Mathematics 118A, Winter 1998

Homework Assignment 1, due Friday, January 16

1. Recall the definition of a one-parameter group of diffeomorphisms {g^{t}} on an open subset M ⊂ ℝ^{n}:

The mapping

g: ℝ × M → M,
g(t, x) = g^{t}x for t ∈ ℝ, x ∈ M,

must satisfy

(a) g ∈ C^{1},

(b) g^{t} is a diffeomorphism for every t ∈ ℝ,

(c) the family {g^{t}: t ∈ ℝ} is a one-parameter group of transformations on M.

Show that requirement (b) is redundant. I.e., show that (a) and (c) imply (b). (Hint: What is the inverse of g^{t}?)

2. Let {g^{t}} be a one-parameter group of diffeomorphisms on an open subset M ⊂ ℝ^{n}, x ∈ M, and v(x) the phase velocity

v(x) := \frac{d}{dt}g^{t}x|_{t=0}.

Prove that if x is a fixed point of the phase flow, then v(x) = 0.

3. Consider the equation

ẋ = x(1 − x).

(a) Find and characterize the critical points. Sketch the flow in the (one-dimensional) phase space.

(b) Solve the equation.

(c) Sketch the solution in the extended phase space.

4. Consider the equation

ẍ − λẋ − (λ − 1)(λ − 2)x = 0

with λ a real parameter.

Find the critical points and characterize these points. Sketch the flow in the phase plane and indicate the direction of the flow.