Abstract
Investigation of some statistical properties for a chaotic transient in a Fermi-Ulam model observed after a boundary crisis is considered in this chapter. The crisis is produced by a crossing of a stable and unstable manifold generated from the same saddle fixed point, hence creating also a homoclinic crossing. The system consists of a classical particle bouncing between two rigid and infinitely heavy walls. One of them is fixed while the other one is moving periodically in time. The dynamics of the system is given by a two-dimensional, nonlinear area-contracting map for the variables velocity of the particle immediately after a collision with the moving wall and time after such a collision. The collisions are assumed to be inelastic, hence a fractional loss of energy is observed leading to the existence of attractors in the phase space.
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Notes
- 1.
Depending on the type of the fixed points born at the bifurcations as well as their stability, the pitchfork bifurcation can be classified as supercritical or subcritical.
- 2.
By local chaos we want to say the chaos observed is confined by invariant spanning curves. In a situation of global chaos the particle can diffuse unlimitedly in the velocity axis therefore leading to a phenomena called as Fermi acceleration. A prototype system exhibiting such phenomena is an analogous 1-D model called as a bouncer. The only difference from the Fermi-Ulam model is the returning mechanism for a further collision. In the Fermi-Ulam it is provided by a fixed wall whereas in the bouncer it is due to the gravitational field only.
- 3.
Indeed it is an asymptotically stable focus.
- 4.
Direct collisions are defined as the collisions the particle has with the moving wall without leaving the collision zone, a region defined by x ∈ [−ε, +ε].
- 5.
In an indirect collision a particle leaves the collision zone, collides with the fixed wall, and is rebounded back for a further collision with the moving wall.
- 6.
This set of control parameters was chosen just before the crisis.
- 7.
Such condition is given by solution of G(ϕ c ) in mapping (6.1).
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Acknowledgements
EDL acknowledges the support from CNPq (303707/2015-1), FAPESP (2017/14414-2), and FUNDUNESP. MFM thanks to CNPq and CAPES for support.
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Leonel, E.D., Marques, M.F. (2019). An Investigation of the Chaotic Transient for a Boundary Crisis in the Fermi-Ulam Model. In: Macau, E. (eds) A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems . Nonlinear Systems and Complexity, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-78512-7_6
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