can be determined by the following relations: - for two-dimensional mappings - for three-dimensional differential systems When choosing which definition of dimension is better to use, one usually proceeds from the possibilities of numerical calculations. In numerical simulation of DS, it is most convenient to use the Lyapunov dimension. To estimate the fractal dimension of an attractor from experimental data, the correlation dimension is best suited.