18.5 Exercises

Exercise 18.1 Verify that \(\displaystyle \vec{s}_{1}(t) = \begin{pmatrix} 0 \\ e^{t} \end{pmatrix}\) is a solution to the following system of equations:

\[\begin{equation} \begin{pmatrix} x' \\ y' \end{pmatrix} =\begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \end{equation}\]

 

Exercise 18.2 (Inspired by Logan and Wolesensky (2009)) Compute the eigenvalues and eigenvectors for the following linear systems. Based on the eigenvalues, classify if the equilibrium solution is stable or unstable. Finally write down the most general solution for the system of equations.

  1. \(\displaystyle \frac{dx}{dt} = -x, \; \frac{dy}{dt} = -y\)
  2. \(\displaystyle \frac{dx}{dt} = 3x-2y, \; \frac{dy}{dt} = 2x-2y\)
  3. \(\displaystyle \frac{dx}{dt} = -4x+2y, \; \frac{dy}{dt} = x-3y\)
  4. \(\displaystyle \frac{dx}{dt}= 4y, \; \frac{dy}{dt}=-9x\)
  5. \(\displaystyle \frac{dx}{dt} = y, \; \frac{dy}{dt} = -x\)
 

Exercise 18.3 Consider the following nonlinear system:

\[\begin{equation} \begin{split} \frac{dx}{dt} &= y-x \\ \frac{dy}{dt} &=-y + \frac{5x^2}{4+x^{2}}, \end{split} \end{equation}\]

  1. Previously you verified that \((x,y)=(1,1)\) is an equilibrium solution for this system. What is the Jacobian matrix at that equilibrium solution?
  2. Generate a phaseplane for the Jacobian matrix.
  3. What are the eigenvalues for the Jacobian matrix at the equilbrium solution?
  4. Based on the eigenvalues, how would you classify the stability of the equilibrium solution?
 

Exercise 18.4 Consider the following nonlinear system:

\[\begin{equation} \begin{split} \frac{dx}{dt} &= 2x+3y+xy \\ \frac{dy}{dt} &= -x + y - 2xy^{3}, \end{split} \end{equation}\]

  1. Previously you verified that \((x,y)=(0,0)\) is an equilibrium solution for this system. What is the Jacobian matrix at that equilibrium solution?
  2. Generate a phaseplane for the Jacobian matrix.
  3. What are the eigenvalues for the Jacobian matrix at the equilbrium solution?
  4. Based on the eigenvalues, how would you classify the stability of the equilibrium solution?

 

Exercise 18.5 Consider the following system:

\[\begin{equation} \begin{split} \frac{dx}{dt} &= y^{2} \\ \frac{dy}{dt} &= -\frac{2}{3} x, \end{split} \end{equation}\]

  1. There is one equilibrium solution to this system of equations. What is it?
  2. What is the Jacobian matrix for this equilibrium solution?
  3. Generate a phaseplane for the Jacobian matrix.
  4. What are the eigenvalues for the Jacobian matrix at the equilbrium solution?
  5. Based on the eigenvalues, how would you classify the stability of the equilibrium solution?

 

Exercise 18.6 Consider the system \(\displaystyle \frac{d}{dt} \vec{x} = A \vec{x}\).

  1. Given the function \(\vec{s}(t)=e^{\lambda t} \vec{v}\), where \(\vec{v}\) is a constant vector, what is an expression for \(\displaystyle \frac{d}{dt} \vec{s}(t)\)?
  2. Given the function \(\vec{s}(t)=e^{\lambda t} \vec{v}\), where \(\vec{v}\) is a constant vector, what is an expression for \(A \vec{s}(t)\)?
  3. Now use the previous results to compare \(\displaystyle \frac{d}{dt} \vec{s}(t) = A \vec{s}(t)\). Explain why it must be the case that \(\lambda \vec{v} = A \vec{v}\)

 

Exercise 18.7 In this section we learned that for a two dimensional matrix \(\displaystyle A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), eigenvalues can be found by solving the characteristic equation \(\det(A-\lambda I)=0\), or \(\lambda^{2} - (a+d) \lambda + ad-bc = 0\). Use the quadratic formula to get an expression for the eigenvalues \(\lambda\) in terms of \(a\),\(b\), \(c\), and \(d\).

References

Logan, J. David, and William Wolesensky. 2009. Mathematical Methods in Biology. 1st ed. Hoboken, N.J: Wiley.