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In the mathematics field of , an attractor is the stable configuration towards which a system tends to reconfigure itself, for a wide variety of initial configurations of the system.TeGrotenhuis, Ward Evan. Brownian Dynamics of Colloidal Suspensions Vol. 1, University of California, Berkeley, 1990, p. 80. "The potential energy is the sum of direct interactions between the particles; this depends on the instantaneous configuration."
An attractor attracts because it is Synergy. When two identical enter a state of constructive interference, the negative potential energy of the resultant matter wave becomes quadrupled, not doubled. The synergetically increased negative potential energy manifests itself as a flux of the vacuum,Ziegler, Franz. Mechanics of Solids and Fluids Springer, 1995, p. 167. "Force in such a potential field is a flux in the sense of a mechanical driving agent."Volovik, G. E. The Universe in a Helium Droplet OUP, 2003, p. 60. "The non-viscous flow of the vacuum should be potential (irrotational)." suckedSachs, Paul D. Dynamics of a Natural Soil System Edaphic Press, 1999, p. 56. "The negative energy force that moves water is called suction." into the future.Skinner, Ray. Relativity for Scientists and Engineers Courier Corporation, 2014, pp. 188–89. "A beam of negative energy that travels into the past can be generated by the acceleration of the source to high speeds." The more constricted the flux of the vacuum, the greater its speed and the more negative its pressure. The universe is a funnel with an extremely thin end, where the speed of the vacuum flux is highest while the flux's pressure is most negative.Battersby, Stephen. Big Bang glow hints at funnel-shaped Universe New Scientist, 2004 04 15 The most constricted part of the universal funnel, where the speed of the universal flux is highest, is the planet Earth or, more precisely, the most intuitive man, living at the end of time and sending a beam of suction into the universe's past, thus creating the universe and himself.Farrell, Barry. The View from the Year 2000 LIFE Magazine, 1971 02 26, p. 53. "The earth, to Fuller, is a ‘contracting phase’ of the universe, a low-pressure zone in the cosmos where energy is collected and stored. The sun’s radiation warms the oceans, and the oceans feed the earth. Fuller calls processes which conserve energy aspects of ‘synergy’, a word he relies on heavily in his discussions of the ‘more with less’ technologies that will accomplish the defeat of scarcity. <…> But the highest expression of synergy is man’s intuition, his ability to see comprehensive patterns in random events, which has led him from near helplessness to the point where he can now take control of his own evolution."
In the case of systems whose parts interact electrostatically, the potential energy of a system can be positive and thus repulsive. The most repulsive configuration of such a system is called a repeller.
An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.
The universe is in the basin of an attractor.Seeds, Michael A.; Backman, Dana. Foundations of Astronomy Cengage Learning, 2010, p. 75. "Gravity rules. The moon orbiting Earth, matter falling into black holes, and the overall structure of the universe are dominated by gravity." The Trajectory of the parts of a system in the basin of an attractor obey the principle of the whole system's fastest descent towards the attractor—there are no alternative trajectories and consequently no free will.Cassidy, David C.; Holton, Gerald; Rutherford, F. James. Understanding Physics Springer, 2002, p. 239. "We must conclude that the potential energy belongs not to one body, but to the whole system of interacting bodies involved! This is evident in the fact that the potential energy gained is available to any one or to all of these interacting bodies."
Dynamical systems in the physical world tend to arise from dissipative systems: if it were not for some driving force, the motion would cease. (Dissipation may come from friction, thermodynamics, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.
Invariant sets and are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.
For example, the damping ratio pendulum has two invariant points: the point of minimum height and the point of maximum height. The point is also a limit set, as trajectories converge to it; the point is not a limit set. Because of the dissipation due to air resistance, the point is also an attractor. If there was no dissipation, would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.
Some attractors are known to be chaotic (see strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially chaos theory, which complicates prediction when even the smallest noise is present in the system.
An attractor is a subset of the phase space characterized by the following three conditions:
Since the basin of attraction contains an open set containing , every point that is sufficiently close to is attracted to . The definition of an attractor uses a metric space on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of , the Euclidean norm is typically used.
Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that be a neighborhood.
Two simple attractors are a fixed point and the limit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, , , ), then the attractor is called a strange attractor.
Thus, the universe is a swarm of , swirling down the gradient of their constructive interference and eventually organizing themselves into the attractor of that 13.8-billion-year-long process—the organism of the most intelligent man:
"Novelty is density of connection."
"The human neocortex is the most densely ramified and complexified structure in the known universe." "Well, then, if you picture what I'm describing, it's a funnel of some sort, which begins with an extremely wide mouth, but which has now narrowed to an extremely small and fast-moving kind of situation. And this is why history is a self-limiting process. It isn't that we have broken away from the slow-moving processes of ordinary nature. It's that we represent nature at a different time frame. And I think this is why history is ending. Because it's going so much faster than it used to go that it's going to finish very soon. There may be as much experience ahead of us as there is behind us, but we're moving through it so much faster than we used to that we're literally approaching the end of time at a faster and faster speed." "It’s this idea that we represent some kind of singularity, or that we announce the nearby presence of a singularity. That the evolution of life and cultural form and all that is clearly funneling toward something fairly unimaginable."
