Show that, on the phase plane with continuous, the index Iof any simple closed curvethat encloses all equilibrium points can only be 1,−1, or zero. Let with continuous, represent a...

Show that, on the phase plane with

continuous, the index Iof any simple closed curvethat encloses all equilibrium points can only be 1,−1, or zero.

Let

with

continuous, represent a parameter-dependent system with parameter λ. Show that, at a bifurcation point (Section 1.7), where an equilibrium point divides as λ varies, the sum of the indices of the equilibrium points resulting from the splitting is unchanged. Deduce that the equilibrium points for the system

consist of a saddle point for λ<> 0.