Attractor

Attractor ( lat. Ad trahere "to pull towards oneself") is a term from the theory of dynamic systems and describes a subset of a phase space (ie a certain number of states) towards which a dynamic system moves in the course of time and which under the dynamics of this system is no longer abandoned. This means that a set of variables approaches ( asymptotically ) a certain value, a curve or something more complex (i.e. a region in n-dimensional space) over time and then remains in the vicinity of this attractor over the further course of time.

An attractor appears as a clearly recognizable structure. Colloquially, one could speak of a kind of “stable state” of a system (whereby periodic , i.e. wave-like recurring states or other recognizable patterns can also be meant), i.e. a state towards which a system is moving.

Well-known examples are the Lorenz attractor , the Rössler attractor and the zeros of a differentiable function, which are attractors of the associated Newton method .

The opposite of an attractor is called a repellor or negative attractor . The terms are used in physics and biology .

Technical term

Example: Lorenz attractor Poisson Saturne

The set of all points of the phase space which strive towards the same attractor under the dynamics is called the attraction or catchment area of this attractor.

Dynamic systems

Dynamic systems describe changes in state as a function of time t . For the mathematical definition, the real system is often viewed in the greatly simplified form of a mathematical model . examples are

the flow behavior of liquids and gases
Movements of celestial bodies under mutual influence by gravity
Population sizes of living things taking into account the predator-prey relationship or
the development of economic parameters under the influence of market laws .

The long-term behavior of a dynamic system can be described by the global attractor, since in physical or technical systems there is often dissipation , especially friction .

A distinction is made between:

continuous dynamic systems; the change of state is defined as a continuous process ( ) ${\ displaystyle t \ in \ mathbb {R}}$
discrete dynamic systems; the change of state is defined in fixed time steps ( ). ${\ displaystyle t \ in \ mathbb {N}}$

Each state is a point in phase space and is represented by any number of state variables that form the dimensions of the phase space.

continuous systems are represented by lines ( trajectories )
discrete systems are represented by sets of isolated points.

A mixed system of continuous and discrete subsystems - with then continuously discrete dynamics - is also referred to as a hybrid dynamic system . Examples of such structurally variable dynamics can be found in process engineering (e.g. dosing systems). Hybrid dynamic systems are described mathematically by hybrid models , e.g. B. switching differential equations . The trajectories in the phase space are generally not steady (they have "kinks" and cracks ).

Attractor

When examining dynamic systems, one is particularly interested in the behavior of - starting from a certain initial state . The limit in this case is called the attractor. Typical and common examples of attractors are: ${\ displaystyle t \ to \ infty}$

asymptotically stable fixed points : the system approaches more and more a certain final state in which the dynamics succumbs; a static system is created. A typical example is a damped pendulum that approaches the state of rest at the lowest point.

(Asymptotically) stable limit cycles : The final state is the sequence of identical states that are passed through periodically ( periodic orbits ). An example of this is the simulation of the predator-prey relationship , which amounts to a periodic increase and decrease in population sizes for certain parameters of the feedback .

For a hybrid dynamic system with chaotic dynamics, the surface of an n- simplex could be identified as an attractor. ${\ displaystyle \ mathbb {R} ^ {n}}$

(Asymptotically stable) boundary tori : If several mutually incommensurable frequencies occur, the trajectory is not closed, and the attractor is a boundary torus which is asymptotically completely filled by the trajectory. The time series corresponding to this attractor is quasi-periodic, i. H. there is no real period , but the frequency spectrum is made up of sharp lines.

These examples are attractors that have an integral dimension in phase space.

The existence of attractors with a more complicated structure had been known for a long time, but they were initially viewed as unstable special cases, the occurrence of which can only be observed with a specific selection of the initial state and the system parameters. This changed with the definition of a new, special type of attractor:

Strange attractor : In its final state, the system often shows a chaotic behavior (there are, however, exceptions, e.g. quasi-periodically driven non-linear systems ). The strange attractor cannot be described in a closed geometric form and has no integral dimension. Attractors of nonlinear dynamic systems then have a fractal structure. An important characteristic is the chaotic behavior, i. H. every change in the initial state, no matter how small, leads to significant state changes in the further course. The most prominent example is the Lorenz attractor , which was discovered while modeling air currents in the atmosphere.

