Chapter 7.P3, Problem 4P

a) Let $c=3$. Find the critical points and the corresponding eigenvalues.

b) Plot the trajectory starting at the point $(1,0,-2)$. Observe that the limit cycle now consists of two loops before itcloses; it is often called a $2$-cycle.

c) Plot $x,y,$ or $z$ versus $t$ and show that the period of motion on the $2$-cycle is very nearly double the period of thesimple limit cycle in Problem $2$. There has been a period-doubling bifurcation of cycles fora certain value of between $1.3$ and $3$.