Idempotent differential equations

We show how to solve a large class of Functional Differential Equations (FDE) of the form xʹ(t) = dJ(t,x(f),x(f(t))) If the argument f(t) has certain properties then these equations are suitable for determining the initial conditions for problems from various fields such as control theory, traffic flow, spread of epidemics, and structured population growth. The required property on f is that it be idempotent, that is, f (f(t)) = t. We extend the methods given here to apply to more general cases for f , solving equations that describe the behavior of a quantity whose rate of change depends upon several types of deviation of the time variable such as reverse time flow, for example.