System analysis near trajectories

In the following we propose another approach to analyze a dynamical system's behavior. It is somewhat similar to the method by de Leeuw and van Wijk [19], as the dynamical system is also transformed into the Frenét-Frame $\Phi$ of a point on the trajectory. Contrary to their approach we use the analysis by eigenvalues and eigenvectors to interpret this transformed Jacobian matrix. Expressing a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in terms of $\Phi$ one gets

 

$$\dot{\mathbf{u}} (\mathbf{g2l}\circ\mathbf{f}\circ\mathbf{l2g})(\mathbf{u}) = \tilde{\mathbf{f}}(\mathbf{u})$$ = (3.15)

$$\ldots{} \textrm{a state of the system in terms of }\Phi$$ u

$$\ldots{} \textrm{transformation from the global coordinate system into }\Phi$$ g2l

$$\ldots{} \textrm{transformation from }\Phi\textrm{ into the global coordinate system}$$ l2g

Near the point of interest p (represented in the global coordinate system) a state of the system can be written as $\mathbf{u}=0\!+\!\mathbf{d}$ in terms of the local coordinate system. Note, that p represented in terms of $\Phi$ is 0. Using a Taylor expansion of $\tilde{\mathbf{f}}(\mathbf{u})$ up to first-order terms, we get

 

$$\dot{\mathbf{u}} \tilde{\mathbf{f}}(\mathbf{u}) = \tilde{\mathbf{f}}(0+\mathbf{d})... ...\nabla\tilde{\mathbf{f}}\right\vert _0\cdot\mathbf{d} \quad\quad\quad\textrm{~}$$ = (3.16)

$$\phi_1 \ldots{} \textrm{unit-vector in terms of }\Phi\textrm{, colinear to the axis}$$ corresponding to the trajectory's tangent

$$\lambda \ldots{}$$ length of f(p)

Note, that $\tilde{\mathbf{f}}(0)$ can be written as $\lambda\!\cdot\!\phi_1$, since the first axis of the local coordinate system (Frenét-Frame) is specified to be colinear to the flow. Transforming the left-most side of Eq. 3.17 by using $\mathbf{u}=0\!+\!\mathbf{d}$ we get a linearized system for small perturbations of p (in terms of $\Phi$), because $\mathrm{d}0/\mathrm{d}t=\tilde{\mathbf{f}}(0)=\lambda\cdot\phi_1$.

$$\dot{\mathbf{u}} = \mathrm{d}(0+\mathbf{d})/\mathrm{d}t = \do... ... = \left.\nabla\tilde{\mathbf{f}}\right\vert _0\cdot\mathbf{d}$$ (3.17)

Now perturbations that are especially useful to analyze are investigated. The elements of d can be separated into a scalar d[1] and a vector $\mathbf{d}[2\ldots]$ that is of one dimension less than d. d[1] is assumed to be 0, since perturbations of p that are not perpendicular to the trajectory's tangent make no sense at all - a state of the system that is represented as a perturbation of p with a component $\mathbf{d}[1]\!\ne\!0$ usually can be more accurately expressed as a (perpendicular) perturbation of another point on exactly the same trajectory. Thus $\dot{\mathbf{d}}$ does not depend on the first row of matrix $\nabla\tilde{\mathbf{f}}\vert _0$. The remaining elements of $\tilde{\mathbf{f}}$'s Jacobian $\nabla\tilde{\mathbf{f}}\vert _0$ can be decomposed into the first line $\nabla\tilde{\mathbf{f}}\vert _0[1,2\ldots]$ and the lower-right sub-matrix $\nabla\tilde{\mathbf{f}}\vert _0[2\ldots,2\ldots]$.

Decomposing $\dot{\mathbf{d}}$ similar to d yields a part parallel to the trajectory's tangent (scalar $\dot{\mathbf{d}}[1]$) and a part perpendicular to a (sub-vector $\dot{\mathbf{d}}[2\ldots]$):

$$\begin{array}{rcrcl} \dot{\mathbf{d}}[1] &=& \left.\nabla\t... ...2\ldots] &=& \mathbf{A}\cdot{\mathbf{d}}[2\ldots] \end{array}$$ (3.18)

We now have a locally linearized system ( $\dot{\mathbf{d}}[2\ldots]=\mathbf{A}\cdot{\mathbf{d}[2\ldots]}$) that describes the evolution of variations orthogonal to the flow. The system has one dimension less than the original system. Matrix A can be analyzed as already shown for continuous systems at the neighborhood of critical points. But we must be careful with the interpretation of this analysis, because all the results hold for the investigated point p only. For example, if the analysis of matrix A reveals that the system's evolution is convergent (critical point is an attractor) the only thing that can be said is that nearby trajectories are locally attracted by the trajectory at the specific location chosen. To detect convergent, divergent, or saddle regions of a trajectory it must be shown that the structural characteristics of matrix A are persistent for a certain region of the trajectory. This might be not simple analytically, but can be done approximately by numerical simulation.

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