The Lorenz attractor is an example of a non-linear dynamical system. The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly This article describes the use of the term nonlinearity in mathematics Studying this system helped give rise to Chaos theory. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake. Note The term model has a different meaning in Model theory, a branch of Mathematical logic.

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. In Mathematics, the real numbers may be described informally in several different ways In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The rule is deterministic: for a given time interval only one future state follows from the current state. In Mathematics, a deterministic system is a system in which no Randomness is involved in the development of future states of the system

## Overview

The concept of a dynamical system has its origins in Newtonian mechanics. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation, difference equation or other time scale. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the "Difference equation" redirects here It should not be confused with a Differential equation. In Mathematics, time scale calculus is a unification of the theory of Difference equations with that of Differential equations ref> Taming nature's numbers ) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. Once the system can be solved, given an initial point it is possible to determine all its future points, a collection known as a trajectory or orbit. In Mathematics, in the study of Dynamical systems an orbit is a collection of points related by the Evolution function of the dynamical system

Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems. A computer is a Machine that manipulates data according to a list of instructions. Numerical methods executed on computers have simplified the task of determining the orbits of a dynamical system.

For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:

• The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. In Mathematics, the notion of Lyapunov stability occurs in the study of Dynamical systems In simple terms if all solutions of the dynamical system that start out In Mathematics, structural stability is an aspect of Stability theory concerning whether a given function is sensitive to a small perturbation. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
• The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood. In a linear dynamical system, the variation of a state vector (an N-dimensional vector denoted \mathbf{x} equals a constant matrix(denoted \mathbf{A} In Mathematics, the Poincaré–Bendixson theorem is a statement about the long term behaviour of orbits of Continuous dynamical systems on the plane
• The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. Bifurcation theory is the mathematical study of changes in the qualitative or Topological structure of a given family For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
• The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems In Mathematics, more particularly in the fields of Dynamical systems and Geometric topology, an Anosov map on a Manifold M is a certain Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that

It was in the work of Poincaré that these dynamical systems themes developed. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician

## Basic definitions

A dynamical system is a manifold M called the phase (or state) space and a smooth evolution function Φ t that for any element of tT, the time, maps a point of the phase space back into the phase space. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.

### Examples

The evolution function Φ t is often the solution of a differential equation of motion

$\dot{x} = v(x) \,.$

The equation gives the time derivative, represented by the dot, of a trajectory x(t) on the phase space starting at some point x0. The vector field v(x) is a smooth function that at every point of the phase space M provides the velocity vector of the dynamical system at that point. (These vectors are not vectors in the phase space M, but in the tangent space TMx of the point x. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since ) Given a smooth Φ t, an autonomous vector field can be derived from it.

There is no need for higher order derivatives in the equation, nor for time dependence in v(x) because these can be eliminated by considering systems of higher dimensions. Other types of differential equations can be used to define the evolution rule:

$G(x, \dot{x}) = 0$

is an example of an equation that arises from the modeling of mechanical systems with complicated constraints.

The differential equations determining the evolution function Φ t are often ordinary differential equations: in this case the phase space M is a finite dimensional manifold. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.

## Linear dynamical systems

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. The logistic map is a Polynomial mapping of degree 2, often cited as an archetypal example of how complex chaotic behaviour can arise from very simple In Horology, a double pendulum is a system of two simple Pendulums on a common mounting which move in anti-phase In mathematics Arnold's cat map is a chaotic map from the Torus into itself named after Vladimir Arnold, who demonstrated its effects in the 1960s using In the Mathematics of Chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself In Dynamical systems theory, the baker's map is a chaotic map from the unit square into itself Piecewise linear may refer to Piecewise linear function Piecewise linear manifold A billiard is a Dynamical system in which a particle alternates between motion in a straight line and Specular reflections off of a boundary Outer Billiards is a Dynamical system based on a convex shape in the plane The Hénon map is a discrete-time Dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly In Mathematics, a circle map is a member of a family of dynamical systems on the circle first defined by Andrey Kolmogorov. RosslerStereopng|thumb|right|344px|Rössler attractor as a Stereogram with a=0 In Mathematics, a chaotic map is a map that exhibits some sort of Chaotic behavior. The Swinging Atwood's machine (SAM is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional In a linear dynamical system, the variation of a state vector (an N-dimensional vector denoted \mathbf{x} equals a constant matrix(denoted \mathbf{A} In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).

### Flows

For a flow, the vector field Φ(x) is a linear function of the position in the phase space, that is,

$\phi(x) = A x + b\,,$

with A a matrix, b a vector of numbers and x the position vector. In Mathematics, a flow formalizes in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in Engineering The solution to this system can be found by using the superposition principle (linearity). The case b ≠ 0 with A = 0 is just a straight line in the direction of b:

$\Phi^t(x_1) = x_1 + b t \,.$

When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x0 = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x0,

$\Phi^t(x_0) = e^{t A} x_0 \,.$

When b = 0, the eigenvalues of A determine the structure of the phase space. In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that

Linear vector fields and a few trajectories.

### Maps

A discrete-time, affine dynamical system has the form

$x_{n+1} = A x_n + b \,,$

with A a matrix and b a vector. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector As in the continuous case, the change of coordinates xx + (1 - A) –1b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system A nx0. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For example, if u1 is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u1, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems. In Mathematics, a chaotic map is a map that exhibits some sort of Chaotic behavior.

