{"history":[],"article":{"id":"42f67754-fa67-46b7-8238-dfad3b5bb484","title":"Attractor","slug":"attractor","content":":::summary\n:::\n\n# \nIn the [mathematical](https://bliptext.com/articles/mathematics) field of [dynamical systems](https://bliptext.com/articles/dynamical-system), an **attractor** is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system.\nSystem values that get close enough to the attractor values remain close even if slightly disturbed.\nIn finite-dimensional systems, the evolving variable may be represented [algebraically](https://bliptext.com/articles/algebra) as an n-dimensional [vector](https://bliptext.com/articles/coordinate-vector).\nThe attractor is a region in n-dimensional space.\nIn [physical systems](https://bliptext.com/articles/physics), the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in [economic systems](https://bliptext.com/articles/economics), they may be separate variables such as the [inflation rate](https://bliptext.com/articles/inflation-rate) and the [unemployment rate](https://bliptext.com/articles/unemployment-rate).\nIf the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented [geometrically](https://bliptext.com/articles/geometry) in two or three dimensions, (as for example in the three-dimensional case depicted to the right).\nAn attractor can be a [point](https://bliptext.com/articles/point-geometry), a finite set of [point](https://bliptext.com/articles/point-geometry)s, a [curve](https://bliptext.com/articles/curve), a [manifold](https://bliptext.com/articles/manifold), or even a complicated set with a [fractal](https://bliptext.com/articles/fractal) structure known as a [strange attractor](https://bliptext.com/articles/) (see [strange attractor](https://bliptext.com/articles/) below).\nIf the variable is a [scalar](https://bliptext.com/articles/scalar-mathematics), the attractor is a subset of the real number line.\nDescribing the attractors of chaotic dynamical systems has been one of the achievements of [chaos theory](https://bliptext.com/articles/chaos-theory).\nA [trajectory](https://bliptext.com/articles/trajectory) of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time.\nThe trajectory may be [periodic](https://bliptext.com/articles/periodic-function) or [chaotic](https://bliptext.com/articles/chaos-theory).\nIf a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).\n# [Visual representation of a strange attractor. Another visualization of the same 3D attractor is\nthis video. Code capable of rendering this is available. | upload.wikimedia.org/wikipedia/commons/7/70/Poisson///saturne///revisited.jpg]\n\n# Motivation of attractors\nA [dynamical system](https://bliptext.com/articles/dynamical-system) is generally described by one or more [differential](https://bliptext.com/articles/differential-equations) or [difference equations](https://bliptext.com/articles/difference-equations).\nThe equations of a given dynamical system specify its behavior over any given short period of time.\nTo determine the system's behavior for a longer period, it is often necessary to [integrate](https://bliptext.com/articles/integral) the equations, either through analytical means or through [iteration](https://bliptext.com/articles/iteration), often with the aid of computers.\nDynamical systems in the physical world tend to arise from [dissipative systems](https://bliptext.com/articles/dissipative-system): if it were not for some driving force, the motion would cease.\n(Dissipation may come from [internal friction](https://bliptext.com/articles/friction), [thermodynamic losses](https://bliptext.com/articles/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.\nThe subset of the [phase space](https://bliptext.com/articles/phase-space) of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.\nInvariant sets and [limit sets](https://bliptext.com/articles/limit-set) are similar to the attractor concept.\nAn invariant set is a set that evolves to itself under the dynamics.\nAttractors may contain invariant sets.\nA 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.\nAttractors 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.\nFor example, the [damped](https://bliptext.com/articles/damping-ratio) [pendulum](https://bliptext.com/articles/pendulum) has two invariant points: the point\nx0\nof minimum height and the point\nx1\nof maximum height.\nThe point\nx0\nis also a limit set, as trajectories converge to it; the point\nx1\nis not a limit set.\nBecause of the dissipation due to air resistance, the point\nx0\nis also an attractor.\nIf there was no dissipation,\nx0\nwould not be an attractor.\nAristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.