Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions —the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Directions of study[link]

The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic type.

The study of the stability of solutions of differential equations is known as stability theory.

Nomenclature[link]

The theory of differential equations is quite developed and the methods used to study them vary significantly with the type of the equation.

• An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.

• Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation

  Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0

(in which Failed to parse (Missing texvc executable; please see math/README to configure.): y

is the dependent variable) is a second-order differential equation. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.

• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

There are very few methods of explicitly solving nonlinear differential equations; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Classification summary[link]

The mathematical definitions for the various classifications of differential equation can be collected as follows.

Ordinary DE classification[link]

See main article: Ordinary differential equation.

In the table below,

All differential equations are of order n and arbitrary degree d.

F is an implicit function of: an independent variable x, a dependent variable y (a function of x), and integer derivatives of y (fractional derivatives are in fact possible, but not considered here).

y may in general be a vector valued function:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{y} = \begin{pmatrix} y_1 (x) \\ y_2 (x) \\ \vdots \\ y_m (x), \end{pmatrix} \,\!

so x is an element of the vector space R, y an element of a vector space of dimension m, Failed to parse (Missing texvc executable; please see math/README to configure.): x \in \mathbf{R}, \mathbf{y} \in \mathbf{R}^m \,\! , where R is the set of real numbers, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}^m = \mathbf{R} \times \mathbf{R} \times \cdots \times \mathbf{R} \,\!

is the cartesian product of R with itself m times to form an m-tuple of real numbers. 

This leads to a system of differential equations to be solved for y₁, y₂,...y_m.

y is characterized by the function mapping Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{y} : \mathbf{R} \rightarrow \mathbf{R}^m \,\! .

r(x) is called a source term in x, and A(x) is an arbitrary function, both assumed continuous in x on defined intervals.

Notice the mapping from Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}^{mn+m+1} \,\!

or Failed to parse (Missing texvc executable; please see math/README to configure.):  \mathbf{R}^{mn+1} \,\!
corresponds to the map from x, y, and the n or (n-1) derivatives of y to the solution, in general implicit.

Examples[link]

In the first group of examples, let u be an unknown function of x, and c and ω are known constants.

• Inhomogeneous first-order linear constant coefficient ordinary differential equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{du}{dx} = cu+x^2.

• Homogeneous second-order linear ordinary differential equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0.

• Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d^2u}{dx^2} + \omega^2u = 0.

• Inhomogeneous first-order nonlinear ordinary differential equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{du}{dx} = u^2 + 1.

• Second-order nonlinear ordinary differential equation describing the motion of a pendulum of length L:

Failed to parse (Missing texvc executable; please see math/README to configure.): L\frac{d^2u}{dx^2} + g\sin u = 0.

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

• Homogeneous first-order linear partial differential equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0.

• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.

• Third-order nonlinear partial differential equation, the Korteweg–de Vries equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}.

Related concepts[link]

• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.

• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.

• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

Connection to difference equations[link]

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Universality of mathematical description[link]

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black–Scholes equation in finance is for instance, related to the heat equation.

Notable differential equations[link]

Biology[link]

• Verhulst equation – biological population growth
• von Bertalanffy model – biological individual growth
• Lotka–Volterra equations – biological population dynamics
• Replicator dynamics – may be found in theoretical biology
• Hodgkin–Huxley model – neural action potentials

Economics[link]

• The Black–Scholes PDE
• Exogenous growth model
• Malthusian growth model
• The Vidale–Wolfe advertising model

See also[link]

Wikibooks has a book on the topic of Ordinary Differential Equations

• Complex differential equation
• Exact differential equation
• Integral equations
• Linear differential equation
• Picard–Lindelöf theorem on existence and uniqueness of solutions
• Numerical methods

References[link]

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (August 2009)

• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.
• W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
• E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956
• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955
• P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006
• P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277
• R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations</ref>

External links[link]

• Lectures on Differential Equations MIT Open CourseWare Videos
• Online Notes / Differential Equations Paul Dawkins, Lamar University
• Differential Equations, S.O.S. Mathematics
• Differential Equation Solver Java applet tool used to solve differential equations.
• Introduction to modeling via differential equations Introduction to modeling by means of differential equations, with critical remarks.
• Mathematical Assistant on Web Symbolic ODE tool, using Maxima
• Exact Solutions of Ordinary Differential Equations
• Collection of ODE and DAE models of physical systems MATLAB models
• Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC
• Khan Academy Video playlist on differential equations Topics covered in a first year course in differential equations.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (July 2011)

The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. 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 springtime in a lake.

