A brief comment on hysteresis

You might hear hysteresis described as a phenomena where the solution to a differential equation depends on its history. But that doesn't make sense: the solution to a differential equation always depends on its history. The solution at any point in time depends (only) on its immediately preceding state. You can take the state at any particular time, say position and velocity, as the initial condition for the solution going forward.

But sometimes there's something about the solution to a differential equation that does not depend on its history. For a driven linear oscillator, the frequency of the solution depends only on the frequency of the driving function. But this is not necessarily true of a nonlinear oscillator. The same driving frequency might correspond to two different solution frequencies, depending on whether the frequency of the driving function increased to its final value or decreased to that value.

Hysteresis is interesting, but it's more remarkable that linear system does not exhibit hysteresis. Why should the solution frequency depend only on the driving frequency and not on the state of the system?

The future of system with hysteresis still depends only on its immediately preceding state. But there may be a thin set of states that lead to solutions with a given frequency, and the easiest way to arrive in such a set may be to sneak up on it slowly.

• Discrete bifurcation
• Continuous bifurcation