1. In 1918, Georg Duffing introduced a nonlinear oscillator with a cubic stiffness term to describe the hardening spring effect observed in many mechanical problems, the equations reads as follows
x + x + x + x3 = cos(!t) ;
where , , , and ! are constants. Show that this system can be expressed as a first order autonomous system of ODEs.
2. Suppose that a new short form video app, KotKit, has been launched. Write down a one-dimensional autonomous system for the growth of the app, taking x(t) to be the fraction of the population that has downloaded the app at time t. Check that the fixed points and their stability match common sense.
3. Suppose that the one-dimensional autonomous ODE
• = f(x);
has a fixed point x so that a = f0(x ) 6= 0.
(a) Write down the linearization of the ODE about x .
(b) Show that the time required for the solution of the linearized equation found in part (a) to increase or decrease its value (depending on the sign of a) by a factor of k > 0 is a constant that depends only on a; k.
(c) The book defines the ‘characteristic timescale’ attendant to the fixed point x to be 1=jaj. Using your answer to part (b), give an interpretation of this quantity.
4. Draw a phase portrait (cf. Figure 1 on p. 37 of Strogatz) for each of the following systems, including the values and stabilities of fixed points. Overlay a sketch of a potential function on each one.
(a) x = x(x 1)2
(b) x = 1 j xj
(c) x = sin(3x)
(d)
(
0
j j
if
x = 0
x =
x ln x
if
x 6= 0
5. For each integer k = 1; 2; 3; : : : and each choice of + or , determine the stability of x = 0 as a fixed point of the equation
x = xk
Restricting to the cases where x = 0 is stable, does making k larger result in faster or slower convergence to the fixed point? Give both a heuristic explanation and one in terms of the exact solutions. (Note that this equation is separable.)
6. (Derived from Strogatz Exercise 2.2.13) The velocity v(t) of a skydiver falling to the ground is governed by
mv = mg kv2;
where m is the mass of the skydiver, g is the acceleration due to gravity, and k > 0 is a constant related to the amount of air resistance.
(a) Find the exact solution for v(t) when v(0) = 0.
(b) Find the limit of v(t) as t ! 1. This limiting velocity is called the terminal velocity.
(c) Draw a phase portrait for this problem, and thereby re-derive a formula for the terminal velocity.
