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A bouncing ball model with two nonlinearities: a prototype for Fermi acceleration. (English) Zbl 1171.70011

Summary: We study some dynamical properties of a bouncing ball model in the presence of an external force modelled by two nonlinear terms. The description of the model is made by the use of a two-dimensional nonlinear measure-preserving map on the variable velocity of the particle and time. We show that raising the straight of a control parameter which controls one of the nonlinearities, the positive Lyapunov exponent decreases in the average and suffers abrupt changes. We also show that for a specific range of control parameters, the model exhibits the phenomenon of Fermi acceleration. The explanation of both behaviours is given in terms of the shape of external force and due to a discontinuity of the moving wall’s velocity.

MSC:

70Q05 Control of mechanical systems
70K40 Forced motions for nonlinear problems in mechanics
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