Duffing’s equation isWhen k > 0 this equation models a vibrating spring, which could be...

Duffing’s equation is

When k > 0 this equation models a vibrating spring, which could be soft (l<0) or hard (l > 0) (See Student Project #1 in Chapter 8). When k<0 the equation arises as a model of the motion of a long thin elastic steel beam that has its top end embedded in a pulsating frame (the F(t) term), and its lower end hanging just above two magnets which are slightly displaced from what would be the equilibrium position. We will be looking at the unforced case (i.e. F(t) = 0), with m = 1. The system corresponding to Duffing’s equation is available in the Gallery menu.

a) This is the case of a hard spring with k = 16, and l = 4. Use pplane6 to plot the phase planes of some solutions with the damping constant c = 0,1, and 4. In particular, find all equilibrium points.

b) Do the same for the soft spring with k = 16 and l = −4. Now there will be a pair of saddle points. Find them and plot the stable/unstable orbits.

c) Now consider the case when k = −1, and l = 1. For each of the cases c = 0, c = 0.2, and c = 1, use pplane6 to analyze the system. In particular, find all equilibrium points and determine their types. Plot stable/unstable orbits where appropriate, and plot typical orbits.

d) With c = 0.2 in part c), there are two sinks. Determine the basins of attraction of each of these sinks. Indicate these regions on a print out of the phase plane.