7.11: Problems

1. Solve the general logistic problem,

\[\dfrac{d y}{d t}=k y-c y^{2}, \quad y(0)=y_{0} \nonumber \]

using separation of variables.

2. Find the equilibrium solutions and determine their stability for the following systems. For each case draw representative solutions and phase lines.
   a. \(y^{\prime}=y^{2}-6 y-16\)
   b. \(y^{\prime}=\cos y\).
   c. \(y^{\prime}=y(y-2)(y+3)\).
   d. \(y^{\prime}=y^{2}(y+1)(y-4)\).

3. For \(y^{\prime}=y-y^{2}\), find the general solution corresponding to \(y(0)=y_{0} .\) Provide specific solutions for the following initial conditions and sketch them: a. \(y(0)=0.25\), b. \(y(0)=1.5\), and c. \(y(0)=-0.5\).

4. For each problem determine equilibrium points, bifurcation points and construct a bifurcation diagram. Discuss the different behaviors in each system.
   a. \(y^{\prime}=y-\mu y^{2}\)
   b. \(y^{\prime}=y(\mu-y)(\mu-2 y)\)
   c. \(x^{\prime}=\mu-x^{3}\)
   d. \(x^{\prime}=x-\dfrac{\mu x}{1+x^{2}}\)

5. Consider the family of differential equations \(x^{\prime}=x^{3}+\delta x^{2}-\mu x\).
   a. Sketch a bifurcation diagram in the \(x \mu\)-plane for \(\delta=0\).
   b. Sketch a bifurcation diagram in the \(x \mu\)-plane for \(\delta>0\).

Hint: Pick a few values of \(\delta\) and \(\mu\) in order to get a feel for how this system behaves.

6. System \(7.52\) can be solved exactly. Integrate the \(r\)-equation using separation of variables. For initial conditions a) \(r(0)=0.25, \theta(0)=0\), and b) \(r(0)=1.5, \theta(0)=0\), and \(\mu=1.0\), find and plot the solutions in the \(x y\)-plane showing the approach to a limit cycle.
7. Consider the system

\[\begin{aligned} x^{\prime} &=-y+x\left[\mu-x^{2}-y^{2}\right] \\ y^{\prime} &=x+y\left[\mu-x^{2}-y^{2}\right] \end{aligned} \nonumber \]

Rewrite this system in polar form. Look at the behavior of the \(r\) equation and construct a bifurcation diagram in \(\mu r\) space. What might this diagram look like in the three dimensional \(\mu x y\) space? (Think about the symmetry in this problem.) This leads to what is called a Hopf bifurcation.

8. Find the fixed points of the following systems. Linearize the system about each fixed point and determine the nature and stability in the neighborhood of each fixed point, when possible. Verify your findings by plotting phase portraits using a computer.
   1. \[\begin{aligned} x^{\prime} &=x(100-x-2 y) \\ y^{\prime} &=y(150-x-6 y) \end{aligned} \nonumber \]

   2. \[\begin{aligned} &x^{\prime}=x+x^{3} \\ &y^{\prime}=y+y^{3} \end{aligned} \nonumber \]

   3. \[\begin{aligned} &x^{\prime}=x-x^{2}+x y \\ &y^{\prime}=2 y-x y-6 y^{2} \end{aligned} \nonumber \]

   4. \[\begin{aligned} &x^{\prime}=-2 x y, \\ &y^{\prime}=-x+y+x y-y^{3} . \end{aligned} \nonumber \]

9. Plot phase portraits for the Lienard system

\[\begin{aligned} &x^{\prime}=y-\mu\left(x^{3}-x\right) \\ &y^{\prime}=-x . \end{aligned} \nonumber \]

for a small and a not so small value of \(\mu\). Describe what happens as one varies \(\mu\).

10. Consider the period of a nonlinear pendulum. Let the length be \(L=1.0\) \(\mathrm{m}\) and \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\). Sketch \(T\) vs the initial angle \(\theta_{0}\) and compare the linear and nonlinear values for the period. For what angles can you use the linear approximation confidently?
11. Another population model is one in which species compete for resources, such as a limited food supply. Such a model is given by

\[\begin{aligned} &x^{\prime}=a x-b x^{2}-c x y, \\ &y^{\prime}=d y-e y^{2}-f x y . \end{aligned} \nonumber \]

In this case, assume that all constants are positive.

1. Describe the effects/purpose of each terms.
2. Find the fixed points of the model.
3. Linearize the system about each fixed point and determine the stability.
4. From the above, describe the types of solution behavior you might expect, in terms of the model.

12. Consider a model of a food chain of three species. Assume that each population on its own can be modeled by logistic growth. Let the species be labeled by \(x(t), y(t)\), and \(z(t)\). Assume that population \(x\) is at the bottom of the chain. That population will be depleted by population \(y\). Population \(y\) is sustained by \(x^{\prime}\) s, but eaten by \(z^{\prime}\) s. A simple, but scaled, model for this system can be given by the system

\[\begin{aligned} &x^{\prime}=x(1-x)-x y \\ &y^{\prime}=y(1-y)+x y-y z \\ &z^{\prime}=z(1-z)+y z . \end{aligned} \nonumber \]

1. Find the equilibrium points of the system.
2. Find the Jacobian matrix for the system and evaluate it at the equilibrium points.
3. Find the eigenvalues and eigenvectors.
4. Describe the solution behavior near each equilibrium point.
5. Which of these equilibria are important in the study of the pop- ulation model and describe the interactions of the species in the neighborhood of these point(s).

