### Author Topic: Real Repeated Eigenvalue  (Read 1674 times)

#### Monika Dydynski

• Full Member
•   • Posts: 26
• Karma: 30 ##### Real Repeated Eigenvalue
« on: December 11, 2018, 03:37:44 PM »
Has anyone encountered an example in which a matrix $A$ has two independent eigenvectors with eigenvalue $\lambda$, and the phase portrait would therefore be an unstable or stable proper node?

If so, please share! If it's in the textbook, a page number is fine!

#### Victor Ivrii ##### Re: Real Repeated Eigenvalue
« Reply #1 on: December 11, 2018, 07:22:53 PM »
What do you want? A linear system? It is boooring (and cannot be any other way)
\begin{aligned} &x' = x,\\ &y' =y \end{aligned}
Nonlinear system? Try this
\begin{aligned} &x' = x-.1y(x^2+y^2),\\ &y' = y+.1x(x^2+y^2) \end{aligned}

The variation appear as this eigenvalue is $0$. But then linear system is simply trivial
\begin{aligned} &x' = 0,\\ &y' =0 \end{aligned}
but non-linear could be entertaining
\begin{aligned} &x' = (x-y)(x^2+y^2),\\ &y' =(x+y)(x^2+y^2) \end{aligned}
or
\begin{aligned} &x' = xy,\\ &y' =(x+y)(x^2+y^2) \end{aligned}
however nothing can be derived from linearization.
« Last Edit: December 11, 2018, 07:26:37 PM by Victor Ivrii »