\nSome attractors are known to be chaotic (see [strange attractor](https://bliptext.com/articles/)), in which case the evolution of any two distinct points of the attractor result in exponentially [diverging trajectories](https://bliptext.com/articles/chaos-theory), which complicates prediction when even the smallest noise is present in the system.\n\n# Mathematical definition\nLet t represent time and let f(t,\\cdot) be a function which specifies the dynamics of the system.\nThat is, if a is a point in an n-dimensional phase space, representing the initial state of the system, then f(0,a)=a and, for a positive value of t, f(t,a) is the result of the evolution of this state after t units of time.\nFor example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane \\R^2 with coordinates (x,v), where x is the position of the particle, v is its velocity, a=(x,v), and the evolution is given by\nAn attractor is a [subset](https://bliptext.com/articles/subset) A of the [phase space](https://bliptext.com/articles/phase-space) characterized by the following three conditions:\nSince the basin of attraction contains an [open set](https://bliptext.com/articles/open-set) containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a [metric](https://bliptext.com/articles/metric-space) on the phase space, but the resulting notion usually depends only on the topology of the phase space.\nIn the case of \\R^n, the Euclidean norm is typically used.\nMany other definitions of attractor occur in the literature.\nFor example, some authors require that an attractor have positive [measure](https://bliptext.com/articles/measure-mathematics) (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood.\n# [Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of the Julia set, which iterates the function f(z) = z2 + c. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points. | upload.wikimedia.org/wikipedia/commons/2/25/Julia///immediate///basin///1///3.png]\n\n# Types of attractors\nAttractors are portions or [subsets](https://bliptext.com/articles/subset) of the [phase space](https://bliptext.com/articles/configuration-space-physics) of a [dynamical system](https://bliptext.com/articles/dynamical-system).\nUntil the 1960s, attractors were thought of as being [simple geometric subsets](https://bliptext.com/articles/geometric-primitive) of the phase space, like [points](https://bliptext.com/articles/point-geometry), [lines](https://bliptext.com/articles/line-mathematics), [surfaces](https://bliptext.com/articles/surface-topology), and simple regions of [three-dimensional space](https://bliptext.com/articles/three-dimensional-space).\nMore complex attractors that cannot be categorized as simple geometric subsets, such as [topologically](https://bliptext.com/articles/topology) wild sets, were known of at the time but were thought to be fragile anomalies.\n[Stephen Smale](https://bliptext.com/articles/stephen-smale) was able to show that his [horseshoe map](https://bliptext.com/articles/horseshoe-map) was [robust](https://bliptext.com/articles/structural-stability) and that its attractor had the structure of a [Cantor set](https://bliptext.com/articles/cantor-set).\nTwo simple attractors are a [fixed point](https://bliptext.com/articles/fixed-point-mathematics) and the [limit cycle](https://bliptext.com/articles/limit-cycle).\nAttractors can take on many other geometric shapes (phase space subsets).\nBut when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. [intersection](https://bliptext.com/articles/intersection-set-theory) and [union](https://bliptext.com/articles/union-set-theory)) of [fundamental geometric objects](https://bliptext.com/articles/geometric-primitive) (e.g. [lines](https://bliptext.com/articles/line-mathematics), [surfaces](https://bliptext.com/articles/surface-topology), [spheres](https://bliptext.com/articles/sphere), [toroids](https://bliptext.com/articles/toroid), [manifolds](https://bliptext.com/articles/manifold)), then the attractor is called a [strange attractor](https://bliptext.com/articles/).\n\n## Fixed point\nA [fixed point](https://bliptext.com/articles/fixed-point-mathematics) of a function or transformation is a point that is mapped to itself by the function or transformation.\nIf 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.\nThe 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 [damped](https://bliptext.com/articles/damping-ratio) [pendulum](https://bliptext.com/articles/pendulum), the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble.\nBut the fixed point(s) of a dynamic system is not necessarily an attractor of the system.\nFor 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.\nThis is equivalent to the difference between [stable and unstable equilibria](https://bliptext.com/articles/stability-theory).