10.E: Center Manifold Theory (Exericses)
- Page ID
- 24194
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Consider the following autonomous vector field on the plane:
\(\dot{x} = x^{2}y-x^3\),
\(\dot{y} = y+x^3, (x, y) \in \mathbb{R}^2\).
Determine the stability of (x, y) = (0, 0) using center manifold theory.
Exercise \(\PageIndex{2}\)
Consider the following autonomous vector field on the plane:
\(\dot{x} = x^2\),
\(\dot{y} = y+x^2, (x, y) \in \mathbb{R}^2\).
Determine the stability of (x, y) = (0, 0) using center manifold theory. Does the fact that solutions of \(\dot{x} = x^2\) "blow up in finite time" influence your answer (why or why not)?
Exercise \(\PageIndex{3}\)
Consider the following autonomous vector field on the plane:
\(\dot{x} = x+y^2\),
\(\dot{y} = -2x^2+2xy^2, (x, y) \in \mathbb{R}^2\).
Show that \(y = x^2\) is an invariant manifold. Show that there is a trajectory connecting (0, 0) to (1, 1), i.e. a heteroclinic trajectory.
Exercise \(\PageIndex{4}\)
Consider the following autonomous vector field on \(\mathbb{R}^3\):
\(\dot{x} = y\),
\(\dot{y} = -x-x^{2}y\),
\(\dot{z} = -z+xz^2, (x, y, z) \in \mathbb{R}^3\)
Determine the stability of (x, y, z) = (0, 0, 0) using center manifold theory.
Exercise \(\PageIndex{5}\)
Consider the following autonomous vector field on \(\mathbb{R}^3\):
\(\dot{x} = y\),
\(\dot{y} = -x-x^{2}y+zxy\),
\(\dot{z} = -z+xz^2, (x, y, z) \in \mathbb{R}^3\).
Determine the stability of (x, y, z) = (0, 0, 0) using center manifold theory.
Exercise \(\PageIndex{6}\)
Consider the following autonomous vector field on \(\mathbb{R}^3\):
\(\dot{x} = y\),
\(\dot{y} = -x+zy^2\),
\(\dot{z} = -z+xz^2, (x, y, z) \in \mathbb{R}^3\).
Determine the stability of (x, y, z) = (0, 0, 0) using center manifold theory.