Sometimes the future state of a system depends not only on the current state (position, velocity, acceleration, etc.) but also on the previous state. Equations for modeling such systems are known as delay differential equations (DDEs), difference differential equations, retarded equations, etc. In a system with hysteresis, it matters not only where you are but how you got there.

The most basic theory of delay differential equations is fairly simple. Suppose you have an equation like the following.

u′(t) + u(t − ω) = f(t).

To uniquely determine a solution, you’d need an initial condition. And we’d need more than the value of u(0). We’d need a function g(t) that give the value of u on the entire interval [0, ω].

So we initially have the value of u over [0, ω]. Next, over the interval [ω, 2ω] the value of u(t − ω) is known. We could replace that term in the DDE with g(t), And after we’ve solved our equation over [ω, 2ω], we can use the solution to solve the equation over [2ω, 3ω]. This process is called the method of steps.

Although you can solve DDEs using the method of steps, this might not be the best approach. It might be more computationally efficient, or theoretically convenient, to use another method to solve such equations, such as Laplace transforms. The method of transforms might convince you that a solution exists, but it might not, for example, be the best way to determine the limiting behavior of solutions.

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