\nIn 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).\nIn 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](https://bliptext.com/articles/nonlinear-dynamics) of [stiction](https://bliptext.com/articles/stiction), [friction](https://bliptext.com/articles/friction), [surface roughness](https://bliptext.com/articles/surface-roughness), [deformation](https://bliptext.com/articles/deformation-engineering) (both [elastic](https://bliptext.com/articles/elastic-deformation) and [plasticity](https://bliptext.com/articles/plastic)), and even [quantum mechanics](https://bliptext.com/articles/quantum-mechanics).\nIn the case of a marble on top of an inverted bowl, even if the bowl seems perfectly [hemispherical](https://bliptext.com/articles/sphere), and the marble's [spherical](https://bliptext.com/articles/sphere) shape, are both much more complex surfaces when examined under a microscope, and their [shapes change](https://bliptext.com/articles/contact-mechanics) or [deform](https://bliptext.com/articles/deformation-physics) during contact.\nAny physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains.\nThere 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](https://bliptext.com/articles/critical-point-mathematics) or fixed points, some of which are categorized as attractors.\n# [Weakly attracting fixed point for a complex number evolving according to a complex quadratic polynomial. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor. | upload.wikimedia.org/wikipedia/commons/0/07/Critical///orbit///3d.png]\n\n## Finite number of points\nIn a [discrete-time](https://bliptext.com/articles/discrete-time-and-continuous-time) system, an attractor can take the form of a finite number of points that are visited in sequence.\nEach of these points is called a [periodic point](https://bliptext.com/articles/periodic-point).\nThis is illustrated by the [logistic map](https://bliptext.com/articles/logistic-map), which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2n points, 3 points, 3×2n points, 4 points, 5 points, or any given positive integer number of points.\n\n## Limit cycle\nA [limit cycle](https://bliptext.com/articles/limit-cycle) is a periodic orbit of a continuous dynamical system that is [isolated](https://bliptext.com/articles/isolated-point).\nIt concerns a [cyclic attractor](https://bliptext.com/articles/attractor-network).\nExamples include the swings of a [pendulum clock](https://bliptext.com/articles/pendulum-clock), and the heartbeat while resting.\nThe 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.\nFor a physical pendulum under friction, the resting state will be a fixed-point attractor.\nThe difference with the clock pendulum is that there, energy is injected by the [escapement](https://bliptext.com/articles/escapement) mechanism to maintain the cycle.\n# [Van der Pol phase portrait: an attracting limit cycle | upload.wikimedia.org/wikipedia/commons/1/1b/VanDerPolPhaseSpace.png]\n\n## Limit torus\nThere may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle.\nFor 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.\nIf two of these frequencies form an [irrational fraction](https://bliptext.com/articles/irrational-number) (i.e. they are [incommensurate](https://bliptext.com/articles/commensurability-mathematics)), the trajectory is no longer closed, and the limit cycle becomes a limit [torus](https://bliptext.com/articles/torus).\nThis kind of attractor is called an\nNt\n-torus if there are\nNt\nincommensurate frequencies.\nFor example, here is a 2-torus:\nA time series corresponding to this attractor is a [quasiperiodic](https://bliptext.com/articles/quasiperiodic) series: A discretely sampled sum of\nNt\nperiodic functions (not necessarily [sine](https://bliptext.com/articles/sine) waves) with incommensurate frequencies.\nSuch a time series does not have a strict periodicity, but its [power spectrum](https://bliptext.com/articles/power-spectrum) still consists only of sharp lines.\n# [300px | upload.wikimedia.org/wikipedia/commons/b/b8/torus.png]\n\n## Strange attractor\nAn attractor is called strange if it has a [fractal](https://bliptext.com/articles/fractal) structure, that is if it has non-integer [Hausdorff dimension](https://bliptext.com/articles/hausdorff-dimension).\nThis is often the case when the dynamics on it are [chaotic](https://bliptext.com/articles/chaos-theory), but strange non[chaotic](https://bliptext.com/articles/chaos-theory) attractors also exist.\nIf a strange attractor is chaotic, exhibiting [sensitive dependence on initial conditions](https://bliptext.com/articles/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.\nThus 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.