Linear dynamical system

From Wikipedia, the free encyclopedia

In a linear dynamical system, the variation of a state vector (an N-dimensional vector denoted \mathbf{x}) equals a constant matrix (denoted \mathbf{A}) multiplied by \mathbf{x}. This variation can take two forms: either as a flow, in which \mathbf{x} varies continuously with time

\frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \cdot \mathbf{x}(t)

or (less commonly) as a mapping, in which \mathbf{x} varies in discrete steps

\mathbf{x}_{m+1} = \mathbf{A} \cdot \mathbf{x}_{m}

These equations are linear in the following sense: if \mathbf{x}(t) and \mathbf{y}(t) are two valid solutions, then so is any linear combination of the two solutions, e.g., \mathbf{z}(t) \ \stackrel{\mathrm{def}}{=}\ \alpha \mathbf{x}(t) + \beta \mathbf{y}(t) where α and β are any two scalars. It is important to note that the matrix \mathbf{A} 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. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.

[edit] Solution of linear dynamical systems

If the initial vector \mathbf{x}_{0} \ \stackrel{\mathrm{def}}{=}\ \mathbf{x}(t=0) is aligned with a right eigenvector \mathbf{r}_{k} of the matrix \mathbf{A}, the dynamics are simple

\frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \cdot \mathbf{r}_{k} = \lambda_{k} \mathbf{r}_{k}

where λk is the corresponding eigenvalue; the solution of this equation is

\mathbf{x}(t) = \mathbf{r}_{k} e^{\lambda_{k} t}

as may be confirmed by substitution.

If \mathbf{A} is diagonalizable, then any vector in an N-dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted \mathbf{l}_{k}) of the matrix \mathbf{A}.

\mathbf{x}_{0} = \sum_{k=1}^{N} \left( \mathbf{l}_{k} \cdot \mathbf{x}_{0} \right) \mathbf{r}_{k}

Therefore, the general solution for \mathbf{x}(t) is a linear combination of the individual solutions for the right eigenvectors

\mathbf{x}(t) = \sum_{k=1}^{n} \left( \mathbf{l}_{k} \cdot \mathbf{x}_{0} \right) \mathbf{r}_{k} e^{\lambda_{k} t}

Similar considerations apply to the discrete mappings.

[edit] 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, λn, to each other may be used to determine the stability of the dynamical system

\frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \mathbf{x}(t).

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:

\lambda_1=\frac{\tau+\sqrt{\tau^2-4\Delta}}{2}
\lambda_2=\frac{\tau-\sqrt{\tau^2-4\Delta}}{2}

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

[edit] See also

Languages