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A Special oscillator. The notion of bifurcation.

The problem we will study here will introduce the notion of bifurcation.

We consider a rod articulated to its foot and in a vertical plan. This rod is submitted to the gravity and to a backforce due to two identical springs (7.8a). The classical modelling lead to this equation (1 is the common stiffness of the springs):

The vertical position is the position of stable equilibre. If the stiffness of the springs is lower to a critical value (1c), the vertical position becomes unstable. Two stable positions appear, symmetrically opposed. To the critical value corresponds the notion of bifurcation. The number and the nature of the equilibrium positions can be represented on a bifurcation diagram, depending on the stiffness (7.8b).

If we consider two different stiffnesses, the is no symmetry. The equation becomes:

The diagram of bifurcation is modified in (7.8c).