\nThe term strange attractor was coined by [David Ruelle](https://bliptext.com/articles/david-ruelle) and [Floris Takens](https://bliptext.com/articles/floris-takens) to describe the attractor resulting from a series of [bifurcations](https://bliptext.com/articles/bifurcation-theory) of a system describing fluid flow.\nStrange attractors are often [differentiable](https://bliptext.com/articles/differentiable-function) in a few directions, but some are [like](https://bliptext.com/articles/homeomorphic) a [Cantor dust](https://bliptext.com/articles/cantor-dust), and therefore not [differentiable](https://bliptext.com/articles/differentiable-function).\nStrange 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.\nExamples of strange attractors include the [double-scroll attractor](https://bliptext.com/articles/double-scroll-attractor), [Hénon attractor](https://bliptext.com/articles/h-non-map), [Rössler attractor](https://bliptext.com/articles/r-ssler-attractor), and [Lorenz attractor](https://bliptext.com/articles/lorenz-attractor).\n# [A plot of Lorenz's strange attractor for values ρ = 28, σ = 10, β = 8/3 | upload.wikimedia.org/wikipedia/commons/5/5b/Lorenz///attractor///yb.svg]\n\n# Attractors characterize the evolution of a system\nThe parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.\nAn example is the well-studied [logistic map](https://bliptext.com/articles/logistic-map),, whose basins of attraction for various values of the parameter r are shown in the figure.\nIf r=2.6, all starting x values of x<0 will rapidly lead to function values that go to negative infinity; starting x values of x>1 will also go to negative infinity.\nBut for 03.6, interspersed with regions of simpler behaviour (white stripes). | upload.wikimedia.org/wikipedia/commons/c/c8/Logistic///Map///Bifurcation///Diagram%2C///Matplotlib.svg]\n\n# Basins of attraction\nAn attractor's basin of attraction is the region of the [phase space](https://bliptext.com/articles/phase-space), over which iterations are defined, such that any point (any [initial condition](https://bliptext.com/articles/initial-condition)) in that region will [asymptotically](https://bliptext.com/articles/asymptotic-behavior) be iterated into the attractor.\nFor a [stable](https://bliptext.com/articles/stability-mathematics) [linear system](https://bliptext.com/articles/linear-system), every point in the phase space is in the basin of attraction.\nHowever, in [nonlinear systems](https://bliptext.com/articles/nonlinear-system), 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.\n\n## Linear equation or system\nAn univariate linear homogeneous difference equation diverges to infinity if |a|>1 from all initial points except 0; there is no attractor and therefore no basin of attraction.\nBut if |a|<1 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.\nLikewise, a linear [matrix difference equation](https://bliptext.com/articles/matrix-difference-equation) in a dynamic [vector](https://bliptext.com/articles/coordinate-vector) X, of the homogeneous form in terms of [square matrix](https://bliptext.com/articles/square-matrix) A will have all elements of the dynamic [vector](https://bliptext.com/articles/coordinate-vector) diverge to infinity if the largest [eigenvalues](https://bliptext.com/articles/eigenvalue) of A is greater than 1 in absolute value; there is no attractor and no basin of attraction.\nBut 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 n-dimensional space of potential initial vectors is the basin of attraction.\nSimilar features apply to linear [differential equations](https://bliptext.com/articles/differential-equation).\nThe scalar equation dx/dt =ax causes all initial values of x except zero to diverge to infinity if a>0 but to converge to an attractor at the value 0 if a<0, making the entire number line the basin of attraction for 0.\nAnd the matrix system dX/dt=AX gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A 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.\n\n## Nonlinear equation or system\nEquations or systems that are [nonlinear](https://bliptext.com/articles/nonlinear-system) can give rise to a richer variety of behavior than can linear systems.\nOne example is [Newton's method](https://bliptext.com/articles/newton-s-method) of iterating to a root of a nonlinear expression.\nIf the expression has more than one [real](https://bliptext.com/articles/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.\nThe 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.\nThe basins of attraction can be infinite in number and arbitrarily small.\nFor example, for the function, the following initial conditions are in successive basins of attraction:\nNewton's method can also be applied to [complex functions](https://bliptext.com/articles/complex-analysis) to find their roots.