14 Facts About Dynamical systems

However, some Dynamical systems are stochastic, in that random events affect the evolution of the state variables.

Study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine.

Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.

Simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories.

Dynamical systems created the modern theory of the stability of a dynamical system.

Dynamical systems outlined a research program carried out by many others.

Dynamical systems are usually defined over a single independent variable, thought of as time.

Such Dynamical systems are useful for modeling, for example, image processing.

Some Dynamical systems have a natural measure, such as the Liouville measure in Hamiltonian Dynamical systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system.

Concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems.

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified.

For nonlinear Dynamical systems this is one of the conditions for chaotic behavior.

Qualitative properties of dynamical systems do not change under a smooth change of coordinates : a singular point of the vector field will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop.

Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems.