Question

Determine the nature of the origin (attractor, repeller, or saddle point) for the dynamical system \({{\rm{x}}_{k + 1}} = A{{\rm{x}}_k}\) if A has the properties described in Exercise 2. Find the directions of greatest attraction or repulsion.

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Short answer

The direction of greatest repulsion is the line through the origin and the eigenvector\((1,0, - 3)\), and \(( - 3,3,7)\).

Step-by-step solution

Find eigenvalue and eigenvector 

Consider the dynamic system

\({{\rm{x}}_{k + 1}} = A{{\rm{x}}_k}\)

Where A is \(3 \times 3\) a matrix with eigenvalue \(3,\frac{4}{5}\;{\rm{and}}\;\frac{3}{5}\).

And eigenvectors are \({{\rm{v}}_1} = \left( {\begin{aligned}{}1\\0\\{ - 3}\end{aligned}} \right),{{\rm{v}}_2} = \left( {\begin{aligned}{}2\\1\\{ - 5}\end{aligned}} \right)\) and \({{\rm{v}}_3} = \left( {\begin{aligned}{}{ - 2}\\{ - 5}\\3\end{aligned}} \right)\).

Here, one of the Eigenvalue 3 is greater than 1 in magnitude and an Eigenvalues are \(\frac{4}{5}\;{\rm{and}}\;\frac{3}{5}\) less than 1 in magnitude, therefore origin is a saddle point for the dynamical system.

Find the direction 

The direction of greatest repulsion is the line through the origin and the eigenvector\((1,0, - 3)\), for the eigenvalue 3.

The direction of greatest attraction is the line through the origin and the eigenvector \(( - 3,3,7)\), for the smallest eigenvalue \(\frac{3}{5}\).