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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a dynamical system is a system in which a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
describes the
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
dependence of a
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
in a
geometrical space
geometrical space
. Examples include the
mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environm ...
s that describe the swinging of a clock
pendulum A pendulum is a weight suspended from a pivot Pivot may refer to: *Pivot, the point of rotation in a lever A lever ( or ) is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '':wikt:fulcrum, fulcrum''. A lever ...

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 State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, Un ...
given by a
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
(a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
) that can be represented by a point in an appropriate
state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence Artificial intelligence (AI) is i ...
(a geometrical
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
). The ''evolution rule'' of the dynamical system is a function that describes what future states follow from the current state. Often the function is
deterministic Determinism is the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the ...
, that is, for a given time interval only one future state follows from the current state. However, some systems are
stochastic Stochastic () refers to the property of being well described by a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...
, in that random events also affect the evolution of the state variables. In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
s involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. The study of dynamical systems is the focus of
dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theo ...
, which has applications to a wide variety of fields such as mathematics, physics,
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

biology
,
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

chemistry
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
,
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

economics
,
history History (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approxima ...
, and
medicine Medicine is the science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (proced ...

medicine
. Dynamical systems are a fundamental part of
chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields ...
,
logistic map The logistic map is a polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only t ...

logistic map
dynamics,
bifurcation theory Bifurcation theory is the Mathematics, mathematical study of changes in the qualitative or topological structure of a given Family of curves, family, such as the integral curves of a family of vector fields, and the solutions of a family of differe ...
, the
self-assembly File:Iron oxide nanocube.jpg, upright=1.2, Transmission electron microscopy image of an iron oxide nanoparticle. Regularly arranged dots within the dashed border are columns of Fe atoms. Left inset is the corresponding electron diffraction pattern. ...

self-assembly
and
self-organization Self-organization, also called (in the social sciences) spontaneous order, is a process where some form of overall order arises from local interactions between parts of an initially disordered system A system is a group of Interaction, inte ...

self-organization
processes, and the
edge of chaos The edge of chaos is a transition space between order and disorder that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engenders a constant dynamic interplay between order ...
concept.


Overview

The concept of a dynamical system has its origins in
Newtonian mechanics Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
,
difference equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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''. If 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 A trajectory or flight path is the path that an with in follows through as a function of time. In , a trajectory is defined by via ; hence, a complete trajectory is defined by position and momentum, simultaneously. The mass might be a or ...

trajectory
'' or ''
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...
''. Before the advent of
computers A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These p ...

computers
, finding an orbit 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 Various types of stability may be discussed for the solutions of differential equation In mathematics, a differential equation is an equation that relates one or more function (mathematics), functions and their derivatives. In applications, the ...
or
structural stability In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit The album-equivalent unit is a measurement unit in music industry to define the consumption of music that equals the purchase of one album copy. This consumpti ...
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
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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
and of
chaos Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (Kinnikuman), Chaos (''Kinnikuman'') * Chaos (Sailor Moon), Chaos (''Sailor Moon'') * Chaos (Sesame Park), Chaos (''Sesame Park'') * Chaos (Warhammer), Chaos ('' ...
.


History

Many people regard French mathematician
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
as the founder of dynamical systems.Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"." ''Physics Reports'' 193.3 (1990): 137-163. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the
Poincaré recurrence theorem In physics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), t ...
, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.
Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – November 3, 1918) was a Russian Russian refers to anything related to Russia, including: *Russians (русские, ''russkiye''), an e ...

Aleksandr Lyapunov
developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. In 1913,
George David Birkhoff George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during ...

George David Birkhoff
proved Poincaré's " Last Geometric Theorem", a special case of the
three-body problem In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
, a result that made him world-famous. In 1927, he published his
Dynamical Systems
'. Birkhoff's most durable result has been his 1931 discovery of what is now called the
ergodic theorem Ergodic theory (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ...
. Combining insights from
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
on the
ergodic hypothesis 150px, This device can trap fruit flies, but if it trapped atoms when placed in gas that already uniformly fills the available Liouville's theorem and the Perpetual_motion#Classification, second law of thermodynamics would be violated. In phys ...
with
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
, this theorem solved, at least in principle, a fundamental problem of
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
. The ergodic theorem has also had repercussions for dynamics.
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theor ...
made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.
Oleksandr Mykolaiovych Sharkovsky Oleksandr Mykolayovych Sharkovsky (also Sharkovskii) ( uk, Олекса́ндр Миколайович Шарко́вський) (born December 7, 1936) is a prominent Ukrainians, Ukrainian mathematician most famous for developing Sharkovsky's th ...
developed
Sharkovsky's theoremIn mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovsky, Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about Dynamical system (definition)#Discrete dynamical system, discrete dynamical systems ...
on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
has a
periodic point In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of period 3, then it must have periodic points of every other period. In the late 20th century, Palestinian mechanical engineer Ali H. Nayfeh applied
nonlinear dynamics In mathematics and science, a nonlinear system is a system in which the change of the output is not proportionality (mathematics), proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, ...
in
mechanical Mechanical may refer to: Machine * Mechanical system A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine A molecular machine, nan ...

