Linear dynamical systems

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).

Flows

For a flow, the vector field Φ(x) is a linear function of the position in the phase space, that is,

with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b ≠ 0 with A = 0 is just a straight line in the direction of b:

When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x₀ = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x₀,

When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.
The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.

Geometrical definition

A dynamical system is the tuple , with a manifold (locally a Banach space or Euclidean space), the domain for time (non-negative reals, the integers, ...) and f an evolution rule t→f ^t (with ) such that f ^t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain .

Measure theoretical definition

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X,Σ,μ,τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X,Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X,Σ,μ) is a probability space. A map is said to be Σ-measurable if and only if, for every , one has . A map τ is said to preserve the measure if and only if, for every , one has μ(τ ^{− 1}σ) = μ(σ). Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X,Σ,μ,τ), for such a τ, is then defined to be a dynamical system.
The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates for integer n are studied. For continuous dynamical systems, the map τ is understood to be finite time evolution map and the construction is more complicated.