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

$\frac{d}{dt} \mathbf{x} = \mathbf{A} \cdot \mathbf{x}$

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

$\frac{d}{dt} \mathbf{x} = \mathbf{A} \cdot \mathbf{r}_{k} = \lambda_{k} \mathbf{r}_{k}$

where $λ k$ 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 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, $λ n$ , to each other may be used to determine the stability of the dynamical system

$\frac{d}{dt} \mathbf{x} = \mathbf{A} \mathbf{x}.$

For a 2-dimensional system, the characteristic polynomial is of the form $λ 2 - τλ + Δ = 0$ 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 $Δ = λ 1 λ 2$ and $τ = λ 1 + λ 2$ . 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).

Categories: Systems | Dynamical systems

Linear dynamical system

From Wikipedia, the free encyclopedia

[edit] Solution of linear dynamical systems

[edit] Classification in two dimensions

[edit] See also

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