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MAT244--2018F => MAT244--Lectures & Home Assignments => Topic started by: Monika Dydynski on December 11, 2018, 03:37:44 PM

Title: Real Repeated Eigenvalue
Post by: Monika Dydynski 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!
Title: Re: Real Repeated Eigenvalue
Post by: Victor Ivrii 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.