Stability of stationary state bifurcations to periodic solutions

Stability of stationary state bifurcations to periodic solutions 

As the parameter A is varied, so the point of intersection of the mapping curve and the straight line in Fig. 13.5 changes. The gradient of the cubic curve at the point of intersection also varies during this process. It turns out that the stationary state represented by the intersection is stable provided the value of the gradient at the point is greater than - 1. The condition for the equivalent of a Hopf bifurcation in the map is thus 

As an example, let us consider A = 4.5. The stationary-state solution given by eqn (3.6) is x ,ss = 0.5286. If we start with any other initial value for x in the range between zero and unity, however, we soon find that the sequence settles to an alternation between x = 5 and . 

With this simple pattern of alternation, the value of x is repeated every second iteration i.e. we have that 

Using our simple cubic form for /(x ), a double application of the mapping is equivalent to a higher-order polynomial form  

---