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**Dynamical systems theory** is an area of mathematics used to describe the behavior of complex dynamical systems, usually 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*. When the time variable runs over a set which is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set then one gets dynamic equations on time scales. Some situations may also be modelled by mixed operators such as differential-difference equations.

This theory deals with the long-term qualitative behavior of dynamical systems, and the studies of the solutions to the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary orbits as well as the behaviour of electronic circuits and the solutions to partial differential equations that arise in biology. Much of modern research is focused on the study of chaotic systems.

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*.

## Contents

## Overview

Dynamical systems theory and chaos theory deal 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 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 system; 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 systems 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 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), Padulo and Arbib (1974), and Strogatz (1994).^{[1]}

## Concepts

### Dynamical systems

*Main article: Dynamical system (definition)*

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

A dynamical system has a *state* determined by a collection of real numbers, or more generally by a set of points in an appropriate *state space*. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. 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: 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 article: Nonlinear system*

In mathematics, a nonlinear system is a system which is not 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 nonhomogeneous

system, which is linear apart from the presence of a function of the independent variables, 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

### Arithmetic dynamics

- Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational,
`p`-adic, and/or algebraic points under repeated application of a polynomial or rational function.

### Chaos theory

- Chaos theory describes the behavior of certain dynamical systems – 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 random. This happens even though these systems are 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 systems 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.

### Control theory

- Control theory is an interdisciplinary branch of engineering and mathematics, that deals with influencing the behavior of dynamical systems.

### Ergodic theory

- Ergodic theory is a branch of mathematics that studies dynamical systems 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 analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word
*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.

### Graph dynamical systems

- The concept of graph dynamical systems (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDS is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.

### Projected dynamical systems

- Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.

### Symbolic dynamics

- Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.

### System dynamics

- System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system.
^{[2]}What makes using system dynamics different from other approaches to studying complex systems is the use of feedback loops and stocks and flows. These elements help describe how even seemingly simple systems display baffling nonlinearity.

### Topological dynamics

- Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.

## Applications

### In biomechanics

In sports 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.^{[3]}

### In cognitive science

Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the neo-Piagetian theories of 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 believed 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. ^{[4]}

Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error.^{[5]}

### In human development

Dynamic systems theory is a psychological theory of human development. Unlike dynamical systems theory which is a mathematical construct, dynamic systems theory is primarily non-mathematical and driven by qualitative theoretical propositions. This psychological theory does, however, apply metaphors derived from the mathematical concepts of dynamical systems theory to attempt to explain the existence of apparently complex phenomena in human psychological and motor development.

As it applies to developmental psychology, this psychological theory was developed by Esther Thelen, Ph.D. at Indiana University-Bloomington. Thelen became interested in developmental psychology through her interest and training in behavioral biology. She wondered if "fixed action patterns," or highly repeatable movements seen in birds and other animals, were also relevant to the control and development of human infants ^{[6]}

According to Miller (2002)^{[7]}, dynamic systems theory is the broadest and most encompassing of all the developmental theories. This theory attempts to encompass all the possible factors that may be in operation at any given developmental moment; it considers development from many levels (from molecular to cultural) and time scales (from milliseconds to years)^{[8]}. Development is viewed as constant, fluid, emergent or non-linear, and multidetermined ^{[9]}. Dynamic systems theory’s greatest impact has been in early sensorimotor development ^{[10]}.

Esther Thelen believed that development involved a deeply embedded and continuously coupled dynamic system. It is unclear however if her utilization of the concept of "dynamic" refers to the conventional dynamics of classical mechanics or to the metaphorical representation of "something that is dynamic" as applied in the colloquial sense in common speech, or both. The typical view presented by R.D. Beer showed that information from the world was given to the nervous system which directs the body, which in turn interacts back on the world. Esther Thelen instead offers a developmental system that has continual and bidirectional interaction between the world, nervous system and body. The exact mechanisms for such interaction, however, remain unspecified.^{[11]}

The dynamic systems view of development has three critical features that separate it from the traditional input-output model. The system must first be multiply causal and self-organizing. This means that behavior is a pattern formed from multiple components in cooperation with none being more privileged than another. The relationship between the multiple parts is what helps provide order and pattern to the system. Why this relation would provide such order and pattern, however, is unclear. Second, a dynamic system is a dependent on time making the current state a function of the previous state and the future state a function of the current state. The third feature is the relative stability of a dynamic system. For a system to change, a loose stability is needed to allow for the components to reorganize into a different expressed behavior. What constitutes a stability as being loose or not-loose, however, is not specified. Parameters that dictate what constitutes one state of organization versus another state are also not specified, as a generality, in dynamic systems theory. The theory contends that development is a sequence of times where stability is low allowing for new development and where stability is stable with less pattern change. The theory contends that in order to make these movements, you must scale up on a control parameter in order to reach a threshold (past a point of stability). Once that threshold is reached, the muscles will begin to form the different movements. This threshold must be reached in order for each different muscle to contract and relax to make the movement. The theory can be seen to present a variant explanation for muscle length-tension regulation but the extrapolation of a vaguely outlined argument for muscle action to a grand theory of human development remains unconvincing and unvalidated.^{[12]}