\nEach root has a basin of attraction in the [complex plane](https://bliptext.com/articles/complex-plane); these basins can be mapped as in the image shown.\nAs can be seen, the combined basin of attraction for a particular root can have many disconnected regions.\nFor many complex functions, the boundaries of the basins of attraction are [fractals](https://bliptext.com/articles/fractal).\n# [A Newton fractal showing basins of attraction in the complex plane for using Newton's method to solve x5 − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge. | upload.wikimedia.org/wikipedia/commons/1/10/newtroot///1///0///0///0///0///m1.png]\n\n# Partial differential equations\n[Parabolic partial differential equations](https://bliptext.com/articles/parabolic-partial-differential-equation) may have finite-dimensional attractors.\nThe diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor.\nThe Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced [Navier–Stokes equations](https://bliptext.com/articles/navier-stokes-equation) are all known to have global attractors of finite dimension.\nFor the three-dimensional, incompressible Navier–Stokes equation with periodic [boundary conditions](https://bliptext.com/articles/boundary-condition), if it has a global attractor, then this attractor will be of finite dimensions.\n\n# [Chaotic hidden attractor (green domain) in Chua's system.\nTrajectories with initial data in a neighborhood of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange). | upload.wikimedia.org/wikipedia/commons/7/7e/Chua-chaotic-hidden-attractor.jpg]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:36:41.977Z","updated_at":"2025-01-25T18:36:41.977Z","search_vector":"'attractor':1","search_content":" \n \n\n# \nIn the [mathematical](https://bliptext.com/articles/mathematics) field of [dynamical systems](https://bliptext.com/articles/dynamical-system), an **attractor** is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system.\nSystem values that get close enough to the attractor values remain close even if slightly disturbed.\nIn finite-dimensional systems, the evolving variable may be represented [algebraically](https://bliptext.com/articles/algebra) as an n-dimensional [vector](https://bliptext.com/articles/coordinate-vector).\nThe attractor is a region in n-dimensional space.\nIn [physical systems](https://bliptext.com/articles/physics), the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in [economic systems](https://bliptext.com/articles/economics), they may be separate variables such as the [inflation rate](https://bliptext.com/articles/inflation-rate) and the [unemployment rate](https://bliptext.com/articles/unemployment-rate).\nIf the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented [geometrically](https://bliptext.com/articles/geometry) in two or three dimensions, (as for example in the three-dimensional case depicted to the right).\nAn attractor can be a [point](https://bliptext.com/articles/point-geometry), a finite set of [point](https://bliptext.com/articles/point-geometry)s, a [curve](https://bliptext.com/articles/curve), a [manifold](https://bliptext.com/articles/manifold), or even a complicated set with a [fractal](https://bliptext.com/articles/fractal) structure known as a [strange attractor](https://bliptext.com/articles/) (see [strange attractor](https://bliptext.com/articles/) below).\nIf the variable is a [scalar](https://bliptext.com/articles/scalar-mathematics), the attractor is a subset of the real number line.\nDescribing the attractors of chaotic dynamical systems has been one of the achievements of [chaos theory](https://bliptext.com/articles/chaos-theory).\nA [trajectory](https://bliptext.com/articles/trajectory) of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time.\nThe trajectory may be [periodic](https://bliptext.com/articles/periodic-function) or [chaotic](https://bliptext.com/articles/chaos-theory).\nIf a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).\n# [Visual representation of a strange attractor. Another visualization of the same 3D attractor is\nthis video. Code capable of rendering this is available. | upload.wikimedia.org/wikipedia/commons/7/70/Poisson///saturne///revisited.jpg]\n\n# Motivation of attractors\nA [dynamical system](https://bliptext.com/articles/dynamical-system) is generally described by one or more [differential](https://bliptext.com/articles/differential-equations) or [difference equations](https://bliptext.com/articles/difference-equations).\nThe equations of a given dynamical system specify its behavior over any given short period of time.