1. Consider the ODE
x = cosh(1 + x) 1 x2 + 2x r :
(a) Find a value rc and corresponding xc at which a bifurcation occurs.
(b) Sketch the bifurcation diagram.
(c) Taking y = x xc and s = r rc, find a function f(y; s) so that
◦ = f(y; s):
(d) Compute the Taylor series of f at (y; s) = (0; 0) to fourth order.
n=0 (2n)!
(e) Use both your bifurcation diagram and Taylor series to explain why this is a new type of bifurcation.
_
N
2. (Strogatz Exercise 3.5.7) Consider the logistic equation N = rN(1
K
), with initial condition N0.
(a) This system has three dimensional parameters r; K; N0. Find the dimensions of each of these parameters.
(b) Show that the system can be rewritten in the dimensionless form
• dx
<
= x(1 x);
d
x(0) = x0:
:
for appropriate choices of the dimensionless variables x, x0, and .
(c) Find a different nondimensionalization in terms of variables u and , were u is chosen such that the initial condition is always u(0) = 1.
3. Show that x = ln(1 + x) rx undergoes a transcritical bifurcation at (x ; r ) = (0; 1). Use the Transcritical Bifurcation Theorem covered in class.
4. Problem 3.5.6, parts a), b), c) and d), from the textbook.