Esther Thelen's early research in infant motor behavior (particularly stepping, kicking, and reaching) led her to become dissatisfied with existing theories and moved her toward a dynamic systems perspective. Prior views of development conceptualized infants as passive and infants’ motor development as the result of a genetically determined developmental plan. Thelen, in her work, contended that infants' body weights and proportions, postures, elastic, and inertial properties of muscle and the nature of the task and environment contribute equally to the motor outcome. None of these contentions have been scientifically validated due in part to the breadth and poor operational definition of the parameters used to represent the phenomena involved. It is theorized that infants can "self-assemble" new motor patterns in novel situations, but what this actually means awaits further and specific clarification. The theory contends that development happens in individual children solving individual problems in their own unique ways ^{[13]}. Thelen used the proposition that because each child is different in terms of his or her body, nervous system, and daily experience, the course of development is nearly impossible to predict, and yet the theory does not account for clear trends and predictability in development for most children, despite there being multiple pathways to development ^{[14]}. Development is supposedly not just the result of genetics or the environment, but rather the interweaving of events at a given moment ^{[15]}. How such interweaving occurs is not specified by the theory in certain terms. Dynamic systems theory proponents claim to have had the greatest impact on early sensorimotor development ^{[16]} .

## See also

- Related subjects

- List of dynamical system topics
- Baker's map
- Biological applications of bifurcation theory
- Dynamical system (definition)
- Embodied Embedded Cognition
- Gingerbreadman map
- Halo orbit
- List of types of systems theory
- Oscillation
- Postcognitivism
- Recurrent neural network
- Combinatorics and dynamical systems
- Synergetics
- Systemography

- Related scientists

- People in systems and control
- Dmitri Anosov
- Vladimir Arnold
- Nikolay Bogolyubov
- Andrey Kolmogorov
- Nikolay Krylov
- Jürgen Moser
- Yakov G. Sinai
- Stephen Smale
- Hillel Fürstenberg
- Grigory Margulis
- Elon Lindenstrauss
- Esther Thelen

## Notes

- ↑ Jerome R. Busemeyer (2008), "Dynamic Systems". To Appear in:
*Encyclopedia of cognitive science*, Macmillan. Retrieved 8 May 2008. - ↑ MIT System Dynamics in Education Project (SDEP)
- ↑ Paul S Glaziera, Keith Davidsb, Roger M Bartlettc (2003). "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research". in: Sportscience 7. Accessdate=2008-05-08.
- ↑ Lewis, Mark D. (2000-02-25). The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development.
*Child Development***71**(1): 36–43. - ↑ Smith, Linda B., Esther Thelen (2003-07-30). Development as a dynamic system.
*TRENDS in Cognitive Sciences***7**(8): 343–8. - ↑ Template:Thelen, E., & Bates, E. (2003). Connectionism and dynamic systems: Are they really different? Developmental Science, 6, 378-391.
- ↑ Template:Miller, P. (2002). Theories of Developmental Psychology (4th ed.). New York, NY: Worth Publishers.
- ↑ Template:Miller, P. (2002). Theories of Developmental Psychology (4th ed.). New York, NY: Worth Publishers.
- ↑ Template:Spencer, J. P., Clearfield, M., Corbetta, D., Ulrich, B., Buchanan, P., & Schöner, G. (2006). Moving toward a grand theory of development: In memory of Esther Thelen. Child Development, 77, 1521-1538.
- ↑ Template:Thelen, E., & Bates, E. (2003). Connectionism and dynamic systems: Are they really different? Developmental Science, 6, 378-391.
- ↑ Thelen, E. (2000), Grounded in the World: Developmental Origins of the Embodied Mind. Infancy, 1: 3–28. DOI:10.1207/S15327078IN0101_02
- ↑ Thelen, E. (2000), Grounded in the World.<r: Developmental Origins of the Embodied Mind. Infancy, 1: 3–28. DOI:10.1207/S15327078IN0101_02
- ↑ Template:Spencer, J. P., Clearfield, M., Corbetta, D., Ulrich, B., Buchanan, P., & Schöner, G. (2006). Moving toward a grand theory of development: In memory of Esther Thelen. Child Development, 77, 1521-1538.
- ↑ Template:Thelen, E. (2005). Dynamic systems theory and the complexity of change. Psychoanalytic Dialogues, 15, 255-283.
- ↑ Template:Thelen, E. (2005). Dynamic systems theory and the complexity of change. Psychoanalytic Dialogues, 15, 255-283.
- ↑ Template:Thelen, E., & Bates, E. (2003). Connectionism and dynamic systems: Are they really different? Developmental Science, 6, 378-391.

## Further reading

- Frederick David Abraham (1990),
*A Visual Introduction to Dynamical Systems Theory for Psychology*, 1990. - Beltrami, E. J. (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

- Dynamic Systems Encyclopedia of Cognitive Science entry.
- Definition of dynamical system in MathWorld.
- DSWeb Dynamical Systems Magazine

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