mechanical
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of
machines A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine A molecular machine, nanite, or nanomachine is a molecular component that produc ...

machines
and
structures A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sy ...

structures
that are common in daily life, such as
ships A ship is a large watercraft Watercraft, also known as water vessels or waterborne vessels, are vehicles A vehicle (from la, vehiculum) is a machine A machine is any physical system with ordered structural and functional propertie ...

ships
, cranes,
bridges A bridge is a Nonbuilding structure, structure built to Span (engineering), span a physical obstacle, such as a body of water, valley, or road, without closing the way underneath. It is constructed for the purpose of providing passage over the ...

bridges
,
buildings A building, or edifice, is a structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a se ...

buildings
,
skyscrapers A skyscraper is a large continuously habitable building having multiple floors. Modern sources currently define skyscrapers as being at least 100 metres or 150 metres in height, though there is no universally accepted definition. Historically, t ...

skyscrapers
,
jet engines A jet engine is a type of reaction engine A reaction engine is an engine or motor that produces thrust Thrust is a described quantitatively by . When a system expels or in one direction, the accelerated mass will cause a force of ...
,
rocket engines A rocket engine uses stored rocket propellant Rocket propellant is the reaction mass of a rocket A rocket (from it, rocchetto, , bobbin/spool) is a projectile that spacecraft, aircraft An aircraft is a vehicle that is able to ...
,
aircraft An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...

aircraft
and
spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite alt=, A full-size model of the Earth observation satellite ERS 2 ">ERS_2.html" ;"title="Earth observation satellite ERS 2">Earth obse ...

spacecraft
.


Basic definitions

A dynamical system is a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
''M'' called the phase (or state) space endowed with a family of smooth evolution functions Φ''t'' that for any element ''t'' ∈ ''T'', the time, map a point of the
phase space In dynamical system theory, a phase space is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called param ...

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 Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on ...
''; 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''. Note: There is a further technical condition that Φ''t'' is an action of ''T'' on ''M''. That includes the facts that Φ0 is the identity function and that Φ''s+t'' is the composition of Φ''s'' and Φ''t''. This is a
monoid action In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
, which doesn't require the existence of negative values for ''t'', and doesn't require the functions Φ''t'' to be invertible.


Examples

The evolution function Φ ''t'' is often the solution of a ''differential equation of motion'' : \dot = 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 ''x''0. The
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

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 In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
''TxM'' 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 In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
can be used to define the evolution rule: : G(x, \dot) = 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 equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s; 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 space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s—in which case the differential equations are
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.


Further examples

*
Arnold's cat map is performed. The lines with the arrows show the direction of the contracting and expanding eigenspaces In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical str ...

Arnold's cat map
*
Baker's map Image:Ising-tartan.png, Example of a measure (mathematics), measure that is invariant under the action of the (unrotated) baker's map: an invariant measure. Applying the baker's map to this image always results in exactly the same image. In dynamica ...
is an example of a chaotic
piecewise linear
piecewise linear
map *
Billiards Cue sports (sometimes written cuesports), also known as billiard sports, are a wide variety of generally played with a , which is used to strike s and thereby cause them to move around a -covered bounded by elastic bumpers known as . Histor ...
and outer billiards * Bouncing ball dynamics *
Circle map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Circle map
*
Complex quadratic polynomial A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathema ...
*
Double pendulum s attached end to end. In physics and mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamical systems, dynamic behavior wi ...

Double pendulum
*
Dyadic transformation The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping ...
*
Hénon mapHénon may refer to: * Hénon, Côtes-d'Armor, France * Michel Hénon (1931–2013), French mathematician * Hénon map, a chaotic dynamical system introduced by Michel Hénon * Guy Hénon (1912–?), French field hockey player * Jacques-Louis H ...
*
Irrational rotation In the mathematical theory of dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples in ...
* Kaplan–Yorke map *
List of chaotic maps In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
Lorenz system The Lorenz system is a system of ordinary differential equations first studied by Edward Norton Lorenz, Edward Lorenz. It is notable for having Chaos theory, chaotic solutions for certain parameter values and initial conditions. In particular, the ...

