Dynamical systems theory

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Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems 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 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 1 Overview 2 History 3 Concepts 3.1 Dynamical systems 3.2 Dynamicism 3.3 Nonlinear system 4 Related fields 4.1 Chaos theory 4.2 Complex systems 4.3 Ergodic theory 4.4 Functional analysis 4.5 Projected dynamical systems 4.6 System dynamics 5 Applications 5.1 In Biomechanics 5.2 In the cognitive science 6 See also 7 References 8 Further reading 9 External links

[edit] Overview

Dynamical systems theory and chaos theory 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 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 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.

[edit] 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), Padula and Arbib (1974), and Strogatz (1994).^[1]

[edit] Concepts

[edit] Dynamical systems

Main article: Dynamical systems

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.

[edit] 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.

[edit] 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 nonhomogenous 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.

[edit] Related fields

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] Applications

[edit] 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.^[3]

[edit] In the cognitive science

Dynamical system theory has recently emerged in the field 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 beliefs 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]

[edit] See also

Related subjects

• List of dynamical system topics
• Baker's map
• Dynamical system (definition)
• Embodied Embedded Cognition
• Gingerbreadman map
• Halo orbit
• List of types of systems theory
• Oscillation
• Recurrent neural network
• Postcognitivism

Related scientists

• People in systems and control
• Dmitri Anosov
• Denis Evans
• Tim van Gelder
• Pim Haselager
• Debra Searles
• Lawrence Ward

[edit] References

[edit] 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

[edit] External links

• Dynamic Systems Encyclopedia of Cognitive Science entry.
• Definition of dynamical system in MathWorld.