This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems and bizarre systems.
This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.
A chaotic solution of the Lorenz system, which is an example of a non-linear dynamical system. Studying the Lorenz system helped give rise to chaos theory.
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 that do not change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.
Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii'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 seemingly random behavior that has been called chaos.[1] The branch of dynamical systems that 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.
The dynamical system concept is a mathematical formalization for any fixed "rule" that 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 may be deterministic (for a given time interval one future state can be precisely predicted given the current state) or stochastic (the evolution of the state can only be predicted with a certain probability).
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.
In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve 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 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 that 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 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 graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
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 systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system.[3] What makes using system dynamics different from other approaches to studying systems is the language used to describe feedback loops with 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 and efficiency. 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.[4] There is no research validation of any of the claims associated to the conceptual application of this framework.
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.[5]
Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error.[6]
Further, since the middle of the 1990s[7]cognitive science, oriented towards a systemtheoretical connectionism, has increasingly adopted the methods from (nonlinear) “Dynamic Systems Theory (DST)“.[8][9][10] A variety of neurosymbolic cognitive neuroarchitectures in modern connectionism, considering their mathematical structural core, can be categorized as (nonlinear) dynamical systems.[11][12][13] These attempts in neurocognition to merge connectionist cognitive neuroarchitectures with DST come from not only neuroinformatics and connectionism, but also recently from developmental psychology (“Dynamic Field Theory (DFT)”[14][15]) and from “evolutionary robotics” and “developmental robotics”[16] in connection with the mathematical method of “evolutionary computation (EC)”. For an overview see Maurer.[17][18]
The application of Dynamic Systems Theory to study second language acquisition is attributed to Diane Larsen-Freeman who published an article in 1997 in which she claimed that second language acquisition should be viewed as a developmental process which includes language attrition as well as language acquisition.[19] In her article she claimed that language should be viewed as a dynamic system which is dynamic, complex, nonlinear, chaotic, unpredictable, sensitive to initial conditions, open, self-organizing, feedback sensitive, and adaptive.
↑ Jerome R. Busemeyer (2008), "Dynamic Systems". To Appear in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008. Archived June 13, 2008, at the Wayback Machine
↑ R.F. Port and T. van Gelder [eds.] (1995). Mind as Motion. Explorations in the Dynamics of Cognition. A Bradford Book. MIT Press, Cambridge/MA.
↑ van Gelder, T. and R.F. Port (1995). It’s about time: an overview of the dynamical approach to cognition. pp. 1-43. In: R.F. Port and T. van Gelder [eds.]: Mind as Motion. Explorations in the Dynamics of Cognition. A Bradford Book. MIT Press, Cambridge/MA.
↑ van Gelder, T. (1998b). The dynamical hypothesis in cognitive science. Behavioral and Brain Sciences 21: 615-628.
↑ Abrahamsen, A. and W. Bechtel (2006). Phenomena and mechanisms: putting the symbolic, connectionist, and dynamical systems debate in broader perspective. pp. 159-185. In: R. Stainton [ed.]: Contemporary Debates in Cognitive Science. Basil Blackwell, Oxford.
↑ Nadeau, S.E. (2014). Attractor basins: a neural basis for the conformation of knowledge. pp. 305-333. In: A. Chatterjee [ed.]: The Roots of Cognitive Neuroscience. Behavioral Neurology and Neuropsychology. Oxford University Press, Oxford.
↑ Leitgeb, H. (2005). Interpreted dynamical systems and qualitative laws: from neural network to evolutionary systems. Synthese 146: 189-202.
↑ Munro, P.W. and J.A. Anderson. (1988). Tools for connectionist modeling: the dynamical systems methodology. Behavior Research Methods, Instruments, and Computers 20: 276-281.
↑ Schöner, G. (2008). Dynamical systems approaches to cognition. pp. 101-126. In: R. Sun [ed.]: The Cambridge Handbook of Computational Psychology. CambridgeUniversity Press, Cambridge.
↑ Schöner, G. (2009) Development as change of systems dynamics: stability, instability, and emergence. pp. 25-31. In: J.P. Spencer, M.S.C. Thomas, and J.L. McClelland. [eds.]: Toward a Unified Theory of Development: Connectionism and Dynamic Systems Theory ReConsidered. Oxford University Press, Oxford.
↑ Schlesinger, M. (2009). The robot as a new frontier for connectionism and dynamic systems theory. pp. 182-199. In: J.P. Spencer, M.S.C. Thomas, and J.L. McClelland. [eds.]: Toward a Unified Theory of Development: Connectionism and Dynamic Systems Theory ReConsidered. Oxford University Press, Oxford.
