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A modification of the one-dimensional Fermi accelerator model is consideredin this work. The dynamics of a classical particle of mass $m$, confined tobounce elastically between two rigid walls where one is described by anon-linear van der Pol type oscillator while the other one is fixed, working asa re-injection mechanism of the particle for a next collision, is carefullymade by the use of a two-dimensional non-linear mapping. Two cases areconsidered: (i) the situation where the particle has mass negligible ascompared to the mass of the moving wall and does not affect the motion of it;(ii) the case where collisions of the particle does affect the movement of themoving wall. For case (i) the phase space is of mixed type leading us toobserve a scaling of the average velocity as a function of the parameter($\c{hi}$) controlling the non-linearity of the moving wall. For large$\c{hi}$, a diffusion on the velocity is observed leading us to conclude thatFermi acceleration is taking place. On the other hand for case (ii), the motionof the moving wall is affected by collisions with the particle. However due tothe properties of the van der Pol oscillation, the moving wall relaxes again toa limit cycle. Such kind of motion absorbs part of the energy of the particleleading to a suppression of the unlimited energy gain as observed in case (i).The phase space shows a set of attractors of different periods whose basin ofattraction has a complicate organization.

A one-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator