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Magazine Problem 1
Consider the following autonomous vector field on the plane: $$ \begin{array}{l} \dot{x}=x^{2} y-x^{3}, \\ \dot{y}=-y+x^{3}, \quad(x, y) \in \mathbb{R}^{2} . \end{array} $$ Determine the stability of \((x, y)=(0,0)\) using center manifold theory7.
Problem 2
Consider the following autonomous vector field on the plane: $$ \begin{array}{l} \dot{x}=x^{2} \\ \dot{y}=-y+x^{2}, \quad(x, y) \in \mathbb{R}^{2} . \end{array} $$ 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)?
Problem 3
Consider the following autonomous vector field on the plane: $$ \begin{array}{l} \dot{x}=-x+y^{2} \\ \dot{y}=-2 x^{2}+2 x y^{2}, \quad(x, y) \in \mathbb{R}^{2} \end{array} $$ 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.
Problem 4
Consider the following autonomous vector field on \(\mathbb{R}^{3}\) : $$ \begin{array}{l} \dot{x}=y \\ \dot{y}=-x-x^{2} y \\ \dot{z}=-z+x z^{2}, \quad(x, y, z) \in \mathbb{R}^{3} . \end{array} $$ Determine the stability of \((x, y, z)=(0,0,0)\) using center manifold theory \(^{8}\).
Problem 5
Consider the following autonomous vector field on \(\mathbb{R}^{3}\) : $$ \begin{array}{l} \dot{x}=y \\ \dot{y}=-x-x^{2} y+z x y \\ \dot{z}=-z+x z^{2}, \quad(x, y, z) \in \mathbb{R}^{3} . \end{array} $$ Determine the stability of \((x, y, z)=(0,0,0)\) using center manifold theory.
Problem 6
Consider the following autonomous vector field on \(\mathbb{R}^{3}\) : $$ \begin{array}{l} \dot{x}=y \\ \dot{y}=-x+z y^{2}, \\ \dot{z}=-z+x z^{2}, \quad(x, y, z) \in \mathbb{R}^{3} . \end{array} $$ Determine the stability of \((x, y, z)=(0,0,0)\) using center manifold theory.