Mathematical definition

Formally, consider a dynamic system consisting of a topological space and a transformation , where a linearly ordered monoid is like or and is normally continuous or at least measurable (or at least it is required that it is continuous / measurable for each ) and fulfilled ${\ displaystyle X}$ ${\ displaystyle f: {\ mathcal {T}} \ times X \ longrightarrow X}$ ${\ displaystyle {\ mathcal {T}}}$ ${\ displaystyle {\ mathcal {T}} = \ mathbb {N}, \ mathbb {Z}, [0, \ infty [}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle f}$ ${\ displaystyle f (t, \ cdot): X \ longrightarrow X}$ ${\ displaystyle t \ in {\ mathcal {T}}}$

{\ displaystyle f (t + s, x) = f (t, f (s, x))}

for all "times" and points .

{\ displaystyle t, s \ in {\ mathcal {T}}}

{\ displaystyle x \ in X}

Definition 1. A subset is called forward invariant if ${\ displaystyle A \ subseteq X}$

{\ displaystyle \ forall {t \ geq 0:} ~ f (t, A) \ subseteq A.}

In other words, once a point enters an attractor, it does not escape the subset.

Definition 2. The collection basin of a subset means the amount ${\ displaystyle A \ subseteq X}$

{\ displaystyle B (A): = \ {x \ in X \ mid \ forall {U \ in {\ mathcal {N}} (A): ~} \ exists {t_ {0} \ geq 0: ~} \ forall {t \ geq t_ {0}: ~} f (t, x) \ in U \},}

where is the set of neighborhoods of . In words, a point is,, in if and only if for all environments from this point from a point in time it is always in this environment. ${\ displaystyle {\ mathcal {N}} (A)}$ ${\ displaystyle A}$ ${\ displaystyle x \ in X}$ ${\ displaystyle B (A)}$ ${\ displaystyle U}$ ${\ displaystyle A}$

Comment. In the case of a compact metric space , this definition is equivalent to ${\ displaystyle (X, d)}$

{\ displaystyle B (A): = \ {x \ in X \ mid \ lim _ {t \ to \ infty} d (f (t, x), A) = 0 \}.}

Comment. Assume that the space can be metrised and is compact. The definition of a reservoir shows that forward is invariant and . Some authors define the reservoir as the (open) set with these two properties. ${\ displaystyle A}$ ${\ displaystyle B (A)}$ ${\ displaystyle \ bigcap \ {f (t, B (A)) \ mid t> 0 \} = {\ overline {A}}}$

Definition 3. An attractor is understood to be a subset that satisfies the following conditions ${\ displaystyle A \ subseteq X}$

1. is forward invariant;

{\ displaystyle A}

2. The reservoir is an environment of ;

{\ displaystyle B (A)}

{\ displaystyle A}

3. is a minimal non-empty subset of with conditions 1 and 2.

{\ displaystyle A}

{\ displaystyle X}

Comment. Condition 1 requires a certain stability of the attractor. Obviously it follows that . On the basis of condition 2 it is further required that and means u. a., every point in a certain vicinity of approaches the attractor arbitrarily. Some authors leave out Condition 2. Condition 3 requires that the attractor cannot be broken down into further components (otherwise, for example, the whole space would trivially be an attractor). ${\ displaystyle A \ subseteq B (A)}$ ${\ displaystyle A \ subseteq B (A) ^ {\ circ}}$ ${\ displaystyle A}$

swell

^ T. Schürmann and I. Hoffmann: The entropy of strange billiards inside n-simplexes. In: J. Phys. Volume A28, 1995, pp. 5033ff. arxiv : nlin / 0208048
↑ Milnor, J. (1985). "On the Concept of Attractor." Comm. Math. Phys 99: 177-195.

literature

G. Jetschke: Mathematics of Self-Organization . Harri-Deutsch-Verlag, Frankfurt / Main, 1989
T. Schürmann and I. Hoffmann: The entropy of strange billiards inside n-simplexes. In: J. Phys. Volume A28, 1995, pp. 5033ff. arxiv : nlin / 0208048

Web links

Commons : Attractors - collection of pictures, videos and audio files

Chaoscope attractor program (freeware)

[1] T. Schürmann and I. Hoffmann: The entropy of strange billiards inside n-simplexes. In: J. Phys. Volume A28, 1995, pp. 5033ff. arxiv : nlin / 0208048

[2] Milnor, J. (1985). "On the Concept of Attractor." Comm. Math. Phys 99: 177-195.