10.E: Center Manifold Theory (Exericses)

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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.