Dynamical Systems
Introduction
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.
Overview
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.
Types of Dynamical Systems
Dynamical systems can be divided into various different types, each with different properties. The main types of dynamical systems are:
Discrete Dynamical Systems
A discrete dynamical system is a system in which the state evolves over discrete points in time. This type of system can be modelled mathematically using difference equations.
Continuous Dynamical Systems
A continuous dynamical system is a system in which the state evolves over continuous points in time. This type of system can be modelled mathemically using differential equations.
Deterministic Dynamical Systems
A deterministic dynamical system is a system in which the state evolves in a deterministic manner for a given time interval. This means that there is only one future state that follows from the current state.
Stochastic Dynamical Systems
A stochastic dynamical system is a system in which the state evolves in a probabilistic manner. This means that there are multiple future states that can follow from the current state, each with a certain probability.
Properties of Dynamical Systems
There are several key properties of dynamical systems that are of interest to mathematicians and physicists. These include:
Stability
The stability of a dynamical system refers to the system's ability to return to a steady state after a small disturbance.
Attractors
An attractor is a set of numerical values toward which a system tends to evolve.
Chaos
Chaos refers to the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect.
Applications of Dynamical Systems
Dynamical systems theory is a broad and active research area with connections to most parts of mathematics. Dynamical systems are also used in many scientific disciplines, including physics, chemistry, biology, economics and engineering. For example, in physics, dynamical systems theory is used to describe the motion of the planets, the behavior of the weather, and the behavior of electrical circuits.