## Local dynamics

The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

### Rectification

A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

### Near periodic orbits

In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x0 in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to v(x0). These points are a Poincaré section S(γ, x0), of the orbit. In Mathematics, particularly in Dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection The flow now defines a map, the Poincaré map F : S → S, for points starting in S and returning to S. In Mathematics, particularly in Dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x0.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J · x + O(x²), so a change of coordinates h can only be expected to simplify F to its linear part

$h^{-1} \circ F \circ h(x) = J \cdot x \,.$

This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ1,…,λν are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form λi – ∑ (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.

### Conjugation results

The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic. In Mathematics, especially in the study of Dynamical system, a hyperbolic equilibrium point or hyperbolic fixed point is a special type of fixed point

In the hyperbolic case the Hartman-Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. In Mathematics, in the study of Dynamical systems, the Hartman-Grobman theorem or linearization theorem is an important theorem about the local behaviour The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.

The Kolmogorov-Arnold-Moser (KAM) theorem gives the behavior near an elliptic point. The Kolmogorov–Arnold–Moser theorem is a result in Dynamical systems about the persistence of quasi-periodic motions under small perturbations

## Bifurcation theory

Main article: Bifurcation theory

When the evolution map Φt (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Bifurcation theory is the mathematical study of changes in the qualitative or Topological structure of a given family In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. Small changes may produce no qualitative changes in the phase space until a special value μ0 is reached. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. In Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point x0 of a system family Fμ can be characterized by the eigenvalues of the first derivative of the system DFμ(x0) computed at the bifurcation point. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes For a map, the bifurcation will occur when there are eigenvalues of DFμ on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory. Bifurcation theory is the mathematical study of changes in the qualitative or Topological structure of a given family

Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle-Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations. In Mathematics, particularly in Dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits of a system In Mathematics, a Period doubling bifurcation in a Dynamical system is a bifurcation in which the system switches to a new behavior with twice the period

## Ergodic systems

Main article: ergodic theory

In many dynamical systems it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φ t(A) and invariance of the phase space means that

$\mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ) \,.$

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. The volume is said to be computed by the Liouville measure. In Physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian

In a Hamiltonian system not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. In Mathematics, the Poincaré recurrence theorem states that certain systems will after a sufficiently long time return to a state very close to the initial state Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms. Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. In Physics and Thermodynamics, the ergodic hypothesis says that over long periods of time the time spent by a particle in some region of the Phase space The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics Koopman approached the study of ergodic systems by the use of functional analysis. Bernard Osgood Koopman (1900&ndash1981 was a French-born American mathematician known for his work in ergodic theory the foundations of probability statistical theory and Operations For functional analysis as used in psychology see the Functional analysis (psychology article An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ t. This introduces an operator U t, the transfer operator,

$(U^t a)(x) = a(\Phi^{-t}(x)) \,.$

By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ t. The transfer operator is different from the transfer homomorphism. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ t gets mapped into an infinite-dimensional linear problem involving U.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

### Chaos theory

Main article: chaos theory

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that Piecewise linear may refer to Piecewise linear function Piecewise linear manifold (Remember that we are speaking of completely deterministic systems!). This seemingly unpredictable behavior has been called chaos. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In Mathematics, more particularly in the fields of Dynamical systems and Geometric topology, an Anosov map on a Manifold M is a certain In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

Note that the chaotic behavior of complicated systems is not the issue. An attractor is a set to which a Dynamical system evolves after a long enough time Meteorology has been known for years to involve complicated—even chaotic—behavior. Meteorology (from Greek grc μετέωρος metéōros, "high in the sky" and grc -λογία -logia) is the Interdisciplinary Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear. The logistic map is a Polynomial mapping of degree 2, often cited as an archetypal example of how complex chaotic behaviour can arise from very simple In the Mathematics of Chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself

### Geometrical definition

A dynamical system is the tuple $\langle \mathcal{M}, f , \mathcal{T}\rangle$, with $\mathcal{M}$ a manifold (locally a Banach space or Euclidean space), $\mathcal{T}$ the domain for time (non-negative reals, the integers, . . . ) and an evolution rule f t (with $t\in\mathcal{T}$) a diffeomorphism of the manifold to itself. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

### Measure theoretical definition

See main article measure-preserving dynamical system. In Mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of Dynamical systems, and Ergodic theory in particular

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X,Σ,μ,τ). In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty Here, X is a set, and Σ is a topology on X, so that (X,Σ) is a sigma-algebra. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of For every element $\sigma \in \Sigma$, μ is its finite measure, so that the triplet (X,Σ,μ) is a probability space. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with A map $\tau:X\to X$ is said to be Σ-measurable if and only if, for every $\sigma \in \Sigma$, one has $\tau^{-1}\sigma \in \Sigma$. In Mathematics, measurable functions are Well-behaved functions between measurable spaces. A map τ is said to preserve the measure if and only if, for every $\sigma \in \Sigma$, one has μ(τ − 1σ) = μ(σ). Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X,Σ,μ,τ), for such a τ, is then defined to be a dynamical system.

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates $\tau^n=\tau \circ \tau \circ \ldots\circ\tau$ for integer n are studied. In Mathematics, iterated functions are the objects of deep study in Computer science, Fractals and Dynamical systems An iterated function is For continuous dynamical systems, the map τ is understood to be finite time evolution map and the construction is more complicated.