2.1: Problem Set

EXERCISE \(\PageIndex{1}\)

Consider an autonomous vector field on the plane having a hyperbolic fixed point with a homoclinic orbit connecting the hyperbolic fixed point, as illustrated in Fig. 1. We assume that existence and uniqueness of solutions holds. Can a trajectory starting at any point on the homoclinic orbit reach the hyperbolic fixed point in a finite time? (You must justify your answer.)

EXERCISE \(\PageIndex{2}\)

Can an autonomous vector field on \(\mathbb{R}\) that has no equilibrium points have periodic orbits? We assume that existence and uniqueness of solutions holds.(You must justify your answer.)

EXERCISE \(\PageIndex{3}\)

Can a nonautonomous vector field on \(\mathbb{R}\) that has no equilibrium points have periodic orbits? We assume that existence and uniqueness of solutions holds.(You must justify your answer.)

EXERCISE \(\PageIndex{4}\)

Can an autonomous vector field on the circle that has no equilibrium points have periodic orbits? We assume that existence and uniqueness of solutions holds. (You must justify your answer.)

EXERCISE \(\PageIndex{5}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = -\omega y\),

\(\dot{y} = \omega x\), \((x, y) \in \mathbb{R}^2\),

where \(\omega > 0\)

• Show that the flow generated by this vector field is given by:

  \[\begin{pmatrix} {x(t)}\\{y(t)} \end{pmatrix} = \begin{pmatrix} {cos \omega t}&{-sin \omega t}\\ {sin \omega t}&{cos \omega t} \end{pmatrix} \begin{pmatrix} {x_{0}}\\ {y_{0}} \end{pmatrix} \nonumber\]

• Show that the flow obeys the time shift property.
• Give the initial condition for the time shifted flow.

EXERCISE \(\PageIndex{6}\)

Consider the following autonomous vector field on the plane:

\(\dot{x} = \lambda y\),

\(\dot{y} = \lambda x\), \((x, y) \in \mathbb{R}^2\),

• Show that the flow generated by this vector field is given by:

  \[\begin{pmatrix} {x(t)}\\{y(t)} \end{pmatrix} = \begin{pmatrix} {cosh \lambda t}&{sinh \lambda t}\\ {sinh \lambda t}&{cosh \lambda t} \end{pmatrix} \begin{pmatrix} {x_{0}}\\ {y_{0}} \end{pmatrix} \nonumber\]

• Show that the flow obeys the time shift property.
• Give the initial condition for the time shifted flow.

EXERCISE \(\PageIndex{7}\)

Show that the time shift property for autonomous vector fields implies that trajectories cannot ''cross each other'', i.e. intersect, in phase space.

EXERCISE \(\PageIndex{8}\)

Show that the union of two invariant sets is an invariant set.

EXERCISE \(\PageIndex{9}\)

Show that the intersection of two invariant sets is an invariant set.

EXERCISE \(\PageIndex{10}\)

Show that the complement of a positive invariant set is a negative invariant set.