| A billiard is defined by a connected region Q ⊂ ℜD with the boundary ∂Q ⊂ ℜ separating Q from its complement. A point-like particle moves freely inside the billiard along geodesic lines until hits the boundary. In this work we revisit the problem of a classical particle bouncing elastically inside a periodically time varying Oval billiard. The problem is described using a four dimensional mapping for the variables velocity of the particle namely: the time immediately after a collision with the moving boundary; the angle that the trajectory of the particle does with the tangent at the position of the hit; and the angular position of the particle along the boundary. Our main goal is to understand and describe the behaviour of the particle's average velocity (and hence its energy) as a function of the number of collisions with the boundary. It was recently shown for a time dependent oval billiard that, in certain cases under the breathing perturbation, the particle does not exhibit unlimited energy growth. As we shall shown in our work, the breathing geometry can indeed lead the particle to experience Fermi acceleration. However, the slope of growth is rather smaller as compared to the non breathing case. The small growing exponent for the average velocity was the main reason to conclude that Fermi acceleration was not observed in the breathing case. Our results reinforce the LRA conjecture. After confirm the existence the mechanism of Fermi acceleration we introduced inelastic collision into the model. We observe that dissipation causes a drastic consequence on the velocity's behaviour. We observed that for short time, the deviation of the average velocity as well as its energy grows according to a power law and suddenly it bends towards a regime of saturation for long enough values of time. It must be emphasized that different values of dissipation generate different behaviours, such kind of behaviours can be usually described using scaling approach. We observed that depending on what kind of dissipation we introduce one can observe different asymptotic behaviors including transients, attracting fixed points and locking, chaotic attractors and even crisis events as the damping coefficients are varied. |
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