Dynamical system

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A dynamical system is a deterministic process in which a variable's value changes over time according to a rule that is defined in terms of the variable's current value.

Table of contents

1 Types of dynamical systems

2 Linear and non-linear systems

3 Dynamical systems and chaos theory

4 Examples of dynamical systems

Types of dynamical systems

A dynamical system is called discrete if time is measured in discrete steps; these are modeled as recursive relations as for instance in the logistic map

x_{t + 1} = ax_t(1 - x_t)

where t denotes the discrete time steps and x is the variable changing over time. If time is measured continuously, the resulting continuous dynamical systems are expressed as ordinary differential equations, for instance

$\frac{dx}{dt} = ax (1-x)$

where x is the variable that changes with time t.

The changing variable x is often a real number, but can also be a vector in R^k.

Linear and non-linear systems

We distinguish between linear dynamical systems and nonlinear dynamical systems. In linear systems, the right-hand-side of the equation is an expression which depends linearly on x, as in

x_{n + 1} = 3x_n

If two solutions to a linear system are given, then their sum is also a solution ("superposition principle"). In general, the solutions form a vector space, which allows to use linear algebra and simplifies the analysis significantly. For linear continuous systems, the method of Laplace transform can also be used to transform the differential equation into an algebraic equation.

The two examples given earlier are nonlinear systems. These are much harder to analyze and often exhibit a phenomenon known as chaos which marks complete unpredictability; see also nonlinearity.

Dynamical systems and chaos theory

Simple nonlinear dynamical systems, and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random (Remember that we are speaking of completely deterministic systems!). This unpredictable behaviour has been called chaos. The branch of dynamical systems which deals with the clean definition and investigation of chaos is called chaos theory.

This branch of mathematics deals 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 systems; these are values of the variable which won't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it will converge towards the fixed point.

Similarly, one is interested in periodic points, states of the system which repeat themselves after several timesteps. Periodic points can also be attractive. Sarkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Note that the chaotic behaviour of complicated systems is not the issue. Meteorology has been known for years to involve complicated - even chaotic - behaviour. Chaos theory has been so surprising because you can find chaos within almost trivial systems. The logistic map is only a second-degree polinomial. The Horseshoe map is piecewise linear.

Examples of dynamical systems

Logistic map
Double pendulum
Horseshoe map is an example of a chaotic piecewise linear map

Topics in mathematics related to change