3.11: Problems

3.1. 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. \(y^{\prime}=y^{2}(y+1)(y-4)\)

3.2. 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. \(y(0)=-0.5\)

3.3. 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. \(x^{\prime}=x-\dfrac{\mu x}{1+x^{2}}\)

3.4. 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.

3.5. Consider the system

\begin{aligned}
x^{\prime} &=-y+x\left[\mu-x^{2}-y^{2}\right] \\[4pt]
y^{\prime} &=x+y\left[\mu-x^{2}-y^{2}\right]
\end{aligned}

Rewrite this system in polar form. Look at the behavior of the \(r\) equation and construct a bifurcation diagram in \(\mu\) 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.

3.6. 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.

a. \begin{aligned}
&x^{\prime}=x(100-x-2 y) \\[4pt]
&y^{\prime}=y(150-x-6 y)
\end{aligned}

b. \begin{aligned}
&x^{\prime}=x+x^{3} \\[4pt]
&y^{\prime}=y+y^{3}
\end{aligned}

c. \begin{aligned}
&x^{\prime}=x-x^{2}+x y \\[4pt]
&y^{\prime}=2 y-x y-6 y^{2}
\end{aligned}

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

3.7. Plot phase portraits for the Lienard system

\begin{aligned}
&x^{\prime}=y-\mu\left(x^{3}-x\right) \\[4pt]
&y^{\prime}=-x
\end{aligned}

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

3.8. 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?

3.9. 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 \\[4pt]
&y^{\prime}=d y-e y^{2}-f x y
\end{aligned}

In this case, assume that all constants are positive.

a. Describe the effects/purpose of each terms.
b. Find the fixed points of the model.
c. Linearize the system about each fixed point and determine the stability.
d. From the above, describe the types of solution behavior you might expect, in terms of the model.

3.10. 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 \(x\)'s, but eaten by \(z\)'s. A simple, but scaled, model for this system can be given by the system

\begin{aligned}
x^{\prime} &=x(1-x)-x y \\[4pt]
y^{\prime} &=y(1-y)+x y-y z \\[4pt]
z^{\prime} &=z(1-z)+y z
\end{aligned}

a. Find the equilibrium points of the system.
b. Find the Jacobian matrix for the system and evaluate it at the equilibrium points.
c. Find the eigenvalues and eigenvectors.
d. Describe the solution behavior near each equilibrium point.
f. Which of these equilibria are important in the study of the population model and describe the interactions of the species in the neighborhood of these point \((\mathrm{s})\)

3.11. 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.