Strange attractor

Thomas attractor with parameters x '= sin (y) -bx, y' = sin (z) -by, z '= sin (x) -bz

A strange attractor is an attractor , i.e. a place in phase space that represents the final state of a dynamic process, whose fractal dimension is not an integer and whose Kolmogorov entropy is really positive. It is a fractal that cannot be described geometrically in a closed form. Occasionally the term chaotic attractor is preferred, as the “weirdness” of this object can be explained with the help of chaos theory . The dynamic process shows an aperiodic behavior.

discovery

The term strange attractor can be traced back to an article by David Ruelle and Floris Takens from 1971 on the mathematical background of the formation of turbulent flows. The Lorenz attractor had been known since 1963 , a mathematical structure that was discovered when modeling air currents under the influence of temperature differences.

Attractors have already been investigated to describe dynamic processes. However, these were usually imagined as classic geometric shapes, for example points or cyclically traversed lines. Acyclic attractors were also known, but they were considered to be special cases, anomalies that can only occur with a certain choice of parameters.

With the introduction of the concept of the strange attractor, it was possible to better understand the laws of chaotic behavior in dynamic systems and to describe them quantitatively. This behavior, for example turbulent flows of liquids and gases, which is characterized by a lack of periodicity and sensitive dependence on the initial conditions ( butterfly effect ), was previously beyond analytical consideration due to its complexity. With constructions like the strange attractors, it is possible to mathematically describe a deterministic, but nevertheless unpredictable behavior ( deterministic chaos ).

definition

We speak of a strange attractor if the following conditions are met:

It is an attractor in a dynamic system with a certain catchment area.
Chaotic behavior: Any small changes in the initial state lead to completely different processes.
Fractal structure: the attractor has a non-integer dimension.
No possibility of splitting up: Each lane that starts in the catchment area approaches any point of the attractor.

Examples

A few classic examples can be used to study the properties of strange attractors quite well. The dynamic system used can be discrete or continuous . Continuous systems are usually described by differential equations, in phase space these systems form a line starting from the initial state, the trajectory . Because of the uniqueness of the derivation, trajectories cannot intersect at any point. From this it can be concluded that attractors can only have a simple structure in two dimensions; Strange attractors and thus chaotic systems only exist in continuous dynamic systems in a phase space with at least three dimensions.

Hénon attractor

A relatively simple example of a strange attractor is the Hénon attractor (named after Michel Hénon ), which is defined as a discrete system in two-dimensional space by the following equations:

{\ displaystyle {\ begin {aligned} x_ {k + 1} & = y_ {k} + 1-ax_ {k} ^ {2} \\ y_ {k + 1} & = bx_ {k} \ end {aligned }}}

Each mapping step can be broken down into three sub-steps: a fold-stretch operation by adding to , a contraction by the factor along the -axis and a mirroring by swapping the - and -axis. When choosing the parameters and , the typical image of the Hénon attractor results if one follows the path of a starting point that is sufficiently close to the attractor. On the surface, the attractor looks like a line that is laid in a few folds, but since it is invariant to the fold-stretch-compress-mirror operation described, each individual line is again divided infinitely often into approximately parallel lines. A cross-section through the attractor has the structure of a Cantor set , the attractor thus has a fractal structure. ${\ displaystyle 1-ax ^ {2}}$ ${\ displaystyle y}$ ${\ displaystyle b}$ ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle a = 1 {,} 4}$ ${\ displaystyle b = 0 {,} 3}$

Looking at the vicinity of a point on the attractor, i. H. a circular disk with a small diameter, this is converted by an imaging step into an elongated ellipse that is stretched along the lines of the attractor, but has a smaller area due to the contraction step. With continued application of the mapping rule, the image of the point environment covers ever larger areas of the attractor, while its area tends to zero.

Rössler attractor

Rössler attractor with parameters , and

{\ displaystyle a = 0 {,} 2}

{\ displaystyle b = 0 {,} 2}

{\ displaystyle c = 5 {,} 7}

The Rössler attractor ( discovered by Otto Rössler in 1976 ) is defined in three-dimensional space by the following system of differential equations:

{\ displaystyle {\ begin {aligned} x ^ {\ prime} & = - (y + z) \\ y ^ {\ prime} & = x + ay \\ z ^ {\ prime} & = b + xz- cz \ end {aligned}}}

The Rössler attractor was defined in order to be able to study the phenomena of the Lorenz attractor, which had been known for a long time, using a simpler example. The dynamics of the underlying system can be illustrated as follows: Basically, the trajectories in the xy-plane run on a circular path counter-clockwise around the zero point and form a fixed, flat band there. The course of the band widens; in the area of high values, the trajectories rise in the z-direction, which means that they are directed towards the interior of the circular path. The outer area of the band is folded inwards, so to speak. As already mentioned, trajectories cannot intersect, so the band consists of two layers lying close to each other, which are getting closer and closer, but are still at a distance from each other. With every further passage of the circular path, the band is folded again, so it consists of an infinite number of layers that are as close as desired to one another. A cross-section through these layers again gives a Cantor set. ${\ displaystyle x}$

Here, too, the fractal structure is created by an infinite series of stretching and folding operations. If you look at the trajectories of two points that are close to each other, they will run relatively close to each other for a while, but increasingly diverge as a result of the stretching until the point is reached that one trajectory runs on the flat part and the other on the one above folded. There is no longer a connection between the two trajectories; we are dealing with chaotic behavior.

