Chapter 7.P3, Problem 3P

The limit cycle found in Problem 2 comes into existence as a result of a Hopf bifurcation at a value $c_1$ of $c$ between $1$ and $1.3$. Determine, or at least estimate more precisely, the value of $c_1$. There are several ways in which you might do this.

a) Draw plots of trajectories for different values of $c$.

b) Calculate eigenvalues at critical points for different values of $c$.

c) Use the result of Problem 3(b) in Section 7.6.

(a) By solving Eq. (9) numerically, show that the real part of complex root changes sign

when $r\cong 24.737$.

(b) show that a cubic polynomial $x^3+A{x}^2+Bx+c$ has one real zero and two pure imaginary zero only if $AB=C.$

(c) By applying the result of part (b) to Eq. (9) show that the real part of the complex root changes sign when $r=\frac{470}{19}.$