Foci and centres Problem set B Worked solutions

A system is \(\dot x=x - 5y,\ \dot y=2x + 3y\).

1. (a) Find the eigenvalues of the system.

   • \(\det\begin{pmatrix}1-\lambda&-5\\2&3-\lambda\end{pmatrix}=0\)
   • \(\lambda^2 - 4\lambda + 13 = 0\)
   • \(\lambda=2\pm 3i\)

2. (b) State the type of equilibrium at the origin.

   • Re(\(\lambda\)) = \(2\): unstable focus
   • The origin is an unstable focus

The system \(\dot x=-x - 8y,\ \dot y=2x - y\) has complex eigenvalues.

1. (a) Find \(\dot x\) at \((0,1)\) and say whether the curves go clockwise or anticlockwise.

   • \(\dot x=-1(0)-8(1)=-8\)
   • x is decreasing on the positive y-axis, so the curves go anticlockwise

2. (b) Is x increasing or decreasing at \((-3,2)\)?

   • \(\dot x=-1(-3)+(-8)(2)=-13\)
   • \(\dot x<0\), so x is decreasing

3. (c) Is y increasing or decreasing at \((-3,2)\)?

   • \(\dot y=(2)(-3)+(-1)(2)=-8\)
   • \(\dot y<0\), so y is decreasing

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

1. (a) \(\lambda=3\pm i\)

   • \(p=3\) is positive
   • Unstable focus: spirals out

2. (b) \(\lambda=-1\pm 6i\)

   • \(p=-1\) is negative
   • Stable focus: spirals in

3. (c) \(\lambda=\pm 4i\)

   • \(p=0\) is zero
   • Neutral focus: closed ovals

A child's swing slows down because of air resistance. Its angle from the vertical is x radians and its angular speed is y radians per second at time t seconds, with \(\dot x=y,\ \dot y=-6.25x-3y\).

1. (a) Find the eigenvalues.

   • \(\det\begin{pmatrix}-\lambda&1\\-6.25&-3-\lambda\end{pmatrix}=0\)
   • \(\lambda^2 + 3\lambda + 6.25 = 0\)
   • \(\lambda=-1.5\pm 2i\)

2. (b) Say what kind of equilibrium the origin is, and what this means for the motion of the swing.

   • Re(\(\lambda\)) = \(-1.5\): stable focus
   • A stable focus: x and y both oscillate and tend to 0, so the swings die away and the swing stops

3. (c) Find \(\dot x\) at \((0,1)\) and state the direction of the curves.

   • \(\dot x=0(0)+1(1)=1\)
   • \(\dot x>0\): clockwise

The deviations of the price (x, in euros) and of the quantity supplied (y, in thousands) of a crop from their long-run levels are modelled by \(\dot x=x-6y,\ \dot y=x\), with time in months.

1. (a) Find the eigenvalues, to 3 significant figures.

   • \(\det\begin{pmatrix}1-\lambda&-6\\1&-\lambda\end{pmatrix}=0\)
   • \(\lambda^2 - \lambda + 6 = 0\)
   • \(\lambda=0.5\pm\frac{\sqrt{23}}{2}\,i=0.5\pm2.40i\)

2. (b) Is x increasing or decreasing when \(x=9\) and \(y=1\)?

   • \(\dot x=1(9)+(-6)(1)=3\)
   • \(\dot x>0\): x is increasing

3. (c) Say what kind of equilibrium the origin is, and comment on the model in the long term.

   • Re(\(\lambda\)) = \(0.5\): unstable focus
   • An unstable focus: the oscillations grow without limit, which cannot happen for real prices, so the model fails in the long term

A system is \(\dot x=kx-4y,\ \dot y=5x - 2y\), where k is a constant. The origin is a neutral focus.

1. (a) Find the value of k.

   • A neutral focus needs \(p=0\), so the trace is zero: \(k+(-2)=0\)
   • \(k=2\)

2. (b) Find the eigenvalues.

   • \(\det=(2)(-2)-(-4)(5)=16\), so \(\lambda^2+16=0\)
   • \(\lambda=\pm4i\)

3. (c) State the direction of the curves.

   • \(\dot x=-4\) at \((0,1)\)
   • Anticlockwise

A system is \(\dot x=-kx-9y,\ \dot y=x-ky\), where k is a constant.

1. (a) Show that the eigenvalues are \(\lambda=-k\pm3i\).

   • \(\det\begin{pmatrix}-k-\lambda&-9\\1&-k-\lambda\end{pmatrix}=0\)
   • \((-k-\lambda)^2+9=0\)
   • \(\lambda=-k\pm3i\)

2. (b) Say what kind of focus the origin is when \(k=2.5\), when \(k=0\) and when \(k=-2.5\).

   • The real part is \(p=-k\)
   • \(k=2.5\): \(p=-2.5\), stable focus
   • \(k=0\): \(p=0\), neutral focus
   • \(k=-2.5\): \(p=2.5\), unstable focus

3. (c) For \(k=2.5\), state the direction of the curves.

   • \(\dot x=-9\) at \((0,1)\)
   • Anticlockwise