At any given time a dynamical system has a state given by a set of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). Small changes in the state of the system create small changes in the numbers. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic; in other words, for a given time interval only one future state follows from the current state.

Overview[link]

The concept of a dynamical system has its origins in Newtonian mechanics. 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.) 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 positions, a collection of points known as a trajectory or orbit.

Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines 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. 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.
• 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. 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. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.

It was in the work of Poincaré that these dynamical systems themes developed.

Basic definitions[link]

A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions Φ^t that for any element of t ∈ T, the time, map a point of the phase space back into the phase space. 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[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 T_xM of the point x.) 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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. 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 the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.

Further examples[link]

Linear dynamical systems[link]

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. 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[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi(x) = A x + b,\,

with A a matrix, b a vector of numbers and x the position vector. 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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,

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi^t(x_0) = e^{t A} x_0. \,

When b = 0, the eigenvalues of A determine the structure of the phase space. 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.

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.

Maps[link]

A discrete-time, affine dynamical system has the form

Failed to parse (Missing texvc executable; please see math/README to configure.): x_{n+1} = A x_n + b, \,

with A a matrix and b a vector. As in the continuous case, the change of coordinates x → x + (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.

Local dynamics[link]

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[link]

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[link]

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. The flow now defines a map, the Poincaré map F : S → S, for points starting in S and returning to S. 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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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, ..., λ_ν 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 λ_i – ∑ (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[link]

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 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. 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.

Bifurcation theory[link]

When the evolution map Φ^t (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. Small changes may produce no qualitative changes in the phase space until a special value μ₀ is reached. 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. 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 F_μ can be characterized by the eigenvalues of the first derivative of the system DF_μ(x₀) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DF_μ 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.

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.

Ergodic systems[link]

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. 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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. The volume is said to be computed by the Liouville measure.

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. 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.

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. 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. Koopman approached the study of ergodic systems by the use of functional analysis. 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,

Failed to parse (Missing texvc executable; please see math/README to configure.): (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. 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. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). 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.

Nonlinear dynamical systems and chaos[link]

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. 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. 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 complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. 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.

Geometrical definition[link]

A dynamical system is the tuple Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \mathcal{M}, f , \mathcal{T}\rangle , with Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{M}

a manifold (locally a Banach space or Euclidean space), Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{T}
the domain for time (non-negative reals, the integers, ...) and f an evolution rule t → f t (with Failed to parse (Missing texvc executable; please see math/README to configure.): t\in\mathcal{T}

) such that f ^t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{T}

into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain Failed to parse (Missing texvc executable; please see math/README to configure.):  \mathcal{T}
.

Measure theoretical definition[link]

See main article Measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X, Σ, μ, τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X, Σ, μ) is a probability space. A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has Failed to parse (Missing texvc executable; please see math/README to configure.): \tau^{-1}\sigma \in \Sigma . A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(\tau^{-1}\sigma ) = \mu(\sigma) . 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \tau^n=\tau \circ \tau \circ \ldots\circ\tau

for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

Examples of dynamical systems[link]

Internal links[link]

• Arnold's cat map
• Baker's map is an example of a chaotic piecewise linear map
• Circle map
• Double pendulum
• Billiards and Outer Billiards
• Hénon map
• Horseshoe map
• Irrational rotation
• List of chaotic maps
• Logistic map
• Lorenz system
• Rossler map

External links[link]

• Interactive applet for the Standard and Henon Maps by A. Luhn

Multidimensional generalization[link]

Dynamical systems are defined over a single independent variable, usually thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.

See also[link]

Systems science portal

• Behavioral modeling
• Dynamical systems theory
• Feedback passivation
• List of dynamical system topics
• Oscillation
• People in systems and control
• Sharkovskii's theorem
• System dynamics
• Systems theory
• Infinite compositions of analytic functions

References[link]

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2007)

Further reading[link]

Works providing a broad coverage:

• Ralph Abraham and Jerrold E. Marsden (1978). Foundations of mechanics. Benjamin–Cummings. ISBN 0-8053-0102-X.  (available as a reprint: ISBN 0-201-40840-6)
• Encyclopaedia of Mathematical Sciences (ISSN 0938-0396) has a sub-series on dynamical systems with reviews of current research.
• Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN 0-521-57557-5. 
• Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana (2005). Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer. ISBN 3-540-22066-6. 
• Stephen Smale (1967). "Differentiable dynamical systems". Bulletin of the American Mathematical Society 73 (6): 747–817. DOI:10.1090/S0002-9904-1967-11798-1. 

