Dynamical systems theory 990632 226140482 2008-07-17T00:20:56Z Blaisorblade 1284957 /* See also */ Add control theory, fix alphabetic sorting '''Dynamical systems theory''' is an area of [[applied mathematics]] used to describe the behavior of [[complex systems|complex]] [[dynamical system]]s by employing [[differential equations]] or [[difference equations]]. When [[differential]] equations are employed, the theory is called ''continuous dynamical systems''. When [[difference]] equations are employed, the theory is called ''discrete dynamical systems''. This theory deals with the long-term qualitative behavior of [[dynamical system]]s, and the studies of the solutions to the [[equations of motion]] of systems that are primarily [[mechanics|mechanical]] in nature; although this includes both [[planetary orbit]]s as well as the behaviour of [[electronic circuit]]s and the solutions to [[partial differential equation]]s that arise in [[biology]]. Much of modern research is focused on the study of [[chaotic system]]s. This field of study is also called just ''Dynamical systems'', ''Systems theory'' or longer as ''Mathematical Dynamical Systems Theory'' and the ''Mathematical theory of dynamical systems''. [[Image:Lorenz attractor yb.svg|thumb|240px|right||The [[Lorenz attractor]] is an example of a [[non-linear]] dynamical system. Studying this system helped give rise to [[Chaos theory]].]] == Overview == Dynamical systems theory and [[chaos theory]] deals with the long-term qualitative behavior of [[dynamical system]]s. 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 steady states?", or "Does the long-term behavior of the system depend on its initial condition?" An important goal is to describe the fixed points, or steady states of a given dynamical systems; these are values of the variable which won't change over time. Some of these fixed points are ''attractive'', meaning that if the system starts out in a nearby state, it will converge towards the fixed point. Similarly, one is interested in ''periodic points'', states of the system which repeat themselves after several timesteps. Periodic points can also be attractive. [[Sarkovskii's theorem]] is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. Even simple [[nonlinear dynamical system]]s often exhibit almost random, completely unpredictable behavior that has been called ''chaos''. The branch of dynamical systems which deals with the clean definition and investigation of chaos is called [[chaos theory]]. == History == The concept of dynamical systems theory 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. Before the advent of [[computer|fast computing machines]], solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems. Some excellent presentations of mathematical dynamic system theory include Beltrami (1987), Luenberger (1979), Padula and Arbib (1974), and Strogatz (1994).<ref>Jerome R. Busemeyer (2008), [http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html "Dynamic Systems"]. To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008.</ref> == Concepts == === Dynamical systems === {{main|Dynamical system}} The [[dynamical system]] concept is a [[mathematics|mathematical]] [[formalization]] for any fixed "rule" which describes the [[time]] dependence of a point's position in its [[ambient space]]. Examples include the [[mathematical model]]s that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake. A dynamical system has a ''state'' determined by a collection of [[real numbers]], or more generally by a [[set]] of [[Point (geometry)|points]] in an appropriate ''state space''. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space&mdash;a [[manifold]]. The ''evolution rule'' of the dynamical system is a [[function (mathematics)|fixed rule]] that describes what future states follow from the current state. The rule is [[Deterministic system (mathematics)|deterministic]]: for a given time interval only one future state follows from the current state. === Dynamicism === [[Dynamicism]], also termed the ''dynamic hypothesis'' or the ''dynamic hypothesis in cognitive science'' or ''dynamic cognition'', is a new approach in [[cognitive science]] exemplified by the work of philosopher [[Tim van Gelder]]. It argues that [[differential equations]] are more suited to modelling [[cognition]] than more traditional [[computer]] models. === Nonlinear system === {{main|Nonlinear system}} In [[mathematics]], a [[nonlinear system]] is a system which is not [[linear system|linear]], i.e. a system which does not satisfy the [[superposition principle]]. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear sum of independent components. A [[homogenous|nonhomogenous]] system, which is linear apart from the presence of a function of the [[independent variable]]s, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known. == Related fields == === Chaos theory === :[[Chaos theory]] describes the behavior of certain [[dynamical system (definition)|dynamical system]]s – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the [[butterfly effect]]). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be [[randomness|random]]. This happens even though these systems are [[deterministic system (philosophy)|deterministic]], meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply ''[[chaos]]''. === Complex systems === :[[Complex systems]] is a scientific field, which studies the common properties of [[system]]s considered [[complex]] in [[nature]], [[society]] and [[science]]. It is also called ''complex systems theory'', ''complexity science'', ''study of complex systems'' and/or ''sciences of complexity''. The key problems of such systems are difficulties with their formal [[modeling]] and [[simulation]]. From such perspective, in different research contexts complex systems are defined on the base of their different attributes. :The study of complex systems is bringing new vitality to many areas of science where a more typical [[reductionist]] strategy has fallen short. ''Complex systems'' is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neurosciences, social sciences, meteorology, chemistry, physics, computer science, psychology, [[artificial life]], [[evolutionary computation]], economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves. === Ergodic theory === :[[Ergodic theory]] is a branch of [[mathematics]] that studies [[dynamical system]]s with an [[invariant measure]] and related problems. Its initial development was motivated by problems of [[statistical physics]]. === Functional analysis === :[[Functional analysis]] is the branch of [[mathematics]], and specifically of [[mathematical analysis|analysis]], concerned with the study of [[vector space]]s and [[operator]]s acting upon them. It has its historical roots in the study of [[functional space]]s, in particular transformations of [[function (mathematics)|functions]], such as the [[Fourier transform]], as well as in the study of [[differential equations|differential]] and [[integral equations|integral]] equations. This usage of the word ''[[functional (mathematics)|functional]]'' goes back to the [[calculus of variations]], implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist [[Vito Volterra]] and its founding is largely attributed to mathematician [[Stefan Banach]]. === Projected dynamical systems === :[[Projected dynamical systems ]] is a [[mathematics|mathematical]] theory investigating the behaviour of [[dynamical system]]s where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of [[Optimization (mathematics)|optimization]] and [[Equilibrium point|equilibrium]] problems and the dynamical world of [[ordinary differential equations]]. A projected dynamical system is given by the [[flow (mathematics)|flow]] to the projected differential equation. === System dynamics === :[[System dynamics]] is an approach to understanding the behaviour of [[complex system]]s over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system.<ref name="sysdyn">[http://sysdyn.clexchange.org MIT System Dynamics in Education Project (SDEP)<!-- Bot generated title -->]</ref> What makes using system dynamics different from other approaches to studying complex systems is the use of [[feedback]] loops and [[Stock and flow|stocks and flows]]. These elements help describe how even seemingly simple systems display baffling [[nonlinearity]]. == Applications == === In biomechanics === In [[biomechanics]], dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems.<ref>Paul S Glaziera, Keith Davidsb, Roger M Bartlettc (2003). [http://www.sportsci.org/jour/03/psg.htm "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research"]. in: Sportscience 7. Accessdate=2008-05-08.</ref> === In cognitive science === Dynamical system theory has recently emerged in the field of [[cognitive science|cognitive development]]. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and [[AI]]. It also believes that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through [[state space]]. In other words, dynamicists argue that [[psychology]] should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of [[chaos theory]] is also frequently adopted. In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. [[Self organization]] (the spontaneous creation of coherent forms) sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process. These links form the structure of a new state of order in the mind through a process called ''scalloping'' (the repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, idiosyncratic and unpredictable. <ref> {{cite journal|title=The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development|journal=Child Development|date=2000-02-25|first=Mark D.|last=Lewis|coauthors=|volume=71|issue=1|pages=36–43|id= |url=http://home.oise.utoronto.ca/~mlewis/Manuscripts/Promise.pdf|format=PDF|accessdate=2008-04-04|doi=10.1111/1467-8624.00116 }}</ref> Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the [[A-not-B error]].<ref> {{cite journal|title=Development as a dynamic system|journal=TRENDS in Cognitive Sciences|date=2003-07-30|first=Lester B.|last=Smith|coauthors=Esther Thelen|volume=7|issue=8|pages=343–8|id= |url=http://www.indiana.edu/~cogdev/labwork/dynamicsystem.pdf|format=PDF|accessdate=2008-04-04|doi=10.1016/S1364-6613(03)00156-6 }}</ref> == See also == ;Related subjects * [[List of dynamical system topics]] * [[Baker's map]] * [[Control theory]] * [[Dynamical system (definition)]] * [[Embodied Embedded Cognition]] * [[Gingerbreadman map]] * [[Halo orbit]] * [[List of types of systems theory]] * [[Oscillation]] * [[Postcognitivism]] * [[Recurrent neural network]] ;Related scientists * [[People in systems and control]] * [[Dmitri Anosov]] * [[Vladimir Arnold]] * [[Andrey Kolmogorov]] * [[Jürgen Moser]] * [[Yakov G. Sinai]] * [[Stephen Smale]] == References == {{reflist}} == Further reading == * Ralph H. Abraham (1990), ''A Visual Introduction to Dynamical Systems Theory for Psychology'', 1990. * Beltrami, E. (1987). ''Mathematics for dynamic modeling''. NY: Academic Press * Otomar Hájek (1968}, ''Dynamical Systems in the Plane''. * Luenberger, D. G. (1979). ''Introduction to dynamic systems''. NY: Wiley. * Anthony N. Michel, Kaining Wang & Bo Hu (2001), ''Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings''. * Padulo, L. & Arbib, M A. (1974). ''System Theory''. Philadelphia: Saunders * Strogatz, S. H. (1994), ''Nonlinear dynamics and chaos''. Reading, MA: Addison Wesley ==External links== *[http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html Dynamic Systems] Encyclopedia of Cognitive Science entry. *[http://mathworld.wolfram.com/DynamicalSystem.html Definition of dynamical system] in MathWorld. [[Category:Dynamical systems| ]] [[Category:Systems theory]]