Predicting the behavior of heterogeneous nonequilibrium systems is currently analytically intractable. Consequently, complex biological systems have resisted unifying principles. Here, I introduce a mapping from dynamical systems to battery-resistor circuits. I show that in these transformed variables (i) arbitrary numbers of heterogeneous dynamical transitions can be reduced to a Thevenin equivalent resistor which is invariant to driving from equilibrium, (ii) resistors (together with the external driving sources) are sufficient to describe system behavior, and (iii) the resistor's directional symmetry leads to universal theorems of nonequilibrium behavior. This mapping is used to derive two general steady-state relations. First, for any cyclic process, the maximum amplification of any state is tightly bounded by the total dissipation of all states; experimental data are used to show that the master signal protein Ras achieves this bound. Second, for any process, the response of any reaction due to driving any other reaction is identical to the reciprocal response rescaled by the ratio of the corresponding Thevenin resistors. This result generalizes Onsager's reciprocal relation to the strongly driven regime and makes a testable prediction about how systems should be designed or evolved to maximize response. These analytic results represent a new perspective applicable to biological complexity and suggest that this mapping provides the natural variables to study heterogeneous nonequilibrium systems.