Comptes Rendus
no. 9-10
Mathematical Problems in Mechanics
Continuous orbit transitions in a one-dimensional inelastic particle system
[Transitions continues entre orbites dans un système de particules inélastique unidimensionnel]

Présenté par : Philippe G. Ciarlet

Rong Yang1 ; Jonathan J. Wylie2, 3
1 Joint Advanced Research Center of University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangsu, China
2 Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
3 Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 593-595.

Nous étudions les transitions continues entre différentes orbites périodiques dans un système unidimensionnel inélastique à deux particules. Nous expliquons pourquoi les transitions continues qui apparaissent lorsque l'on ajoute ou enlève une collision sont, en général, de codimension 2. Cependant, nous montrons qu'il existe un ensemble infini de transitions dégénérées de codimension 1. Nous fournissons une méthode qui, en se basant uniquement sur l'ensemble des collisions qui interviennent dans les orbites, donne un critère simple pour déterminer quelles transitions sont dégénérées.

Continuous transitions between different periodic orbits in a one-dimensional inelastic particle system with two particles are investigated. We explain why continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. However, we show that there are an infinite set of degenerate transitions of co-dimension 1. We provide an analysis that gives a simple criteria to classify which transitions are degenerated purely from the discrete set of collisions that occur in the orbits.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.04.016
Rong Yang 1 ; Jonathan J. Wylie 2, 3

1 Joint Advanced Research Center of University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangsu, China
2 Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
3 Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA 
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Rong Yang; Jonathan J. Wylie. Continuous orbit transitions in a one-dimensional inelastic particle system. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 593-595. doi : 10.1016/j.crma.2010.04.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.016/

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[2] S. McNamara; J. Barrat Energy flux into a fluidized granular medium at a vibrating wall, Phys. Rev. E, Volume 55 (1997), pp. 7767-7770

[3] S. McNamara; W.R. Young Inelastic collapse and clumping in a one-dimensional granular medium, Phys. Fluids A, Volume 4 (1992), pp. 496-504

[4] J.J. Wylie; Q. Zhang Periodic orbits of a one-dimensional inelastic particle system, C. R. Math., Volume 339 (2004), pp. 603-606

[5] J.J. Wylie; Q. Zhang Collapse of periodic orbits in a driven inelastic particle system, Phys. Rev. E, Volume 74 (2006), p. 011305

[6] J. Yang Dynamics of a one-dimensional inelastic particle system, Phys. Rev. E, Volume 61 (2000), pp. 2920-2923

[7] R. Yang, J.J. Wylie, Degenerate orbit transitions in a one-dimensional inelastic particle system, submitted for publication

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