Main results

In this work, we consider mainly adiabatic differential equations of the form

εdx/dτ = F(x,λ(τ)),

where x is the vector of dynamic variables, τ = εt is the slow time, ε the small adiabatic parameter, and λ(τ) the slowly varying parameter. The main idea is to establish connections between solutions of this equation and those of the associated family of frozen dynamical systems dx/dt = F(x,λ).

Detailed results are contained in my Ph.D. thesis, and a summary of mathematical results can be found in this proceedings. A non-technical description is given here, several physical applications are found in this paper, while chaotic hysteresis is briefly described in this letter.

The rotating pendulum is mounted on a table rotating with angular frequency Ω. It experiences friction, gravity and an intertial torque. When the frequency Ω is varied periodically, this system sometimes displays chaotic hysteresis: each time the frequency becomes large, the pendulum leaves the origin for one of two possible equilibria at an angle. The sequence of visited equilibria is chaotic for some values of the parameters.