Typical types of bifurcation

We would start with the simple examples of (a) tangent bifurcation (b) pitchfork bifurcation and (c) Hopf bifurcation [Figs. 8.3 and 8.4]. 

Let us consider the graph of function f x) = A e where A is some constant. Depending on the value of A, the curve would be a tangent at f x) = 1 x = 1 passing through the origin as shown below. 

Three situations can arise when (i) A 1 /e, (ii) A = 1 /e and (iii) A 1/e. In case (i) the curve will not touch OC, while in case (iii), the curve will cut OC at two points. On the other hand for case (ii), OC will be tangent to the curve. This type of bifurcation is called tangent bifurcation and A is called bifurcation parameter. 

It is obvious that dx/dt = 0 when x = 0 or x = The unique fixed point x = 0 existing for 0 is stable while for 0 is unstable. Furthermore, for ju, 0, the fixed points x = are stable. This result can be expressed diagrammatically as follows. 

The bifurcation diagram is represented by the curve A B C D, which looks like a tuning fork. The stable branch A B bifurcates to branches B C and B D at the point B which is called the bifurcation point. 

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