Normal form of nonlinear oscillator model relevant to spin-torque nano-oscillator theory

We introduce a normal form for the equation which describes nonlinear oscillations of spin-torque nano-oscillators (STNO). The general form of the equation might be reduced to the equation for STNO with uniformly magnetized free layer or to the equation for STNO with free layer in a magnetic vortex state. The equation is used to study the injection-locking problem. The bifurcation diagram for this problem is derived by using as bifurcation parameters the frequency and the amplitude of the ac excitation. Bifurcations lines corresponding to local bifurcations, i.e. the Saddle-node and the Hopf bifurcations, are derived by using linearization, while the lines corresponding to the Homoclinic bifurcation, which is global in nature, are derived numerically. The diagram exhibits universal features in terms of the shape and position of bifurcation lines. When the frequency of the ac excitation is relatively close to the free-running frequency of the oscillator the system is in synchronized state. For small ac amplitudes, the synchronization is lost for values of the ac frequency which imply the crossing of saddle-node bifurcations. For sufficiently larger values of the ac amplitude and for ac frequency smaller that the free-running frequency, the loss of the synchronized state occurs through a more complex bifurcation pattern which involves a sequence of two homoclinic bifurcations and an Hopf bifurcation. The value of frequency corresponding to loss of stability of the synchronized state can be determined analytically through linearization. The above sequence of bifurcations is connected to the presence of a special point in the bifurcation diagram, referred to as Takens-Bogdanov bifurcation.