A second-order ordinary differential equation is outlined for the temporal dynamics of the respiratory central pattern generator (RCPG). Recurrent interactions between central excitation and inhibition confine the breathing cycle to the interior of a heteroclinic orbit between switching points (saddle equilibria) located at end expiration (E-I) and end inspiration (I-E). Dynamics depend on four eigenvalues that control inspiratory drive (lambda), excitability of inspiratory off switch (omega 1; stage 1 expiration), rate of central excitation disinhibition (omega 2; stage 2 expiration), and damping of the oscillator (epsilon). Ratios omega 2/lambda and omega 1/lambda determine local E-I and I-E phase switching, whereas inspiratory-to-expiratory balance varies as omega 2/(lambda omega 1). Stable apnea is seen when (lambda omega 2)/epsilon is near zero; inspiratory apneusis is seen when (lambda omega 1)/epsilon is low. The equations provide formalisms for discussing phase switching, apneas, apneuses, phase resetting and singularities, rapid shallow breathing, postinhibitory rebound excitation, redundancy, gating within the RCPG, and behavioral control of breathing. The model is offered as an explicit alternative to the harmonic oscillator models that have been used in the past to describe RCPG function.