\nTo determine the system's behavior for a longer period, it is often necessary to [integrate](https://bliptext.com/articles/integral) the equations, either through analytical means or through [iteration](https://bliptext.com/articles/iteration), often with the aid of computers.\nDynamical systems in the physical world tend to arise from [dissipative systems](https://bliptext.com/articles/dissipative-system): if it were not for some driving force, the motion would cease.\n(Dissipation may come from [internal friction](https://bliptext.com/articles/friction), [thermodynamic losses](https://bliptext.com/articles/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.\nThe subset of the [phase space](https://bliptext.com/articles/phase-space) of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.\nInvariant sets and [limit sets](https://bliptext.com/articles/limit-set) are similar to the attractor concept.\nAn invariant set is a set that evolves to itself under the dynamics.\nAttractors may contain invariant sets.\nA 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.\nAttractors 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.\nFor example, the [damped](https://bliptext.com/articles/damping-ratio) [pendulum](https://bliptext.com/articles/pendulum) has two invariant points: the point\nx0\nof minimum height and the point\nx1\nof maximum height.\nThe point\nx0\nis also a limit set, as trajectories converge to it; the point\nx1\nis not a limit set.\nBecause of the dissipation due to air resistance, the point\nx0\nis also an attractor.\nIf there was no dissipation,\nx0\nwould not be an attractor.\nAristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.\nSome attractors are known to be chaotic (see [strange attractor](https://bliptext.com/articles/)), in which case the evolution of any two distinct points of the attractor result in exponentially [diverging trajectories](https://bliptext.com/articles/chaos-theory), which complicates prediction when even the smallest noise is present in the system.\n\n# Mathematical definition\nLet t represent time and let f(t,\\cdot) be a function which specifies the dynamics of the system.\nThat is, if a is a point in an n-dimensional phase space, representing the initial state of the system, then f(0,a)=a and, for a positive value of t, f(t,a) is the result of the evolution of this state after t units of time.\nFor example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane \\R^2 with coordinates (x,v), where x is the position of the particle, v is its velocity, a=(x,v), and the evolution is given by\nAn attractor is a [subset](https://bliptext.com/articles/subset) A of the [phase space](https://bliptext.com/articles/phase-space) characterized by the following three conditions:\nSince the basin of attraction contains an [open set](https://bliptext.com/articles/open-set) containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a [metric](https://bliptext.com/articles/metric-space) on the phase space, but the resulting notion usually depends only on the topology of the phase space.\nIn the case of \\R^n, the Euclidean norm is typically used.\nMany other definitions of attractor occur in the literature.\nFor example, some authors require that an attractor have positive [measure](https://bliptext.com/articles/measure-mathematics) (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood.\n# [Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of the Julia set, which iterates the function f(z) = z2 + c. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points. | upload.wikimedia.org/wikipedia/commons/2/25/Julia///immediate///basin///1///3.png]\n\n# Types of attractors\nAttractors are portions or [subsets](https://bliptext.com/articles/subset) of the [phase space](https://bliptext.com/articles/configuration-space-physics) of a [dynamical system](https://bliptext.com/articles/dynamical-system).\nUntil the 1960s, attractors were thought of as being [simple geometric subsets](https://bliptext.com/articles/geometric-primitive) of the phase space, like [points](https://bliptext.com/articles/point-geometry), [lines](https://bliptext.com/articles/line-mathematics), [surfaces](https://bliptext.com/articles/surface-topology), and simple regions of [three-dimensional space](https://bliptext.com/articles/three-dimensional-space).\nMore complex attractors that cannot be categorized as simple geometric subsets, such as [topologically](https://bliptext.com/articles/topology) wild sets, were known of at the time but were thought to be fragile anomalies.\n[Stephen Smale](https://bliptext.com/articles/stephen-smale) was able to show that his [horseshoe map](https://bliptext.com/articles/horseshoe-map) was [robust](https://bliptext.com/articles/structural-stability) and that its attractor had the structure of a [Cantor set](https://bliptext.com/articles/cantor-set).\nTwo simple attractors are a [fixed point](https://bliptext.com/articles/fixed-point-mathematics) and the [limit cycle](https://bliptext.com/articles/limit-cycle).\nAttractors can take on many other geometric shapes (phase space subsets).\nBut when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. [intersection](https://bliptext.com/articles/intersection-set-theory) and [union](https://bliptext.com/articles/union-set-theory)) of [fundamental geometric objects](https://bliptext.com/articles/geometric-primitive) (e.g. [lines](https://bliptext.com/articles/line-mathematics), [surfaces](https://bliptext.com/articles/surface-topology), [spheres](https://bliptext.com/articles/sphere), [toroids](https://bliptext.com/articles/toroid), [manifolds](https://bliptext.com/articles/manifold)), then the attractor is called a [strange attractor](https://bliptext.com/articles/).\n\n## Fixed point\nA [fixed point](https://bliptext.com/articles/fixed-point-mathematics) of a function or transformation is a point that is mapped to itself by the function or transformation.\nIf 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.\nThe 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 [damped](https://bliptext.com/articles/damping-ratio) [pendulum](https://bliptext.com/articles/pendulum), the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble.\nBut the fixed point(s) of a dynamic system is not necessarily an attractor of the system.\nFor 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.\nThis is equivalent to the difference between [stable and unstable equilibria](https://bliptext.com/articles/stability-theory).\nIn 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).\nIn 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](https://bliptext.com/articles/nonlinear-dynamics) of [stiction](https://bliptext.com/articles/stiction), [friction](https://bliptext.com/articles/friction), [surface roughness](https://bliptext.com/articles/surface-roughness), [deformation](https://bliptext.com/articles/deformation-engineering) (both [elastic](https://bliptext.com/articles/elastic-deformation) and [plasticity](https://bliptext.com/articles/plastic)), and even [quantum mechanics](https://bliptext.com/articles/quantum-mechanics).\nIn the case of a marble on top of an inverted bowl, even if the bowl seems perfectly [hemi[spherical](https://bliptext.com/articles/sphere)](https://bliptext.com/articles/sphere), and the marble's [spherical](https://bliptext.com/articles/sphere) shape, are both much more complex surfaces when examined under a microscope, and their [shapes change](https://bliptext.com/articles/contact-mechanics) or [deform](https://bliptext.com/articles/deformation-physics) during contact.\nAny physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains.\nThere 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](https://bliptext.com/articles/critical-point-mathematics) or fixed points, some of which are categorized as attractors.\n# [Weakly attracting fixed point for a complex number evolving according to a complex quadratic polynomial. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor. | upload.wikimedia.org/wikipedia/commons/0/07/Critical///orbit///3d.png]\n\n## Finite number of points\nIn a [discrete-time](https://bliptext.com/articles/discrete-time-and-continuous-time) system, an attractor can take the form of a finite number of points that are visited in sequence.\nEach of these points is called a [periodic point](https://bliptext.com/articles/periodic-point).\nThis is illustrated by the [logistic map](https://bliptext.com/articles/logistic-map), which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2n points, 3 points, 3×2n points, 4 points, 5 points, or any given positive integer number of points.\n\n## Limit cycle\nA [limit cycle](https://bliptext.com/articles/limit-cycle) is a periodic orbit of a continuous dynamical system that is [isolated](https://bliptext.com/articles/isolated-point).\nIt concerns a [cyclic attractor](https://bliptext.com/articles/attractor-network).\nExamples include the swings of a [pendulum clock](https://bliptext.com/articles/pendulum-clock), and the heartbeat while resting.\nThe 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.\nFor a physical pendulum under friction, the resting state will be a fixed-point attractor.\nThe difference with the clock pendulum is that there, energy is injected by the [escapement](https://bliptext.com/articles/escapement) mechanism to maintain the cycle.\n# [Van der Pol phase portrait: an attracting limit cycle | upload.wikimedia.org/wikipedia/commons/1/1b/VanDerPolPhaseSpace.png]\n\n## Limit torus\nThere may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle.\nFor 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.