Example 4.2.1:

For a typical dynamic system the frequency-amplitude relation is given by the following equation.

...............................................................................(4.2.6)

................................................................(4.2.7)

Here, . are fixed system parameters. The saddle node bifurcation points have been shown in Fig. 4.2.2. (b)

Fig. 4.2.2: Saddle-node bifurcation point corresponding to (a) , (b) example of a typical second order system.

Pitchfork bifurcation: The normal form for a generic pitchfork bifurcation of a fixed point is ...................................................................................................(4.2.8)

In this case the equilibrium point is obtained by substituting .

So, ..................................................................................(4.2.9)

So, One can obtain two solutions viz., the trivial solution i.e., x=0 and the non trivial solution

The solutions are plotted in Figure 4.2.3.