Lorenz attractor

The Lorenz attractor was discovered in 1963 by Edward N. Lorenz while modeling air currents. It is defined by the following system of equations:

{\ displaystyle {\ begin {aligned} x ^ {\ prime} & = - \ sigma x + \ sigma y \\ y ^ {\ prime} & = Rx-y-xz \\ z ^ {\ prime} & = - Bz + xy \ end {aligned}}}

The three parameters are determined by the underlying physical model and were specified by Lorenz with and . ${\ displaystyle \ sigma = 10, B = {\ tfrac {8} {3}}}$ ${\ displaystyle R = 28}$

The attractor is symmetrical to the -axis and consists of two disk-like parts that are slightly tilted towards the -axis. Here, too, as with the Rössler attractor, the trajectories form flat circular bands that widen as you walk around the respective center of the circle. The direction of travel goes outwards on the underside of the "discs", towards the middle the bands coming from above approach each other more and more and then lie flat on top of each other. One half of the band then forms the band of one circle, the other that of the other circle. So there is a continued stretch-fold-part operation. A trajectory can remain one or more revolutions in one subsystem, then one or more in the other, the respective number is practically arbitrary and depends on the initial value. Here, too, trajectories of points that are close together will run in parallel for a certain time and execute the same orbits until a trajectory ends up on one side and the other on the other side, and a completely different further course occurs. ${\ displaystyle z}$ ${\ displaystyle z}$

Lyapunov exponent

The Lyapunov exponents are usually used to describe the behavior of a dynamic system quantitatively . These describe the dynamic behavior of the surroundings of a point on the attractor: First, one expects that a point in the surroundings is attracted by the attractor, which is expressed by a negative Lyapunov exponent, the amount of which is a measure of the strength of the attraction. If it is a strange attractor, as can be seen in the examples, a repulsion of points which are close together is observed, which corresponds to a positive Lyapunov exponent. In fact, the behavior is dependent on the direction the two points have towards each other. If one imagines the surroundings of a point as a circular disk or sphere, then this is deformed into a narrowed and elongated image in the further course. To illustrate this, a dynamical system has as many Lyapunov exponents as there are dimensions of phase space.

If one carries out calculation steps for a point on the attractor and a different point in its vicinity, the first Lyapunov exponent is carried out ${\ displaystyle n}$

{\ displaystyle \ lambda = \ lim _ {n \ to \ infty} \ lim _ {E_ {0} \ to 0} {\ frac {1} {n}} \ sum _ {k = 1} ^ {n} \ log \ left | {\ frac {E_ {k}} {E_ {k-1}}} \ right |}

Are defined. is the deviation in the calculation step, i.e. the average error gain for large is determined. ${\ displaystyle E_ {k}}$ ${\ displaystyle k}$ ${\ displaystyle n}$

The first Lyapunov exponent always gives the value of the greatest error gain, i.e. the strongest repulsion. This is achieved by determining the limit value for : With any choice of the disturbed starting point, the direction of its deviation is generally a combination of a disturbance in the direction of the greatest error amplification and other disturbances with a lower amplification. If the direction of the disturbance is retained in further calculation steps, after several steps the greatest error amplification prevails. ${\ displaystyle n \ to \ infty}$

When calculating the first Lyapunov exponent numerically, a precaution must be taken so that any number of steps can actually be carried out: a renormalization is carried out after each step, i. H. the newly calculated disturbed point is replaced before the next step by a point that has the same direction from the undisturbed point, but the same distance as before the calculation step. This prevents the amplification of the initial error from reaching an order of magnitude at which the geometrical properties of the attractor, which in any case has a finite extent, falsify the result.

The other Lyapunov exponents are defined analogously: If it is determined by the maximum change in distance , then follows from the maximum change in area in the vicinity of an attractor point, from the maximum change in space and so on. Often the other Lyapunov exponents can be calculated with the help of the first, since the factors of surface and space contraction can be derived from the definition of the dynamic system. In the case of an attractor, the sum of all Lyapunov's exponents is generally negative, but in the case of a strange attractor, at least the first is positive. ${\ displaystyle \ lambda _ {1}}$ ${\ displaystyle e ^ {\ lambda _ {1}}}$ ${\ displaystyle \ lambda _ {2}}$ ${\ displaystyle e ^ {\ lambda _ {1} + \ lambda _ {2}}}$ ${\ displaystyle \ lambda _ {3}}$ ${\ displaystyle e ^ {\ lambda _ {1} + \ lambda _ {2} + \ lambda _ {3}}}$

dimension

An important figure for a fractal, and thus for a strange attractor, is the dimension. There are several possibilities to extend the concept of dimension, which in classical geometry can only assume integer values, to fractals. In the case of classical geometric objects, all these definitions must necessarily result in their known dimensions, for example 1 for lines and 2 for surfaces. For a fractal, however, different definitions of the fractal dimension can also result in different values.

