Scaling properties of the Fermi-Ulam accelerator model

Abstract

The chaotic low energy region (chaotic sea) of the Fermi-Ulam accelerator model is discussed within a scaling framework near the integrable to non-integrable transition. Scaling results for the average quantities (velocity, roughness, energy etc.) of the simplified version of the model are reviewed and it is shown that, for small oscillation amplitude of the moving wall, they can be described by scaling functions with the same characteristic exponents. New numerical results for the complete model are presented. The chaotic sea is also characterized by its Lyapunov exponents.

Describing Fermi acceleration with a scaling approach: The Bouncer model revisited

“…Such an approximation also retains the nonlinearity of the problem. This is a common approach used to speed up the numerical simulations [21][22][23][24] since no transcendental equations [25] need to be solved, as they must be in the complete model [5,26].…”

Scaling properties for a family of discontinuous mappings

“…As examples, we can mention some well known mappings having in common the choice of f (θ n , I n+1 ) = sin(θ n ) and h(θ n , I n+1 ) = 0: Chirikov's standard map [2,3], g(I n+1 ) = I n+1 , also known as Taylor-Chirikov's map; the bouncer model [4], g(I n+1 ) = ξ I n+1 ; the logistic twist map [5], g(I n+1 ) = I n+1 + ξ I 2 n+1 ; the Fermi-Ulam accelerator model [6,7], g(I n+1 ) = 2/I n+1 ; a generalized Fermi-Ulam accelerator (FU) model [8][9][10][11],…”

A simplified Fermi Accelerator Model under quadratic frictional force

“…This model is sometimes referred as to the Fermi-Ulam Model (FUM) [4][5][6] and was studied in many different versions and considering different approaches as well as external fields and damping forces [7][8][9][10]. Additionally, this subject has been object of intense study in last years [11][12][13][14] and many tools used to characterize such system can be extended to encompass to much more complex billiard problems. One of the approaches commonly used is the well known simplified version presented in [15].…”