Exercises 2.2.4, 2.2.13 (a,b,c), 2.3.4, 2.4.2, 2.4.8, 2.7.6
Computer work: Exercises 2.8.2, 2.8.3
Matlab program 1 - to assist you with problem
2.8.2. This program draws the slope field of the equation in Example 2.8.1
(page 35). You only need to modify the velocity field (denoted by S in the
code) and perhaps the meshgrid [t,x].
Matlab program 2 and Matlab program 3 - to assist you with problem
2.8.3. This program solves numerically the equation in Example 2.8.1
(page 35) using the Euler method. To solve other equations you will have to
modify the rhseuler.m file by entering the desired right-hand-side.
Exercises 3.1.2, 3.2.3, 3.2.4, 3.3.2, 3.4.2, 3.4.6, 3.4.11
Hw2 - Solutions (selected problems)
Exercises 3.6.2, 3.7.3, 3.7.5.
For parts d) and e) in Ex 3.7.5 I expect a detailed discussion on the stability diagram (behaviour of r and s in terms of the curve parameter x, location of the cusp point, range of the curve parameter x). Also, in addition to what Ex 3.7.5 is asking for, you are asked to choose values of the parameters r and s that correspond to three fixed points, and initialize the ode at different locations to obtain numerical time evolutions of the model toward the two stable equilibria.
Exercises for Chapter 4: 4.1.2, 4.1.5, 4.3.3, 4.3.4, 4.5.3.
Additional work required for Ex 4.5.3: Illustrate numerically the bottleneck phenomenon for the equation in the problem. Discuss (and illustrate in plots) where the bottleneck occurs, as well as the scaling law for the time needed to pass through the bottleneck.
Matlab codes:
Program biochemical.m - solves numerically the ode in Ex 3.7.5. Also, to help with Ex 3.7.5, here is a program stabdiag.m which draws a parameterized two-dimensional curve. Read the comments on top of the files!
Program bottleneck.m - solves numerically the ode in Ex 4.5.3.
Exercises 5.1.10 (a,c,e), 5.2.2, 5.2.4, 5.2.6, 5.2.8, 5.2.10, 5.2.13, 5.3.4.
For all problems above I want you to illustrate the results with a phase portrait. Most exercises are explicitly asking for it in fact. You can certainly use a computer-generated phase portait using the Matlab program below. Make sure phase portraits have details provided (fast and slow eigendirections for nodes, stable and unstable manifolds for saddles, etc) to show that you have a clear understading of the qualitative behaviour of the solution.
Matlab code pplane7.m - solves
numerically systems (linear or nonlinear) of two odes. It may ocasionally break
or give inaccurate answers due to coarse numerical integration. It is quite
robust and reliable however. The Java applet is available at http://math.rice.edu/~dfield/dfpp.html. Personally, I have not
downloaded the applet and I have not run the codes there.
CAUTION: Use this computer software very
wisely. Use it only to check your answers. At the exam this tool will not be available, and you will have to draw
phase portraits by hand!
Exercises for Chapter 6: 6.2.2, 6.3.4, 6.3.9, 6.4.3, 6.5.1, 6.5.6, 6.6.1, 6.6.6
Use of the Matlab program pplane7.m (linked above) is certainly very recommended.
Extra computational work: for Exercise 6.3.9(e) I want you to find numerically (see 2d Euler Matlab code below) a time-backward trajectory and investigate its asymptotic behaviour. In the same figure, plot this time-backward trajectory, along with an approximation of the curve that the trajectories asymptote to as time approaches minus infinity.
Matlab code euler2d.m - solves numerically (by Euler method) a system of two odes (specifically, this code solves the system in Ex 6.3.9).
Additional problem for Chapter 7. This problem deals with the model from Example 7.3.1 for values of the parameter mu>1. I ask you to do the following. i) Write the system in rectangular coordinates and investigate the existence of fixed points. Also, discuss the linear stability theory of the fixed point at the origin. ii) Use the polar coordinate formulation and investigate the applicability of the Poincare-Bendixson theorem for mu>1. Provide phase portraits for several (two is enough) values of mu and explain the behaviour of the solutions. Identify numerically the limit cycles. Particular attention should be paid to the behaviour of the trajectories near the origin.
I do not recommend using pplane7.m to draw phase portraits for the additional problem. The results are correct, but it could be confusing. I recommend using the 2d Euler integration code linked above in homework 5 (just change the right-hand-side of the ode system, initial conditions, and possibly the time step).
Exercises for Chapter 9: 9.2.1, 9.2.2, 9.2.6
Computer work:
(i) Choose appropriate initial conditions and produce results similar to those presented in Example 9.5.1 (Figures 9.5.2 and 9.5.3). Use r = 21, sigma = 10, b = 8/3. Explain why this behavior is called "transient chaos".
(ii) Use sigma = 10, b = 8/3 and choose a value for r and initial conditions to produce
results similar to those in Example 9.5.2 (Figures 9.5.4 and 9.5.5). Compare with the results in part (i).
Exercises 9.3.8., 9.4.2, 10.1.2, 10.1.6, 10.1.11, 10.3.4, 10.3.6