Transcritical Bifurcation

Example 2: Consider the planar system x = px-x1, y = -y, (x,y)pR2, ppR [3]

For p <0, there is a stable node point O (0, 0) and a saddle A (p, 0), and for p = 0, there is a nonhyperbolic equilibrium point O (0, 0) (saddle-node point). For p >0, there is a

Saddles

\ Unstable nodes

Stable nodes

Saddles

Figure 4 Saddle-node bifurcation of x = p—x2, y=x—cy.

Figure 5 Transcritical bifurcation of x = px-x2, y = -y.

stable node point A (p, 0) and a saddle 0 (0, 0). It is easy to know that the stability of 0 and A at p = 0 have commuted. The bifurcation diagram is displayed in Figure 5.

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