Exercises for Chapter 2 (pages 36-43): 2.2.4, 2.2.13 (a,b,c), 2.3.4, 2.4.2, 2.4.8, 2.7.6
Exercises for Chapter 3 (pages 79-92): 3.1.2, 3.2.4, 3.3.2, 3.4.2, 3.4.6
Computer work: page 42: 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 for Chapter 3, pages 79-92: 3.4.11, 3.6.2, 3.7.3, 3.7.5
Exercises for Chapter 4, pages 113-119: 4.1.2, 4.1.5, 4.3.3, 4.3.4, 4.5.3.
Exercises for Chapter 5 (pages 140-144): 5.1.10 (a,c,e), 5.2.4, 5.2.6, 5.2.8, 5.2.10, 5.3.4.
Exercises for Chapter 6, Sections 6.1-6.6 (pages 181-191): 6.3.4, 6.3.9 (a-d),
6.4.3, 6.5.1, 6.5.6, 6.6.1, 6.6.6
Matlab program pplane7.m - it 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.
CAUTION: Use this computer software very
wisely. Use it only to check your answers. At the exam you will not even be allowed to
use calculators, not to mention this sort of tools. So, please be very careful
how you use it. Again, at the exam you will have to draw phase portraits by
hand, so you need practice!
Exercises for Chapter 7, Sections 7.1-7.3 (pages 227-240): 7.1.1, 7.1.4, 7.2.6, 7.2.10,
7.3.3, 7.3.4.
Exercises for Chapter 8, Sections 8.1-8.2 and 8.4 (pages 284-297): 8.1.1, 8.1.6, 8.2.2, 8.2.8, 8.2.12, 8.4.2.
Computer work: 8.2.3, 8.2.6.
Exercises for Chapter 9 (pages 342-347): 9.2.1, 9.2.2, 9.2.6, 9.3.8., 9.4.2
Exercises for Chapter 10 (pages 388-397): 10.1.2, 10.1.6, 10.1.11, 10.3.4, 10.3.6
Exercises for Section 8.7 (page 295): 8.7.1, 8.7.2
Computer work:
(i) Choose appropriate initial conditions and produce results similar to those presented in Figures 9.5.2 and 9.5.3 on page 332. Use r = 21, sigma = 10, b = 8/3. Explain why this behavior is called "transient chaos".
(ii) Use sigma = 10, b = 8/3 and pick a value for r such that you produce
results similar to those in Figures 9.3.1 and 9.3.2 on pages 318 and 319. The Hopf bifurcation value
for these values of sigma and b is approx. 24.74. Explain why we call this a
"chaotic" bahavior. Compare with the results in part (i).
Matlab programs: lorenz.m and matlorenz.m - to assist you with the
computer work. The program matlorenz.m represents the subroutine that gives the
right-hand-side of the Lorenz system. The parameters sigma,r and b are defined
in this file. The program lorenz.m solves the Lorenz system and plots the solution. The
Matlab routine ode23 (see the line "ode23('matlorenz',[0,100],[-3;44;17]);"
in the lorenz.m program) is a built-in ODE solver. The first argument in ode23
is the right-hand-side routine 'matlorenz', the second argument is the time
interval [0,100] and the third argument is the initial condition (-3,44,17).
The command plot3 produces a 3D plot - the trajectory (x(t),y(t),z(t)). To plot,
let's say, the second component y(t) as a function of time - as in Figures
9.5.3 or 9.3.1 -
you can type "plot(t,y(:,2))" in the Matlab command line, after you run the lorenz.m code.