Dynamics of damped coupled oscillators near resonance

We study the dynamics of two conservative oscillators with perturbations from a linear displacement coupling and non-Hamiltonian forces such as damping. We examine the dynamics of these systems when they are near the primary resonance using secular perturbation theory. We show that near resonance a large class of driven oscillators and two coupled oscillators can be transformed to the same ordinary differential equations (ODEs). This common type of dynamics near the resonance is a generalization of the standard Hamiltonian dynamics of two coupled conservative oscillators. We derive expressions for the parameters in these ODEs. From these parameters, we derive analytical expressions for the parameters in these ODEs. From these parameters, we derive analytical expressions for the linear fixed point behavior of these oscillators near resonance. We find a relation between the amplitude frequency coupling of the oscillators and their phase-locking behavior. In particular, we show that two hard oscillators lock in phase and two soft oscillators lock out of phase. We compare our theoretical predictions with computer simulations of two examples: a sinusoidally driven X3 force oscillator and two coupled van der Pol oscillators with X3 force.