Dynamical systems theory is an area of mathematics used to describe the behavior of the complex dynamical systems , usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems.
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From a physical point of view, continuous dynamical systems is a generalization of classical mechanics , a generalization where the equations of motion are postulated directly and are not constrained to be Euler—Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set , one gets dynamic equations on time scales.
Some situations may also be modeled by mixed operators, such as differential-difference equations. 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. This field of study is also called just dynamical systems , mathematical dynamical systems theory or the mathematical theory of dynamical systems. 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?
An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't 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.
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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. 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 only one future state follows from the current state or stochastic the evolution of the state is subject to random shocks.
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. 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.
In human development , dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. According to this model, stage transitions between age intervals represent self-organization processes at multiple levels e.
For example, at the stage transition from adolescence to young adulthood , and after reaching the critical point of 18 years of age young adulthood , a peak in testosterone is observed in males  and the period of optimal fertility begins in females.
Giovanni Gallavotti | Sapienza Università di Roma - nodehehoosi.ga
These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern.
This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA  and self-organizing properties of the Fibonacci numbers that converge on the golden ratio. 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. In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems.
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. Complexity and self-organization in biology and physiology Workshop organized by: P. Paradisi and R. Sociophysics and Econophysics Workshop organized by: M. Bertotti and V. Special Sessions A1. Talbot and C.
Kinetic Theory and its applications Special Session organized by: G. Palasanzas and A. Information Geometry Special Session organized by: D.
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