It may be noted that while supercritical pitchfork and Hopf bifurcation, respectively results in stable fixed-point and periodic responses, the sub-critical pitchfork and Hopf bifurcation, respectively results in unstable fixed-point and periodic responses. Hence, these sub critical bifurcation points are dangerous bifurcation points.

In case of nonlinear vibration many phenomena such as jump-up, jump-down, saturation, multi-stable region along with different types of responses such as fixed-point, periodic, quasi-periodic and chaotic are observed. Many bifurcation phenomena such as sub and super critical pitchfork, Hopf, saddle-point, period doubling etc are observed. One may observe different type of crisis phenomena in chaotically modulated system.

Fig. 4.2.6 (a) super-critical Hopf bifurcation, (b) sub-critical Hopf bifurcation.

Example 4.2.4: Stability analysis of multi-degree of freedom system

The reduced equations for a based excited cantilever beam with attached mass can be given by

(4.2.17)