13. Derive the first integral of the Lotka-Volterra system, \(a \ln y+d \ln x-\) \(c x-b y=C\).
14. Show that the system \(x^{\prime}=x-y-x^{3}, y^{\prime}=x+y-y^{3}\), has a unique limit cycle by picking an appropriate \(\psi(x, y)\) in Dulac’s Criteria.
15. The Lorenz model is a simple model for atmospheric convection developed by Edward Lorenz in \(1963 .\) The system is given by the three equations

\[ \begin{aligned} &\dfrac{d x}{d t}=\sigma(y-x) \\ &\dfrac{d y}{d t}=x(\rho-z)-y \\ &\dfrac{d z}{d t}=x y-\beta z \end{aligned} \label{ \]

1. Find the equilibrium points of the system.
2. Find the Jacobian matrix for the system and evaluate it at the equilibrium points.
3. Determine any bifurcation points and describe what happens near the bifurcation point(s). Consider \(\sigma=10, \beta=8 / 3\), and vary \(\rho\).
4. This system is know to exhibit chaotic behavior. Lorenz found a so-called strange attractor for parameter values \(\sigma=10, \beta=8 / 3\), and \(\rho=28 .\) Using a computer, locate this strange attractor.

16. The Michaelis-Menten kinetics reaction is given by

\[E+S \stackrel{k_{3}}{k_{1}} \longrightarrow E S \underset{k_{2}}{\longrightarrow} E+P \nonumber \]

The resulting system of equations for the chemical concentrations is

\[ \begin{aligned} \dfrac{d[S]}{d t} &=-k_{1}[E][S]+k_{3}[E S] \\ \dfrac{d[E]}{d t} &=-k_{1}[E][S]+\left(k_{2}+k_{2}\right)[E S] \\ \dfrac{d[E S]}{d t} &=k_{1}[E][S]-\left(k_{2}+k_{2}\right)[E S] \\ \dfrac{d[P]}{d t} &=k_{3}[E S] \end{aligned} \label{7.95} \]

In chemical kinetics one seeks to determine the rate of product formation \(\left(v=d[P] / d t=k_{3}[E S]\right)\). Assuming that \([E S]\) is a constant, find \(v\) as a function of \([S]\) and the total enzyme concentration \(\left[E_{T}\right]=[E]+[E S] .\) As a nonlinear dynamical system, what are the equilibrium points?

17. In Equation \((6.58)\) we saw a linear version of an epidemic model. The commonly used nonlinear SIR model is given by

    \[ \begin{aligned} \dfrac{d S}{d t} &=-\beta S I \\ \dfrac{d I}{d t} &=\beta S I-\gamma I \\ \dfrac{d R}{d t} &=\gamma I \end{aligned}\label{7.96} \]

    where \(S\) is the number of susceptible individuals, \(I\) is the number of infected individuals, and \(R\) are the number who have been removed from the the other groups, either by recovering or dying.

    1. Let \(N=S+I+R\) be the total population. Prove that \(N=\) constant. Thus, one need only solve the first two equations and find \(R=N-S-I\) afterwards.
    2. Find and classify the equilibria. Describe the equilibria in terms of the population behavior.
    3. Let \(\beta=0.05\) and \(\gamma=0.2\). Assume that in a population of 100 there is one infected person. Numerically solve the system of equations for \(S(t)\) and \(I(t)\) and describe the solution being careful to determine the units of population and the constants.
    4. How does this affect any equilibrium solutions?
    5. Again, let \(\beta=0.05\) and \(\gamma=0.2\). Let \(\mu=0.1\) For a population of 100 with one infected person numerically solve the system of equations for \(S(t)\) and \(I(t)\) and describe the solution being careful to determine the units of population and the constants.
18. An undamped, unforced Duffing equation, \(\ddot{x}+\omega^{2} x+\epsilon x^{3}=0\), can be solved exactly in terms of elliptic functions. Using the results of Exercise 7.10.1, determine the solution of this equation and determine if there are any restrictions on the parameters.
19. Determine the circumference of an ellipse in terms of an elliptic integral.
20. Evaluate the following in terms of elliptic integrals and compute the values to four decimal places.
    1. \(\int_{0}^{\pi / 4} \dfrac{d \theta}{\sqrt{1-\dfrac{1}{2} \sin ^{2} \theta}}\).
    2. \(\int_{0}^{\pi / 2} \dfrac{d \theta}{\sqrt{1-\dfrac{1}{4} \sin ^{2} \theta}}\).
    3. \(\int_{0}^{2} \dfrac{d x}{\sqrt{\left(9-x^{2}\right)\left(4-x^{2}\right)}}\).
    4. \(\int_{0}^{\pi / 2} \dfrac{d \theta}{\sqrt{\cos \theta}}\).
    5. \(\int_{1}^{\infty} \dfrac{d x}{\sqrt{x^{4}-1}}\).