Interpreting linear dynamical systems

As already stated previously, linear dynamical systems are especially simple to analyze. Since we need this procedure for the rest of this chapter, we briefly discuss some different approaches of analyzing the matrix of a linear and autonomous dynamical system's matrix A [83].

Eigenvalues and eigenvectors

Continuous dynamical systems ( $\dot{\mathbf{x}}=\mathbf{A}\cdot\mathbf{x}$) as well as discrete systems ( $\mathbf{x}_{n+1}=\mathbf{A}\cdot\mathbf{x}_n$) that are autonomous and linear can be entirely analyzed by investigating the matrix A and its characteristics. One possibility is to compute A's eigenvalues and its eigenvectors from Eqs. 3.8 and 3.9, respectively [72,83]. The knowledge of A's eigenvalues and eigenvectors is always sufficient to describe the qualitative behavior of a linear system.

  

$$\mathrm{det}\,(\mathbf{A}-\lambda_i\cdot\mathbf{I})$$ = 0 (3.7)

$$\mathbf{A}\cdot\mathbf{e}_i \lambda_i\cdot\mathbf{e}_i$$ = (3.8)

Interpreting matrix A as a (linear) transformation the eigenvectors of A specify exactly those lines (containing e_i), which are invariant under the transformation. The way such a line itself is transformed is given by the corresponding eigenvalue $\lambda_i$.

The interpretation of the eigenvalues $\lambda_i$ - they can be either real or complex - is different for continuous and discrete dynamical systems, because a continuous system is specified by the change of the current state, whereas a discrete dynamical system is specified by giving the next state of the system (see Tab. 3.1).

 

Table 3.1: Interpreting the eigenvalues of a linear system.

	continuous case	discrete case
convergence	$\mathrm{Re}\,\lambda_i < 0$	$\left\vert\lambda_i\right\vert < 1$
divergence	$\mathrm{Re}\,\lambda_i > 0$	$\left\vert\lambda_i\right\vert > 1$
rotation	$\mathrm{Im}\,\lambda_{j,k} \neq 0$	$\mathrm{Im}\,\lambda_{j,k} \neq 0$

 

Convergence, divergence, and rotation are to be interpreted relatively to the origin of the coordinate system. Note, that a critical point of a continuous dynamical system is called hyperbolic, if its eigenvalues do not lie on the imaginary axis ( $\mathrm{Re}\,\lambda_i \neq 0$). Critical points of discrete dynamical systems are hyperbolic, if $\left\vert\lambda_i\right\vert \neq 1$ for all eigenvalues.

Decomposing matrix A

Another possibility of analyzing matrix A of a linear and autonomous system is by decomposing it into a symmetric matrix A⁺ and an asymmetric matrix A^- as follows [19]:

$$\mathbf{A}^{+} = \frac{\left(\mathbf{A} + \mathbf{A}^\mathrm... ...} = \frac{\left(\mathbf{A} - \mathbf{A}^\mathrm{T}\right)}{2}$$ (3.9)

The elements of A⁺ and A^- can be interpreted rather straightforwardly [75]:

$$\mathbf{A}^{+} = \left(\begin{array}{ccc} d_x & \cdot & \cdo... ...rm{and } (d_x+d_y+d_z) = \mathrm{div}\,\mathbf{f}(\mathbf{x})$$ (3.10)

The elements of A⁺ marked with `$\cdot$' built up the shear strain portion of this linear system.

$$\mathbf{A}^{-} = \frac{1}{2}\left(\begin{array}{ccc} 0 & -r_... ..._z \end{array}\right) = \mathrm{rot}\,\mathbf{f}(\mathbf{x})$$ (3.11)

Analysis in the local coordinate system

A third possibility of linear system analysis is especially useful while investigating a flow's Jacobian J. It can be transformed into the local Frenét-frame at some point of a trajectory ( $\mathbf{J}\to\mathbf{J}^\mathrm{local}$). Then the elements of J^local as given in Eq. 3.13 allow a detailed characterization of the underlying flow [19].

 

$$\left(\begin{array}{ccc} \hat{a} & \hat{s} & \hat{s} \\ \hat{c} ... ...d} & \hat{d},\hat{t} \\ \hat{c} & \hat{d},\hat{t} & \hat{d} \end{array}\right)$$ J^local = (3.12)

$$\hat{a}, \hat{c}, \hat{d}, \hat{s}, \hat{t} \ldots{}$$ markups of J^local's elements.

Elements of matrix J^local that are marked with `$\hat{a}$', `$\hat{s}$', or `$\hat{c}$' specify changes of the flow that are parallel to f(x). The element marked with `$\hat{a}$' gives the acceleration of the flow, whereas the elements marked with `$\hat{s}$' give the shear strain at this state of the system. Elements that are marked with `$\hat{c}$' give the curvature of the flow.

Remaining elements of matrix J^local, that are marked with either `$\hat{d}$' alone or `$\hat{d}$' and `$\hat{t}$', specify the changes of the flow that are perpendicular to f(x). Splitting the bottom-right 2$\times$2-matrix into a symmetric and an asymmetric one gives the divergence (by the elements marked with `$\hat{d}$') and the torsion (by the elements marked with `$\hat{t}$') of the flow.

mailto:helwig@cg.tuwien.ac.at.