5.17 Phase portraits questions for IB Maths AI

Find the eigenvalues of the matrix $\left(\begin{array}{cc}2& 1\\ 3& 4\end{array}\right)$.

Find an eigenvector for each eigenvalue.

Hence write down the general solution of the system.

Write down the matrix $M$ such that $\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=M\left(\begin{array}{c}x\\ y\end{array}\right)$.

Show that the eigenvalues, $\lambda$, of $M$ satisfy ${\lambda }^2+0.2\lambda +1.01=0$.

Find the eigenvalues of $M$, giving your answers in the form $a+b\mathrm{i}$.

Write the eigenvalue with positive imaginary part in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

State, giving a reason, what your answer to part (c) shows about the shape of the trajectories in the phase portrait.

State, giving a reason, what happens to the populations in the long term.

Find the direction of the trajectory at the point $(x,y)=(1,0)$, and hence state whether the trajectories spiral clockwise or anticlockwise.

A change in the habitat alters the model to ${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=kx-2y$, ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=0.5x+ky$, where $k$ is a constant. The eigenvalues are now $k\pm \mathrm{i}$.

Describe the long-term behaviour of the populations if $k>0$.

At one moment there are 6000 aphids and 100 ladybirds in the greenhouse.

Find the value of $\frac{\mathrm{d}x}{\mathrm{d}t}$ and the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at this moment, and interpret your answers in context.

Consider the case in which there are no ladybirds in the greenhouse.

Write down the differential equation satisfied by $x$.

Hence find the long-term number of aphids when there are no ladybirds, justifying your answer.

Show that $(4,2)$ is an equilibrium point of the system.

Write down the coordinates of the other two equilibrium points.

To investigate the behaviour close to $(4,2)$, let $x=4+p$ and $y=2+q$, where $p$ and $q$ are small. You may assume that terms in $p^2$, $q^2$ and $pq$ can be ignored.

Show that $\frac{\mathrm{d}p}{\mathrm{d}t}\approx -0.4p-1.6q$.

Find a similar approximation for $\frac{\mathrm{d}q}{\mathrm{d}t}$.

The approximations can be written as $\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\mathit{M}\left(\begin{array}{c}p\\ q\end{array}\right)$.

Find the eigenvalues of $\mathit{M}$.

Hence describe the behaviour of the aphid and ladybird populations when they start close to $(4,2)$.

At the start of one season there are 2000 aphids and 100 ladybirds in the greenhouse.

Use Euler's method with a step length of 0.25 weeks to estimate the number of aphids and the number of ladybirds after 1 week.

Sketch, on a set of axes with $x$ on the horizontal axis and $y$ on the vertical axis, the trajectory that starts at $(2,1)$. Show the direction of motion along the trajectory.

Selin considers spraying a pesticide that kills aphids and ladybirds at the same rate. She modifies her model to

Find the equilibrium point of the modified model at which both species are present.

Comment on whether spraying the pesticide is a sensible way to reduce the aphid population in the long term.

Write the system in the form $\dot{\mathit{x}}=\mathit{M}\mathit{x}$, where $\mathit{M}$ is a $2\times 2$ matrix.

Find the eigenvalues of $\mathit{M}$.

Find an eigenvector corresponding to each eigenvalue.

Describe the behaviour of the trajectories near the equilibrium point at the origin. Justify your answer.

Find the eigenvalues of the matrix $\left(\begin{array}{cc}1& 4\\ 2& -1\end{array}\right)$.

Find an eigenvector for each eigenvalue.

Write down the general solution of the system.

Find the particular solution.

Find the value that ${\displaystyle \frac{x}{y}}$ approaches as $t$ increases. Justify your answer.

State, with a reason, the type of equilibrium point at the origin.

By letting $\omega =\frac{\mathrm{d}\theta }{\mathrm{d}t}$, write the differential equation as a system of first-order equations in the form $\left(\begin{array}{c}\dot{\theta }\\ \dot{\omega }\end{array}\right)=\mathit{M}\left(\begin{array}{c}\theta \\ \omega \end{array}\right)$, where $\mathit{M}$ is a $2\times 2$ matrix in terms of $c$.

The closer is first set with $c=5$.

Show that the eigenvalues of $\mathit{M}$ are $-1$ and $-4$.

Find an eigenvector for each eigenvalue.

Hence show that $\theta =1.6{\mathrm{e}}^{-t}-0.4{\mathrm{e}}^{-4t}$.

The door is considered shut once $\theta <0.05$.

Find the time taken for the door to be shut.

Show that, with this setting, the door never swings past its closed position.

Sketch the phase portrait for this system, with $\theta$ on the horizontal axis and $\omega$ on the vertical axis. Your sketch should show the eigenvector lines and the trajectory that starts at $(1.2,0)$.

After some years the closer wears and the damping setting falls to $c=2$.

Find the eigenvalues of $\mathit{M}$ for this setting.

Hence describe the motion of the door with this setting.

Use Euler's method with a step length of 0.1 seconds to estimate the value of $\theta$ when $t=0.5$ for the worn closer.

A computer solution of the differential equation gives $\theta =0.7916$ when $t=0.5$.

Calculate the percentage error in your answer to part (h).

Find the set of values of $c$ for which $\mathit{M}$ has two distinct real eigenvalues.

Explain why the manufacturer would not choose a very large value of $c$.