Box dimension

The box dimension is most commonly used for fractals . The basic idea is to subdivide the surrounding space into space elements of equal size ("boxes"), the length of which is reduced with each step. It is counted in how many of these spatial elements a part of the fractal lies. For the side length s is this number N ( s ) is to be expected then the following relationship: . is the dimension of the fractal. If this has the value 1, for example, then the number of spatial elements in which part of the examined structure is located is proportional to the reciprocal of the side length, i.e. exactly what one would expect with linear structures. As the side length decreases, the examined fractal is displayed more and more precisely, so that increasing accuracy is expected. ${\ displaystyle N (s) \ propto s ^ {- D_ {f}}}$ ${\ displaystyle D_ {f}}$

For the calculation of the dimension of a strange attractor, however, this method does not prove to be very helpful. The smaller the side length, the more space elements come into consideration; a lot of calculation steps of the attractor have to be carried out without knowing whether all space elements in which the attractor is located have already been recorded. The calculation error increases particularly in the size ranges that should actually provide increasingly more precise values.

Information dimension

In order to get the problems with the box dimension at least partially under control, this dimension concept can be refined a little. In the case of the box dimension, the only thing that counts is whether a part of the fractal is actually located in a space element, then the size of this part is determined first (it is not particularly useful to speak of area or space contents here). In the case of the strange attractor, this can be done simply by counting the iteration steps whose end point lies in the relevant spatial element. This proportion (a number between 0 and 1) is called the natural measure . The information , measured in bits, that a certain point of the attractor lies in a certain space element, is calculated as a binary logarithm from the reciprocal of the natural measure : The smaller the proportion of the attractor that lies in this space element, the greater the information, if the point being examined lies therein. ${\ displaystyle B_ {k}}$ ${\ displaystyle \ mu (B_ {k})}$

The information dimension is obtained by replacing the logarithm with the information of the overall system when determining the box dimension : ${\ displaystyle D_ {I}}$ ${\ displaystyle N (s)}$

{\ displaystyle I (s) = \ sum _ {k = 1} ^ {N (s)} \ mu (B_ {k}) \ log _ {2} {\ frac {1} {\ mu (B_ {k })}}.}

This is the mean value of the information for the individual spatial elements, weighted according to their natural measure, or the mean value of the information of all calculated points of the attractor.

Spatial elements that are recognized as part of the attractor only very late in the course of the calculation also contain only a small portion of the attractor and provide only little to the overall information of the system. In this way, the calculation error due to an early termination of the calculation is greatly reduced in contrast to determining the box dimensions.

The information dimension is not always the same as the box dimension; the inequality applies . ${\ displaystyle D_ {I} \ leq D_ {f}}$

Lyapunov dimension

Another dimension concept is based on the conjecture of Kaplan-Yorke . This conjecture claims that the information dimension is identical to the Lyapunov dimension , a quantity that can be calculated relatively easily from the Lyapunov exponent. To determine this Lyapunov dimension, the value is drawn in a coordinate system over each n and the points are connected with straight lines. In the case of a strange attractor, the first Lyapunov exponent is positive, but the sum of all Lyapunov exponents is negative, so a piece of this line intersects the x-axis. The x-value of this intersection is the Lyapunov dimension. In the case of a fixed-point attractor, all Lyapunov exponents are negative, the point of intersection with the x-axis is therefore at the origin, the Lyapunov dimension is 0. In the case of a cyclic, linear attractor , all other Lyapunov exponents are negative. Here the intersection at the x-value is 1, which confirms the correctness of the Lyapunov dimension in this classic case as well. ${\ displaystyle \ lambda _ {1} + \ cdots + \ lambda _ {n}}$ ${\ displaystyle \ lambda _ {1} = 0}$

The importance of the Lyapunov dimension lies in the possibility of its numerical calculation. While the determination of the box and information dimensions, especially in the case of higher-dimensional phase spaces, soon reaches its limits, the determination of Lyapunov exponents and thus the Lyapunov dimension is often still possible.

literature

H.-O. Peitgen , H. Juergens, D. Saupe: Chaos: building blocks of order , Klett-Cotta / Springer (1994), ISBN 3-608-95435-X
D. Ruelle and F. Takens: On the nature of turbulence. In: Communications in Mathematical Physics , 20/1971, pp. 167-192, ISSN 0010-3616
D. Ruelle: Small random perturbations of dynamical systems and the definition of attractors. In: Communications in Mathematical Physics , 82/1981, pp. 137-151, ISSN 0010-3616
John Milnor: On the concept of attractor. In: Communications in Mathematical Physics 99/1985, pp. 177-195, ISSN 0010-3616
David Ruelle: Elements of Differentiable Dynamics and Bifurcation Theory. , Academic Press (1989), ISBN 0126017107
R. Temam: Infinite dimensional dynamical systems in mechanics and physics. , Springer (1997), ISBN 038794866X
Manfred Schroeder: Fractals, Chaos, Power Laws. , WH Freeman and Company (1991), ISBN 0716721368