↑ Maurer, H. (2021). Cognitive science: Integrative synchronization mechanisms in cognitive neuroarchitectures of the modern connectionism. CRC Press, Boca Raton/FL, chap. 1.4, 2., 3.26, 11.2.1, ISBN 978-1-351-04352-6. https://doi.org/10.1201/9781351043526
↑ Maurer, H. (2016). „Integrative synchronization mechanisms in connectionist cognitive Neuroarchitectures“. Computational Cognitive Science. 2: 3. https://doi.org/10.1186/s40469-016-0010-8
Cognitive science is the interdisciplinary, scientific study of the mind and its processes. It examines the nature, the tasks, and the functions of cognition. Mental faculties of concern to cognitive scientists include language, perception, memory, attention, reasoning, and emotion; to understand these faculties, cognitive scientists borrow from fields such as linguistics, psychology, artificial intelligence, philosophy, neuroscience, and anthropology. The typical analysis of cognitive science spans many levels of organization, from learning and decision to logic and planning; from neural circuitry to modular brain organization. One of the fundamental concepts of cognitive science is that "thinking can best be understood in terms of representational structures in the mind and computational procedures that operate on those structures."
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.
Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
Systems theory is the transdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or artificial. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" when it expresses synergy or emergent behavior.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Connectionism is the name of an approach to the study of human mental processes and cognition that utilizes mathematical models known as connectionist networks or artificial neural networks. Connectionism has had many 'waves' since its beginnings.
A cognitive model is an approximation of one or more cognitive processes in humans or other animals for the purposes of comprehension and prediction. There are many types of cognitive models, and they can range from box-and-arrow diagrams to a set of equations to software programs that interact with the same tools that humans use to complete tasks. In terms of information processing, cognitive modeling is modeling of human perception, reasoning, memory and action.
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to test scientific theories. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms are sometimes interchanged.
Tim van Gelder is the co-founder of Austhink Software, an Australian software development company, and the Managing Director of Austhink Consulting. He was born in Australia, and was educated at the University of Melbourne. He went on to receive his PhD from the University of Pittsburgh (1989). He has held academic positions at Indiana University and the Australian National University before returning to Melbourne as an Australian Research Council QEII Research Fellow. In 1998, he transitioned to part-time academic work allowing him to pursue private training and consulting, and in 2005 began working full-time at Austhink Software. In 2009 he transitioned to Managing Director of Austhink Consulting. He co-leads The SWARM Project at the University of Melbourne.
Dynamicism, also termed dynamic hypothesis or dynamic cognition, is an approach in cognitive science popularized by the work of philosopher Tim van Gelder. It argues that differential equations and dynamical systems are more suited to modeling cognition rather than the commonly used ideas of symbolicism, connectionism, or traditional computer models. It is closely related to dynamical neuroscience.
In mathematics, symbolic dynamics is the practice of modeling 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. Formally, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.
Computational cognition is the study of the computational basis of learning and inference by mathematical modeling, computer simulation, and behavioral experiments. In psychology, it is an approach which develops computational models based on experimental results. It seeks to understand the basis behind the human method of processing of information. Early on computational cognitive scientists sought to bring back and create a scientific form of Brentano's psychology.
Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. The mathematical approach is used with the goal of deriving hypotheses that are more exact and thus yield stricter empirical validations. There are five major research areas in mathematical psychology: learning and memory, perception and psychophysics, choice and decision-making, language and thinking, and measurement and scaling.
Neurophilosophy or philosophy of neuroscience is the interdisciplinary study of neuroscience and philosophy that explores the relevance of neuroscientific studies to the arguments traditionally categorized as philosophy of mind. The philosophy of neuroscience attempts to clarify neuroscientific methods and results using the conceptual rigor and methods of philosophy of science.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems.
Neural computation is the information processing performed by networks of neurons. Neural computation is affiliated with the philosophical tradition known as Computational theory of mind, also referred to as computationalism, which advances the thesis that neural computation explains cognition. The first persons to propose an account of neural activity as being computational was Warren McCullock and Walter Pitts in their seminal 1943 paper, A Logical Calculus of the Ideas Immanent in Nervous Activity.
A coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear systems. They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases.
Valentin Afraimovich was a Soviet, Russian and Mexican mathematician. He made contributions to dynamical systems theory, qualitative theory of ordinary differential equations, bifurcation theory, concept of attractor, strange attractors, space-time chaos, mathematical models of non-equilibrium media and biological systems, travelling waves in lattices, complexity of orbits and dimension-like characteristics in dynamical systems.
The dynamical systems approach to neuroscience is a branch of mathematical biology that utilizes nonlinear dynamics to understand and model the nervous system and its functions. In a dynamical system, all possible states are expressed by a phase space. Such systems can experience bifurcation as a function of its bifurcation parameters and often exhibit chaos. Dynamical neuroscience describes the non-linear dynamics at many levels of the brain from single neural cells to cognitive processes, sleep states and the behavior of neurons in large-scale neuronal simulation.
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