Lorenz system
*
Quadratic map simulation system
Quadratic map simulation system
* Rössler map *
Swinging Atwood's machine 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 plane, producing a dynamical system In mathematics, a dynamical system is a syste ...
*
Tent map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Tent map


Linear dynamical systems

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 The superposition principle, also known as superposition property, states that, for all linear system In systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent parts that ...
: 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 Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on ...
, the vector field v(''x'') is an
affine Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology *Affine cipher, a special case of the more general substitution cipher *Aff ...
function of the position in the phase space, that is, : \dot = v(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'': : \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 ''x''0 = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the
exponential of a matrix In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix expo ...
: for an initial point ''x''0, : \Phi^t(x_0) = e^ x_0. When ''b'' = 0, the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

eigenvalue
s of ''A'' determine the structure of the phase space. From the eigenvalues and the
eigenvector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

eigenvector
s 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

A
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time. Discrete time Discrete time views values of variables as occurring at distinct, separate "points ...
,
affine Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology *Affine cipher, a special case of the more general substitution cipher *Aff ...
dynamical system has the form of a
matrix difference equation A matrix difference equation is a difference equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
: : x_ = 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'') –1''b'' removes the term ''b'' from the equation. In the new
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coordinate system
, the origin is a fixed point of the map and the solutions are of the linear system ''A'' ''n''''x''0. 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 ''u''1 is an eigenvector of ''A'', with a real eigenvalue smaller than one, then the straight lines given by the points along ''α'' ''u''1, 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

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 ''x''0 in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to ''v''(''x''0). These points are a Poincaré section ''S''(''γ'', ''x''0), 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 ''x''0. 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(''x''2), so a change of coordinates ''h'' can only be expected to simplify ''F'' to its linear part : h^ \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 ''λ''1, ..., ''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'' – Σ (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 ''x''0 of ''F'' is called ''Hyperbolic fixed point, 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 theorem, Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.


Bifurcation theory

When the evolution map Φ''t'' (or the
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

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 In dynamical system theory, a phase space is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called param ...

phase space
until a special value ''μ''0 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 (mathematics), 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 ''x''0 of a system family ''Fμ'' can be characterized by the eigenvalues of the first derivative of the system ''DF''''μ''(''x''0) computed at the bifurcation point. 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. 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, Bifurcation diagram, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.


Ergodic systems

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 : \mathrm (A) = \mathrm ( \Phi^t(A) ). In the Hamiltonian mechanics, 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's theorem (Hamiltonian), 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 Poincaré recurrence theorem, 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 Ernst Zermelo, Zermelo to object to Ludwig Boltzmann, 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
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. 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
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and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Bernard Koopman, 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, : (U^t a)(x) = a(\Phi^(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 Statistical mechanics, equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Statistical mechanics#Canonical ensemble, Boltzmann factor exp(−β''H''). 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

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 Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (Kinnikuman), Chaos (''Kinnikuman'') * Chaos (Sailor Moon), Chaos (''Sailor Moon'') * Chaos (Sesame Park), Chaos (''Sesame Park'') * Chaos (Warhammer), Chaos ('' ...
''. Anosov diffeomorphism, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
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
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logistic map
is only a second-degree polynomial; the horseshoe map is piecewise linear.


Geometrical definition

A dynamical system is the tuple \langle \mathcal, f , \mathcal\rangle , with \mathcal a manifold (locally a Banach space or Euclidean space), \mathcal the domain for time (non-negative reals, the integers, ...) and ''f'' an evolution rule ''t'' → ''f'' ''t'' (with t\in\mathcal) such that ''f t'' is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain \mathcal 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 \mathcal .


Measure theoretical definition

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 (mathematics), set, and Σ is a sigma-algebra on ''X'', so that the pair (''X'', Σ) is a measurable space. μ is a finite measure (mathematics), measure on the sigma-algebra, so that the triplet (''X'', Σ, μ) is a measure space, probability space. A map τ: ''X'' → ''X'' is said to be measurable function, Σ-measurable if and only if, for every σ ∈ Σ, one has \tau^\sigma \in \Sigma. A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has \mu(\tau^\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 iterated function, iterates \tau^n=\tau \circ \tau \circ \cdots\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.


Multidimensional generalization

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

* Behavioral modeling * Cognitive model#Dynamical systems, Cognitive modeling * Complex dynamics * Dynamic approach to second language development * Feedback passivation * Infinite compositions of analytic functions * List of dynamical system topics * Oscillation * People in systems and control * Sharkovskii's theorem * System dynamics * Systems theory * Principle of maximum caliber


References


Further reading

Works providing a broad coverage: * (available as a reprint: ) * ''Encyclopaedia of Mathematical Sciences'' () has a sub-series on dynamical systems with reviews of current research. * * Introductory texts with a unique perspective: * * * * * Textbooks * * * * * * * * * * * * * * Popularizations: * * * *


External links


Arxiv preprint server
has daily submissions of (non-refereed) manuscripts in dynamical systems.
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

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.

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.

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).

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.
UPC Dynamical Systems Group Barcelona
Polytechnical University of Catalonia.
Center for Control, Dynamical Systems, and Computation
University of California, Santa Barbara. {{DEFAULTSORT:Dynamical System Dynamical systems, Systems theory Mathematical and quantitative methods (economics)