  Recent experiments on spin torque oscillators have revealed interactions
between multiple magnetodynamic modes, including mode-coexistence,
mode-hopping, and temperature-driven cross-over between modes. Initial
multimode theory has indicated that a linear coupling between several dominant
modes, arising from the interaction of the subdynamic system with a magnon
bath, plays an essential role in the generation of various multimode behaviors,
such as mode hopping and mode coexistence. In this work, we derive a set of
rate equations to describe the dynamics of coupled magnetodynamic modes in a
nano-contact spin torque oscillator. Expressions for both linear and nonlinear
coupling terms are obtained, which allow us to analyze the dependence of the
coupled dynamic behaviors of modes on external experimental conditions as well
as intrinsic magnetic properties. For a minimal two-mode system, we further map
the energy and phase difference of the two modes onto a two-dimensional phase
space, and demonstrate in the phase portraits, how the manifolds of periodic
orbits and fixed points vary with external magnetic field as well as with
temperature.
