I am trying to understand what kind of ODE or system of ODE would result in a monotonically increasing solution with multiple steady states. For lack of better terminology, a solution that increases in "steps", and moves to the next (higher) steady state after a previous (lower) steady state has been obtained.
For a single "step" that reaches a single steady state, a good candidate ODE can be the following:
$$A \rightleftharpoons B$$ $$ \frac{dB}{dT} = \theta_1A - \theta_2B$$
To achieve a step-like solution with multiple, increasing, steady states I have tried adding other steps to the ODE, turning it to a system of ODE such as: $$A \rightleftharpoons B \rightleftharpoons C$$ $$ \frac{dB}{dT} = \theta_1A - \theta_2B - \theta_3B + \theta_4C $$ $$ \frac{dC}{dT} = \theta_3B - \theta_4C $$
with the hope that the sum of B and C would lead to the desired step-like response. Sadly, the value $B(t )+C(t)$ obtained from solving such system results in a single steady state.
What kind of differential equation could be resulting in the desired multi-step response?