In a linear dynamical system, the variation of a state vector (an N-dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by . This variation can take two forms: either as a flow, in which varies continuously with time Look up vector in Wiktionary, the free dictionary. ... In mathematics, flow refers to the group action of a one-parameter group on a set. ...

or (less commonly) as a mapping, in which varies in discrete steps Discrete time is non-continuous time. ...

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 α and β are any two scalars. It is important to note that the matrix need not be symmetric. In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ...

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. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems. In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...

Solution of linear dynamical systems

If the initial vector is aligned with a right eigenvector of the matrix , the dynamics are simple Look up matrix in Wiktionary, the free dictionary. ...

where λ_k is the corresponding eigenvalue; the solution of this equation is In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

as may be confirmed by substitution.

If A is diagonalizeble, then any vector in an N-dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted ) of the matrix . In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

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 det(A - λI)are the eigenvalues of A. The sign and relation of these roots, λ_n, to each other may be used to determine the stability of the dynamical system In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...

For a 2-dimensional system, the characteristic polynomial is of the form λ² − τλ + Δ = 0 where τ is the trace and Δ is the determinant of A. Thus the two roots are in the form: 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 (the diagonal from the upper left to the lower right) of A, i. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Note also that Δ = λ₁λ₂ and τ = λ₁ + λ₂. Thus if Δ < 0 then the eigenvalues are of opposite sign, and the fixed point is a saddle. If Δ > 0 then the eigenvalues are of the same sign. Therefore if τ > 0 both are positive and the point is unstable, and if τ < 0 then both are negative and the point is stable. The descriminant will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).

See also

• Linear system
• Dynamic systems
• List of dynamical system topics