ODE solutions form a hierarchy. (a) The bottom of the hierarchy consists of a single pair of and time courses. These are the empirically accessible objects of the theory and would be tested against real and time courses to determine how well the predicted time courses fit experimentally measured time courses. The three columns of time courses derive from the phase planes in B and represent monostable sublethal, bistable lethal, and monostable lethal time courses, respectively. The monostable time courses and top bistable time course are from initial conditions . The middle bistable time course is from initial conditions (0, 0.16) and indicates that preactivating stress responses to 16% of their maximum value are not sufficient to flip state. However, at 17% of stress responses preactivation, the system flips state and survives an insult that would be lethal from initial conditions (0, 0). (b) The middle of the hierarchy consists of phase planes showing trajectories at all possible initial conditions for a given set of parameters. Trajectories are converted to pairs of and time courses by well-established methods (e.g., Runge-Kutta). The phase planes shown correspond to the and time course pairs in A. The middle phase plane is bistable. The survival and death attractors are shown as green and red circles, respectively. The three trajectories on the middle phase plane correspond to the above time courses, as indicated. (c) The top level of the hierarchy is a bifurcation diagram. When the control parameter for the bifurcation diagram is injury magnitude, , we call the resulting bifurcation diagram an injury course. The phase planes in B are indicated by dashed lines, labeled accordingly. The system parameters in A give rise to a doubly bistable injury course. The bifurcation diagrams shown would constitute the “answer” to and would fully characterize the injury system represented by the parameter set in A. In our deductive theory, the cause of cell death is always . The injury course (bifurcation diagram) becomes the way to formulate any injury system, and it provides a basis for a comprehensive and systematic approach to therapeutics. The bistable region is indicated by the open circles, which are unstable repeller fixed points. The areas marked by purple boxes are the therapeutic region, those injury states where it is possible in principle to flip state and prevent a system that would normally die from dying. That equation (1) not only predicts bistability but also provides a systematic and quantitative understanding of it is perhaps the most important novel contribution of the nonlinear dynamical theory of cell injury.