\nIf two of these frequencies form an [irrational fraction](https://bliptext.com/articles/irrational-number) (i.e. they are [incommensurate](https://bliptext.com/articles/commensurability-mathematics)), the trajectory is no longer closed, and the limit cycle becomes a limit [torus](https://bliptext.com/articles/torus).\nThis kind of attractor is called an\nNt\n-torus if there are\nNt\nincommensurate frequencies.\nFor example, here is a 2-torus:\nA time series corresponding to this attractor is a [quasiperiodic](https://bliptext.com/articles/quasiperiodic) series: A discretely sampled sum of\nNt\nperiodic functions (not necessarily [sine](https://bliptext.com/articles/sine) waves) with incommensurate frequencies.\nSuch a time series does not have a strict periodicity, but its [power spectrum](https://bliptext.com/articles/power-spectrum) still consists only of sharp lines.\n# [300px | upload.wikimedia.org/wikipedia/commons/b/b8/torus.png]\n\n## Strange attractor\nAn attractor is called strange if it has a [fractal](https://bliptext.com/articles/fractal) structure, that is if it has non-integer [Hausdorff dimension](https://bliptext.com/articles/hausdorff-dimension).\nThis is often the case when the dynamics on it are [chaotic](https://bliptext.com/articles/chaos-theory), but strange non[chaotic](https://bliptext.com/articles/chaos-theory) attractors also exist.\nIf a strange attractor is chaotic, exhibiting [sensitive dependence on initial conditions](https://bliptext.com/articles/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.\nThus 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.\nThe term strange attractor was coined by [David Ruelle](https://bliptext.com/articles/david-ruelle) and [Floris Takens](https://bliptext.com/articles/floris-takens) to describe the attractor resulting from a series of [bifurcations](https://bliptext.com/articles/bifurcation-theory) of a system describing fluid flow.\nStrange attractors are often [differentiable](https://bliptext.com/articles/differentiable-function) in a few directions, but some are [like](https://bliptext.com/articles/homeomorphic) a [Cantor dust](https://bliptext.com/articles/cantor-dust), and therefore not [differentiable](https://bliptext.com/articles/differentiable-function).\nStrange 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.\nExamples of strange attractors include the [double-scroll attractor](https://bliptext.com/articles/double-scroll-attractor), [Hénon attractor](https://bliptext.com/articles/h-non-map), [Rössler attractor](https://bliptext.com/articles/r-ssler-attractor), and [Lorenz attractor](https://bliptext.com/articles/lorenz-attractor).\n# [A plot of Lorenz's strange attractor for values ρ = 28, σ = 10, β = 8/3 | upload.wikimedia.org/wikipedia/commons/5/5b/Lorenz///attractor///yb.svg]\n\n# Attractors characterize the evolution of a system\nThe parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.\nAn example is the well-studied [logistic map](https://bliptext.com/articles/logistic-map),, whose basins of attraction for various values of the parameter r are shown in the figure.\nIf r=2.6, all starting x values of x 1 will also go to negative infinity.\nBut for 0 3.6, interspersed with regions of simpler behaviour (white stripes). | upload.wikimedia.org/wikipedia/commons/c/c8/Logistic///Map///Bifurcation///Diagram%2C///Matplotlib.svg]\n\n# Basins of attraction\nAn attractor's basin of attraction is the region of the [phase space](https://bliptext.com/articles/phase-space), over which iterations are defined, such that any point (any [initial condition](https://bliptext.com/articles/initial-condition)) in that region will [asymptotically](https://bliptext.com/articles/asymptotic-behavior) be iterated into the attractor.\nFor a [stable](https://bliptext.com/articles/stability-mathematics) [linear system](https://bliptext.com/articles/linear-system), every point in the phase space is in the basin of attraction.\nHowever, in [nonlinear systems](https://bliptext.com/articles/nonlinear-system), 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.\n\n## Linear equation or system\nAn univariate linear homogeneous difference equation diverges to infinity if |a|>1 from all initial points except 0; there is no attractor and therefore no basin of attraction.\nBut if |a| 0 but to converge to an attractor at the value 0 if a \n# [Chaotic hidden attractor (green domain) in Chua's system.\nTrajectories with initial data in a neighborhood of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange). | upload.wikimedia.org/wikipedia/commons/7/7e/Chua-chaotic-hidden-attractor.jpg]","revisionCount":0},"hasMore":false,"nextCursor":null}

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