Dynamical systems theory

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.

Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Some excellent presentations of mathematical dynamic system theory include Beltrami (1998), Luenberger (1979), Padulo & Arbib (1974), and Strogatz (1994).

[2] 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.

In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance and efficiency.

From psychophysiological 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 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.

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.)

[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.

[8][9][10] A variety of neurosymbolic cognitive neuroarchitectures in modern connectionism, considering their mathematical structural core, can be categorized as (nonlinear) dynamical systems.