Foci and centres Problem set A One question at a time

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

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

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

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

   • Re(\(\lambda\)) = \(1\): unstable focus
   • The origin is an unstable focus

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.

   • \(\dot x=1(0)+6(1)=6\)
   • x is increasing on the positive y-axis, so the curves go clockwise

2. (b) Is x increasing or decreasing at \((7,-2)\)?

   • \(\dot x=1(7)+(6)(-2)=-5\)
   • \(\dot x<0\), so x is decreasing

3. (c) Is y increasing or decreasing at \((7,-2)\)?

   • \(\dot y=(-3)(7)+(-2)(-2)=-17\)
   • \(\dot y<0\), so y is decreasing

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

1. (a) \(\lambda=2\pm 5i\)

   • \(p=2\) is positive
   • Unstable focus: spirals out

2. (b) \(\lambda=-4\pm i\)

   • \(p=-4\) is negative
   • Stable focus: spirals in

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

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

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.

   • \(\det\begin{pmatrix}-\lambda&1\\-10&-2-\lambda\end{pmatrix}=0\)
   • \(\lambda^2 + 2\lambda + 10 = 0\)
   • \(\lambda=-1\pm 3i\)

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

   • Re(\(\lambda\)) = \(-1\): stable focus
   • A stable focus: x and y both oscillate and tend to 0, so the swings die away and the mass comes to rest

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, 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.

   • \(\det\begin{pmatrix}1-\lambda&-4\\1&-\lambda\end{pmatrix}=0\)
   • \(\lambda^2 - \lambda + 4 = 0\)
   • \(\lambda=0.5\pm\frac{\sqrt{15}}{2}\,i=0.5\pm1.94i\)

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

   • \(\dot x=1(2)+(-4)(1)=-2\)
   • \(\dot x<0\): x is decreasing

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 populations, so the model fails 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.

   • A neutral focus needs \(p=0\), so the trace is zero: \(k+(-3)=0\)
   • \(k=3\)

2. (b) Find the eigenvalues.

   • \(\det=(3)(-3)-(-5)(2)=1\), so \(\lambda^2+1=0\)
   • \(\lambda=\pm i\)

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

   • \(\dot x=-5\) at \((0,1)\)
   • Anticlockwise

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

1. (a) Show that the eigenvalues are \(\lambda=-k\pm2i\).

   • \(\det\begin{pmatrix}-k-\lambda&4\\-1&-k-\lambda\end{pmatrix}=0\)
   • \((-k-\lambda)^2+4=0\)
   • \(\lambda=-k\pm2i\)

2. (b) Say what kind of focus the origin is when \(k=1.5\), when \(k=0\) and when \(k=-1.5\).

   • The real part is \(p=-k\)
   • \(k=1.5\): \(p=-1.5\), stable focus
   • \(k=0\): \(p=0\), neutral focus
   • \(k=-1.5\): \(p=1.5\), unstable focus

3. (c) For \(k=1.5\), state the direction of the curves.

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