Linear dynamical systems are a special type of dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical 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 springtime in a...

 where the equation governing the system's evolution is linear

Linear

In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general linear systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.

Introduction

In a linear dynamical system, the variation of a state vector
(an -dimensional vector

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 denoted ) equals a constant matrix
(denoted ) multiplied by
. This variation can take two forms: either
as a flow

Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...

, in which varies
continuously with time


or as a mapping, in which
varies in discrete

Discrete time

Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...

 steps


These equations are linear in the following sense: if
and
are two valid solutions, then so is any linear combination
of the two solutions, e.g.,

where and
are any two scalars

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

. The matrix
need not be symmetric.

Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.

Solution of linear dynamical systems

If the initial vector
is aligned with a right eigenvector  of
the matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 , the dynamics are simple


where is the corresponding eigenvalue;
the solution of this equation is
as may be confirmed by substitution.

If is diagonalizable

Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

, then any vector in an -dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted ) of the matrix .


Therefore, the general solution for is
a linear combination of the individual solutions for the right
eigenvectors

Similar considerations apply to the discrete mappings.

Classification in two dimensions

The roots of the characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 det(A - λI) are the eigenvalues of A. The sign and relation of these roots, , to each other may be used to determine the stability of the dynamical system
For a 2-dimensional system, the characteristic polynomial is of the form where is the trace

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

 and is the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of A. Thus the two roots are in the form:
Note also that and . Thus if then the eigenvalues are of opposite sign, and the fixed point is a saddle. If then the eigenvalues are of the same sign. Therefore if both are positive and the point is unstable, and if then both are negative and the point is stable. The discriminant

Discriminant

In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).