Lorenz Equations with Mathematica.......click here
For Tuesday, Nov 9
(1) Find the equilibrium points for the Lorenz's equations, both by hand and using Mathematica's "Solve" .
(2) Try to get a plot of a few trajectories of the Lorenz flow starting near the equilibrium points. Might be hard to make sense of what you get.... try your best. Our goal for the next classes is to understand the local flow around the equilibrium points.
(3) This is review material from Calculus, we will need it for the rest of the course. Recall from calculus what the "linear approximation" (or linearization) of a function is. If needed, consult books.
a) Find the linear approximation (or linearization of f) of the function f(x) = x^4 -2 x^3 + Sqrt(x-1) at x =5 . (The answer should be a linear function g(x) of the form g(x) = a x + b
b) Find the linear approximation (or linearization of f) of the function f(x, y) = x e^(x y) at (x,y) = (1,0). (The answer should be a linear function g(x,y) of the form g(x,y) = a x + b y + c
c) Find the linearization of the function f(x, y) = sqrt( y + cos^2(x)) at (0,0). (Answer: g(x,y) = 1 + .5 y )
d) Find the linearization of f(x,y, z) = y z + x y^3 + 3(y-x) at (x,y,z) = (1,2,3)
(4) Start reading Chapter 3
For Tuesday, Nov 16.
We are working in class with material that is written as chapters 3 and 5 in your text. We will skip chapter 4.
Now is best now to work on 5.1 first and then on chapter 3:
1) Study 5.1 (important) and 5.2 (less important). Practice with the exercises. Your book provides you with solutions to the odd ones. Bring questions to class, any and many. Be prepared to voice voice voice your qestions.
2) Apply what you learn in 5.1 to the pendulum equations we had in class. Bring your results.
3) 3.1: At this point, all exercises in 3.1 should be ok. Bring questions of the ones that you have difficulties with.
4) 3.2: Work on 2 and 4 in detail. Submit these two on Thursday.