Exercise2

We will consider matrices that have the form \(A=PDP^{-1}\) where

\begin{equation*} D = \mattwo p00{\frac12}, P = \mattwo 2{-2}11 \end{equation*}

where \(p\) is a parameter that we will vary. Sketch phase portraits for \(D\) and \(A\) below when

  1. \(p=\frac12\text{.}\)

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  2. \(p=1\text{.}\)

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  3. \(p=2\text{.}\)

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  4. For the different values of \(p\text{,}\) determine which types of dynamical system results. For what range of \(p\) values do we have an attractor? For what range of \(p\) values do we have a saddle? For what value does the transition between the two types occur?

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