8.2: Problem Set
- Page ID
- 24180
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
Exercise \(\PageIndex{1}\)
Consider the following autonomous vector fields on the plane depending on a scalar parameter \(\mu\). Verify that each vector field has a fixed point at (x, y) = (0, 0) for \(\mu = 0\). Determine the linearized stability of this fixed point. Determine the nature (i.e. stability and number) of the fixed points for μ in a neighborhood of zero. (In other words, carry out a bifurcation analysis.) Sketch the flow in a neighborhood of each fixed point for values of μ corresponding to changes in stability and/or numbers of fixed points.
- \(\dot{x} = \mu+10x^2\),
\(\dot{y} = x-5y\).
- \(\dot{x} = \mu x+10x^2\),
\(\dot{y} = x-2y\).
- \(\dot{x} = \mu x+x^5\),
\(\dot{y} = y\).