Practice Final

Problem 1

The Van der Pol equation has been used to model a number of biological processes such as the heart beat, circadian rhythms, biochemical oscillators, and pacemaker neurons. The equation is as follows:

Where x is that state variable or measurement (which might represent voltage or chemical concentrations, depending upon the application), and μ is a parameter that is proportional to the damping in the system.

A. Show how this equation can be rewritten into the following system of equations:

Answer:

B. Is this equation linear or nonlinear?

The system is nonlinear because of the term.


C. What is the critical point of this system?

D. What is the Jacobian matrix J(x, v)?

E. Find the eigenvalues of the Jacobian matrix evaluated at the critical point.

F. Classify the critical point for each of the following cases.

1)

2)

3)

4)


5)

G. Match the five cases above to the phase planes on the following pages.

Case 3


Case 2

Case 1


Case 4

Case 5

H. Consider the case where μ = 4. Note that wherever we start, we always end up on the same cycle (closed trajectory) as . This is called a stable or attractive limit cycle.

Some plots of x vs. t for several initial conditions follow:

Explain why this case would make a good design for a heart (note that x represents the voltage as a function of time in the cardiac tissues, which signals for contraction).

Heart beats are generated by a regular oscillation in the voltage difference between the inside and outside of the cardiac cells. In this case, small changes (or noise) does not disrupt the regular oscillations, the system returns to a stable cycle. This is good because a return to rest would represent a flat line, and loss of regular oscillations would represent a heart attack.

Problem 2

One model for chemical kinetics is the Schnakenberg reaction. Let A and B denote the concentration of two chemicals A and B. Let A be added at a constant rate k1, let B be added at a constant rate k4, let A break down at a rate k2, and let B be converted into A at a rate proportional to k3.

A. Let and let .

1) Find the equilibrium.

2) Find the Jacobian matrix J(A, B).

3) Find the eigenvalues of the Jacobian matrix evaluated at the critical point.

4) Classify the critical point.

The critical point (2.2, 0.413) is an stable spiral point.

B. Label the graph below.

1) Label the nullclines.

2) Draw the direction arrows on the nullclines.

3) Draw and label the trajectory starting at A = 3, B = 4.

4) Draw and label the trajectory starting at A = 0.75, B = 2.

5) Sketch a graph of A and B vs. time starting at A = 0.75, B = 2.


Problem 3

Consider the following system of differential equations:

A. Rewrite this system in the form x’ = Ax.

B. Find the eigenvalues of A.

C. Use the eigenvalues to classify the critical point at (0,0).

The critical point is an unstable saddle point.

D. Which of the following graphs is the phase plane for this system?