Chapter 7.P3, Problem 5P

a) Let $c=3.8$. Find the critical points and the corresponding eigenvalues.

b) Plot the trajectory starting at the point $(1,0,-2)$. Observe that the limit cycle is now a $4$-cycle. Find the period of motion. Another period-doubling bifurcation has occurred for $c$ between $3$ and $3.8$.

c) For $c=3.85$, show that the limit cycle is an $8$-cycle. Verify that its period is very close to eight times the period of the simple limit cycle in Problem $2$.

Note: As $c$ increases further, there is an accelerating cascade of period-doubling bifurcations. The bifurcation values of $c$ converge to a limit, which marks the onset of chaos.