# American Institute of Mathematical Sciences

October  2017, 37(10): 5271-5284. doi: 10.3934/dcds.2017228

## A locally integrable multi-dimensional billiard system

 Steklov Mathematical Institute, 8 Gubkina St. Moscow, 119991, Russia

Received  January 2017 Revised  May 2017 Published  June 2017

Fund Project: The research is supported by the RNF grant 14-50-00005

We consider a multi-dimensional billiard system in an $(n+1)$-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $γ$ of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $γ$) conjugated to the dynamics of a linear map?

Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions $± f$, where $f$ is even and defined in a neighborhood of the origin on the "horizontal" coordinate hyperplane. We prove that $f$ exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.

Citation: Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
##### References:

show all references

##### References:
The graph of $b^{-1/2}_\infty$ as a function of $\alpha/(2\pi)$. Two "gaps" correspond to the resonances $\frac\alpha{2\pi} = 3/10$ and $\frac\alpha{2\pi} = 1/3$
 [1] Nicolas Bedaride. Entropy of polyhedral billiard. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89 [2] Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov. Perfect retroreflectors and billiard dynamics. Journal of Modern Dynamics, 2011, 5 (1) : 33-48. doi: 10.3934/jmd.2011.5.33 [3] Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557 [4] David Cowan. A billiard model for a gas of particles with rotation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101 [5] Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310 [6] David Cowan. Rigid particle systems and their billiard models. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 111-130. doi: 10.3934/dcds.2008.22.111 [7] Alexey Glutsyuk, Yury Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories. Journal of Modern Dynamics, 2012, 6 (3) : 287-326. doi: 10.3934/jmd.2012.6.287 [8] Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096 [9] Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345 [10] Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775 [11] Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 [12] Salah Drabla, Salim A. Messaoudi, Fairouz Boulanouar. A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1329-1339. doi: 10.3934/dcdsb.2017064 [13] M. Bauer, A. Lopes. A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 107-116. doi: 10.3934/dcds.1997.3.107 [14] W. Patrick Hooper. Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons. Journal of Modern Dynamics, 2007, 1 (4) : 649-663. doi: 10.3934/jmd.2007.1.649 [15] Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993 [16] Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278 [17] Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 [18] Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038 [19] Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652 [20] Gerald Sommer, Di Zang. Parity symmetry in multi-dimensional signals. Communications on Pure & Applied Analysis, 2007, 6 (3) : 829-852. doi: 10.3934/cpaa.2007.6.829

2018 Impact Factor: 1.143