MAT244--2018F > MAT244--Lectures & Home Assignments

Real Repeated Eigenvalue

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**Monika Dydynski**:

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**:

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.

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