Introductory texts with a unique perspective:

• V. I. Arnold (1982). Mathematical methods of classical mechanics. Springer-Verlag. ISBN 0-387-96890-3. 
• Jacob Palis and Wellington de Melo (1982). Geometric theory of dynamical systems: an introduction. Springer-Verlag. ISBN 0-387-90668-1. 
• David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press. ISBN 0-12-601710-7. 
• Tim Bedford, Michael Keane and Caroline Series, eds. (1991). Ergodic theory, symbolic dynamics and hyperbolic spaces. Oxford University Press. ISBN 0-19-853390-X. 
• Ralph H. Abraham and Christopher D. Shaw (1992). Dynamics—the geometry of behavior, 2nd edition. Addison-Wesley. ISBN 0-201-56716-4. 

Textbooks

• Stephen Lynch (2010). Dynamical Systems with Applications using Maple 2nd Ed.. Springer. ISBN 0-8176-4389-3. 
• Stephen Lynch (2007). Dynamical Systems with Applications using Mathematica. Springer. ISBN 0-8176-4482-2. 
• Stephen Lynch (2004). Dynamical Systems with Applications using MATLAB. Springer. ISBN 0-8176-4321-4. 
• James Meiss (2007). Differential Dynamical Systems. SIAM. ISBN 0-89871-635-7. 
• Steven H. Strogatz (1994). Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering. Addison Wesley. ISBN 0-201-54344-3. 
• Kathleen T. Alligood, Tim D. Sauer and James A. Yorke (2000). Chaos. An introduction to dynamical systems. Springer Verlag. ISBN 0-387-94677-2. 
• Morris W. Hirsch, Stephen Smale and Robert Devaney (2003). Differential Equations, dynamical systems, and an introduction to chaos. Academic Press. ISBN 0-12-349703-5. 
• Julien Clinton Sprott (2003). Chaos and time-series analysis. Oxford University Press. ISBN 0-19-850839-5. 
• Oded Galor (2011). Discrete Dynamical Systems. Springer. ISBN 978-3-642-07185-0. 

Popularizations:

• Florin Diacu and Philip Holmes (1996). Celestial Encounters. Princeton. ISBN 0-691-02743-9. 
• James Gleick (1988). Chaos: Making a New Science. Penguin. ISBN 0-14-009250-1. 
• Ivar Ekeland (1990). Mathematics and the Unexpected (Paperback). University Of Chicago Press. ISBN 0-226-19990-8. 
• Ian Stewart (1997). Does God Play Dice? The New Mathematics of Chaos. Penguin. ISBN 0-14-025602-4. 

External links[link]

• A collection of dynamic and non-linear system models and demo applets (in Monash University's Virtual Lab)
• Arxiv preprint server has daily submissions of (non-refereed) manuscripts in dynamical systems.
• DSWeb provides up-to-date information on dynamical systems and its applications.
• Encyclopedia of dynamical systems A part of Scholarpedia — peer reviewed and written by invited experts.
• Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
• Oliver Knill has a series of examples of dynamical systems with explanations and interactive controls.
• Sci.Nonlinear FAQ 2.0 (Sept 2003) provides definitions, explanations and resources related to nonlinear science

Online books or lecture notes:

• Geometrical theory of dynamical systems. Nils Berglund's lecture notes for a course at ETH at the advanced undergraduate level.
• Dynamical systems. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.
• Chaos: classical and quantum. An introduction to dynamical systems from the periodic orbit point of view.
• Modeling Dynamic Systems. An introduction to the development of mathematical models of dynamic systems.
• Learning Dynamical Systems. Tutorial on learning dynamical systems.
• Ordinary Differential Equations and Dynamical Systems. Lecture notes by Gerald Teschl

Research groups:

• Dynamical Systems Group Groningen, IWI, University of Groningen.
• Chaos @ UMD. Concentrates on the applications of dynamical systems.
• Dynamical Systems, SUNY Stony Brook. Lists of conferences, researchers, and some open problems.
• Center for Dynamics and Geometry, Penn State.
• Control and Dynamical Systems, Caltech.
• Laboratory of Nonlinear Systems, Ecole Polytechnique Fédérale de Lausanne (EPFL).
• Center for Dynamical Systems, University of Bremen
• Systems Analysis, Modelling and Prediction Group, University of Oxford
• Non-Linear Dynamics Group, Instituto Superior Técnico, Technical University of Lisbon
• Dynamical Systems, IMPA, Instituto Nacional de Matemática Pura e Applicada.
• Nonlinear Dynamics Workgroup, Institute of Computer Science, Czech Academy of Sciences.

Simulation software based on Dynamical Systems approach:

• FyDiK
• iDMC, simulation and dynamical analysis of nonlinear models