American Institute of Mathematical Sciences

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

A locally integrable multi-dimensional billiard system

Dmitry Treschev *,

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.

Keywords: Billiard system, integrable billiard system, multi-dimensional billiard system.
Mathematics Subject Classification: Primary: 37J10, 37J40; Secondary: 65P10.
Citation: Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
References:
[1]

A. Avila, J. De Simoi and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558.  doi: 10.4007/annals.2016.184.2.5.  Google Scholar

[2]

M. Bialy and A. E. Mironov, Angular Billiard and Algebraic Birkhoff conjecture, Adv. Math., 313 (2017), 102-126, arXiv: 1601.03196 doi: 10.1016/j.aim.2017.04.001.  Google Scholar

[3]

G. D. Birkhoff, Dynamical Systems American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. 1966  Google Scholar

[4]

S. V. Bolotin, Integrable Birkhoff billiards, Vestnik Moskov. Univ. Ser. I Mat. Mekh. , 2 (1990), 33-36, (in Russian); translated in Mosc. Univ. Mech. Bull., 2 (1990), 10-13.  Google Scholar

[5]

S. V. Bolotin and D. V. Treschev, The anti-integrable limit, Russian Math. Surveys, 70 (2015), 975-1030.  doi: 10.4213/rm9692.  Google Scholar

[6]

B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery, Products of polynomials in many variables, Journal of Number Theory, 36 (1990), 219-245.  doi: 10.1016/0022-314X(90)90075-3.  Google Scholar

[7]

A. Delshams, Yu. Fedorov and R. Ramirez-Ros, Homoclinic billiard orbits inside symmetrically perturbed ellipsoids, Nonlinearity, 14 (2001), 1141-1195.  doi: 10.1088/0951-7715/14/5/313.  Google Scholar

[8]

A. Glutsyuk and E. Shustin On polynomially integrable planar outer billiards and curves with symmetry property, preprint arXiv: 1607.07593. Google Scholar

[9]

V. V. Kozlov, Two-link billiard trajectories: Extremal properties and stability, J. Appl. Math. Mech., 64 (2000), 903-907.  doi: 10.1016/S0021-8928(00)00121-0.  Google Scholar

[10]

V. V. Kozlov, Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard, Proc. Steklov Inst. Math., 273 (2011), 196-213.  doi: 10.1134/S0081543811040092.  Google Scholar

[11]

V. V. Kozlov, Polynomial conservation laws for the Lorentz gas and the Boltzmann-Gibbs gas, Russian Math. Surveys, 71 (2016), 253-290.  doi: 10.4213/rm9707.  Google Scholar

[12]

V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Translations of Mathematical Monographs, 89 Amer. Math. Soc., Providence, RI, 1991.  Google Scholar

[13]

S. Tabachnikov, Geometry and Billiards Student Mathematical Library, 30 Providence, RI -Amer. Math. Soc, 2005. doi: 10.1090/stml/030.  Google Scholar

[14]

D. Treschev, Billiard map and rigid rotation, Phys. D, 255 (2013), 31-34.  doi: 10.1016/j.physd.2013.04.003.  Google Scholar

[15]

D. V. Treschev, On a conjugacy problem in billiard dynamics, Proc. Steklov Inst. Math., 289 (2015), 291-299.  doi: 10.1134/S0081543815040173.  Google Scholar

[16]

H. Whitney, Analytic extensions of functions defined in closed sets, Transactions of the American Mathematical Society, American Mathematical Society, 36 (1934), 63-89. doi: 10. 1090/S0002-9947-1934-1501735-3.  Google Scholar

show all references

References:
[1]

A. Avila, J. De Simoi and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558.  doi: 10.4007/annals.2016.184.2.5.  Google Scholar

[2]

M. Bialy and A. E. Mironov, Angular Billiard and Algebraic Birkhoff conjecture, Adv. Math., 313 (2017), 102-126, arXiv: 1601.03196 doi: 10.1016/j.aim.2017.04.001.  Google Scholar

[3]

G. D. Birkhoff, Dynamical Systems American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. 1966  Google Scholar

[4]

S. V. Bolotin, Integrable Birkhoff billiards, Vestnik Moskov. Univ. Ser. I Mat. Mekh. , 2 (1990), 33-36, (in Russian); translated in Mosc. Univ. Mech. Bull., 2 (1990), 10-13.  Google Scholar

[5]

S. V. Bolotin and D. V. Treschev, The anti-integrable limit, Russian Math. Surveys, 70 (2015), 975-1030.  doi: 10.4213/rm9692.  Google Scholar

[6]

B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery, Products of polynomials in many variables, Journal of Number Theory, 36 (1990), 219-245.  doi: 10.1016/0022-314X(90)90075-3.  Google Scholar

[7]

A. Delshams, Yu. Fedorov and R. Ramirez-Ros, Homoclinic billiard orbits inside symmetrically perturbed ellipsoids, Nonlinearity, 14 (2001), 1141-1195.  doi: 10.1088/0951-7715/14/5/313.  Google Scholar

[8]

A. Glutsyuk and E. Shustin On polynomially integrable planar outer billiards and curves with symmetry property, preprint arXiv: 1607.07593. Google Scholar

[9]

V. V. Kozlov, Two-link billiard trajectories: Extremal properties and stability, J. Appl. Math. Mech., 64 (2000), 903-907.  doi: 10.1016/S0021-8928(00)00121-0.  Google Scholar

[10]

V. V. Kozlov, Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard, Proc. Steklov Inst. Math., 273 (2011), 196-213.  doi: 10.1134/S0081543811040092.  Google Scholar

[11]

V. V. Kozlov, Polynomial conservation laws for the Lorentz gas and the Boltzmann-Gibbs gas, Russian Math. Surveys, 71 (2016), 253-290.  doi: 10.4213/rm9707.  Google Scholar

[12]

V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Translations of Mathematical Monographs, 89 Amer. Math. Soc., Providence, RI, 1991.  Google Scholar

[13]

S. Tabachnikov, Geometry and Billiards Student Mathematical Library, 30 Providence, RI -Amer. Math. Soc, 2005. doi: 10.1090/stml/030.  Google Scholar

[14]

D. Treschev, Billiard map and rigid rotation, Phys. D, 255 (2013), 31-34.  doi: 10.1016/j.physd.2013.04.003.  Google Scholar

[15]

D. V. Treschev, On a conjugacy problem in billiard dynamics, Proc. Steklov Inst. Math., 289 (2015), 291-299.  doi: 10.1134/S0081543815040173.  Google Scholar

[16]

H. Whitney, Analytic extensions of functions defined in closed sets, Transactions of the American Mathematical Society, American Mathematical Society, 36 (1934), 63-89. doi: 10. 1090/S0002-9947-1934-1501735-3.  Google Scholar

$\frac\alpha{2\pi} = 3/10$ and $\frac\alpha{2\pi} = 1/3$">Figure 1.  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 $\frac\alpha{2\pi} = 1/3$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[2]

Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021017
[3]

Shoufeng Ji, Jinhuan Tang, Minghe Sun, Rongjuan Luo. Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021051
[4]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198
[5]

Zhisong Chen, Shong-Iee Ivan Su. Assembly system with omnichannel coordination. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021047
[6]

Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018
[7]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[8]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[9]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[10]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008
[11]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026
[12]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
[13]

Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021018
[14]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[15]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[16]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[17]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[18]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[19]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
[20]

Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (69)
  • HTML views (64)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences