Foci and centres Problem set A

A system is \(\dot x=2x - y,\ \dot y=5x\).

1. (a) Find the eigenvalues of the system.
2. (b) State the type of equilibrium at the origin.

The system \(\dot x=x + 6y,\ \dot y=-3x - 2y\) has complex eigenvalues.

1. (a) Find \(\dot x\) at \((0,1)\) and say whether the curves go clockwise or anticlockwise.
2. (b) Is x increasing or decreasing at \((7,-2)\)?
3. (c) Is y increasing or decreasing at \((7,-2)\)?

For each pair of eigenvalues, say what kind of equilibrium the origin is, and describe the curves.

1. (a) \(\lambda=2\pm 5i\)
2. (b) \(\lambda=-4\pm i\)
3. (c) \(\lambda=\pm 3i\)

A mass on a spring is slowed by friction. Its displacement from rest is x cm and its velocity is y cm/s at time t seconds, with \(\dot x=y,\ \dot y=-10x-2y\).

1. (a) Find the eigenvalues.
2. (b) Say what kind of equilibrium the origin is, and what this means for the motion of the mass.
3. (c) Find \(\dot x\) at \((0,1)\) and state the direction of the curves.

The deviations, in hundreds, of the numbers of hares (x) and lynx (y) from their usual levels are modelled by \(\dot x=x-4y,\ \dot y=x\), with time in years.

1. (a) Find the eigenvalues, to 3 significant figures.
2. (b) Is x increasing or decreasing when \(x=2\) and \(y=1\)?
3. (c) Say what kind of equilibrium the origin is, and comment on the model in the long term.

A system is \(\dot x=kx-5y,\ \dot y=2x - 3y\), where k is a constant. The origin is a neutral focus.

1. (a) Find the value of k.
2. (b) Find the eigenvalues.
3. (c) State the direction of the curves.

A system is \(\dot x=-kx+4y,\ \dot y=-x-ky\), where k is a constant.

1. (a) Show that the eigenvalues are \(\lambda=-k\pm2i\).
2. (b) Say what kind of focus the origin is when \(k=1.5\), when \(k=0\) and when \(k=-1.5\).
3. (c) For \(k=1.5\), state the direction of the curves.