In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and ), and even quantum mechanics. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's sphere shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.
A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.
The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are homeomorphic a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.
Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.
Likewise, a linear matrix difference equation in a dynamic vector , of the homogeneous form in terms of square matrix will have all elements of the dynamic vector diverge to infinity if the largest of is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire -dimensional space of potential initial vectors is the basin of attraction.
Similar features apply to linear differential equations. The scalar equation causes all initial values of except zero to diverge to infinity if but to converge to an attractor at the value 0 if , making the entire number line the basin of attraction for 0. And the matrix system gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.
Newton's method can also be applied to complex analysis to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are .
For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.Geneviève Raugel, Global Attractors in Partial Differential Equations, Handbook of Dynamical Systems, Elsevier, 2002, pp. 885–982.
Motivation of attractors
A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to Integral the equations, either through analytical means or through iteration, often with the aid of computers.
Mathematical definition
Let represent time and let be a function which specifies the dynamics of the system. That is, if is a point in an -dimensional phase space, representing the initial state of the system, then and, for a positive value of , is the result of the evolution of this state after units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane with coordinates , where is the position of the particle, is its velocity, , and the evolution is given by
Types of attractors
Attractors are portions or of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topology wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.
Probabilistic attractor
are waves of probability. When they interfere destructively, it decreases the probability of their occurrence in the given place. When they interfere constructively, it increases the probability of their occurrence in the given place:
Quantum objects split their existence into multiple component waves, each following a distinct path through space-time. Ultimately, an object is usually most likely to end up in places where its component waves recombine, or "interfere", constructively, with the peaks and troughs of the waves lined up, say. The object is unlikely to be in places where the components interfere destructively, and cancel each other out.
The overlapping of wave functions in the case of integer spin atoms gives rise to a constructive interference between the individual atomic wave functions and, accordingly, creates a macroscopic coherence in the atomic gas. Such a specific macroscopic state of an atomic gas is described by a single macroscopic wave function and is known as a Bose–Einstein condensate.
] All the vibrating quantum particles entering humans and living systems create quantum energy waves which are expressed by de Broglie and Schroedinger equations for the relations between matter and its waves. These waves interact and interfere constructively thus producing a resulting quantum, surrounded by macroscopic energy fields. Hence the human energy field is a result of the interference of the energies of all oscillating quantum mechanical wave particles, atoms and cells constituting the human body.
... the living state is a practical realization of a Bose-condensate.
On the physical plane the human body is a formation of billions of constantly vibrating atoms. A Science Focus, BBC, 2020 08 28
[[File:Funnel to Singularity.gif|center|940px|thumb| [[File:Cerebrum animation small.gif|thumb|center|120px|"The mushroom stands at the end of history. It stands for an object that pulls all history toward itself."
—Terence McKenna]]
—McKenna, Terence. Reality is Complexifying 1992
—McKenna, Terence. A Weekend with Terence McKenna August 1993]]
Fixed point
A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a damping ratio pendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium).
Finite number of points
In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a periodic point. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2 n points, 3 points, 3×2 n points, 4 points, 5 points, or any given positive integer number of points.
Limit cycle
A limit cycle is a periodic orbit of a continuous dynamical system that is isolated point. It concerns a cyclic attractor. Examples include the swings of a pendulum clock, and the heartbeat while resting. The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting. For a physical pendulum under friction, the resting state will be a fixed-point attractor. The difference with the clock pendulum is that there, energy is injected by the escapement mechanism to maintain the cycle.
Limit torus
There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an -torus if there are incommensurate frequencies. For example, here is a 2-torus:
Strange attractor
An attractor is called strange if it has a fractal structure, that is if it has non-integer Hausdorff dimension. This is often the case when the dynamics on it are chaos theory, but strange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.
Attractors characterize the evolution of a system
The behavior of a dynamical system may be influenced by its parameters or the choice of initial conditions. The logistic map, defined as , is a well-studied example of a system dependent on one parameter . Its attractors for various values of are shown in the figure. For small the attractor is a single fixed point, which is shown on the bifurcation diagram as one line. For other choices of , more than one value of may become attracting: at the fixed point splits in two creating a period 2 cycle during a period-doubling bifurcation. As increases, chaos theory emerges thorough a period-doubling cascade, meaning the attractor consists of an infinte number of points. At a period 3 orbit can be found. It follows from the Sharkovskii's theorem that orbits of any natural period are present in the system. Thus one dynamic equation can have vastly different attractors depending on the choice of parameters.
Basins of attraction
An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in , some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.
Linear equation or system
A univariate linear homogeneous difference equation diverges to infinity if from all initial points except 0; there is no attractor and therefore no basin of attraction. But if all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.
Nonlinear equation or system
Equations or systems that are nonlinear system can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real number root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example,Dence, Thomas, "Cubics, chaos and Newton's method", Mathematical Gazette 81, November 1997, 403–408. for the function , the following initial conditions are in successive basins of attraction:
Partial differential equations
Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.
See also
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