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doi: 10.3934/jcd.2022014
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Detecting and determining preserved measures and integrals of birational maps

1. 

Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway

2. 

Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus

3. 

Department of Mathematical and Physical Sciences, La Trobe University, Bundoora, VIC 3083, Australia

*Corresponding author: D. I. McLaren

Received  June 2021 Revised  May 2022 Early access May 2022

In this paper we use the method of discrete Darboux polynomials to calculate preserved measures and integrals of rational maps. The approach is based on the use of cofactors and Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps, thus lending weight to a previous ansatz [2]. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.

Citation: Elena Celledoni, Charalambos Evripidou, David I. McLaren, Brynjulf Owren, G. R. W. Quispel, Benjamin K. Tapley. Detecting and determining preserved measures and integrals of birational maps. Journal of Computational Dynamics, doi: 10.3934/jcd.2022014
References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishstadt, Mathematical Aspects of Classical and Celestial Mechanics, Encyclopedia of Mathematical Sciences, 3, Springer, Berlin, 1993. doi: 10.1007/978-3-642-61237-4.

[2]

E. Celledoni, C. Evripidou, D. I. McLaren, B. Owren, G. R. W. Quispel, B. K. Tapley and P. H. van der Kamp, Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps, J. Phys. A, 52 (2019), 31LT01, 11 pp. doi: 10.1088/1751-8121/ab294b.

[3]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202, 20 pp. doi: 10.1088/1751-8113/47/36/365202.

[4]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Discretization of polynomial vector fields by polarization, Proc. Roy. Soc. A, 471 (2014), 20150390, 10 pp. doi: 10.1098/rspa.2015.0390.

[5]

E. Celledoni, D. I. McLaren, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: The preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9 pp. doi: 10.1088/1751-8121/aafb1e.

[6]

E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2013), 025201, 12 pp. doi: 10.1088/1751-8113/46/2/025201.

[7]

G. Cheze and T. Combot, Symbolic Computations of first integrals for polynomial vector fields, JFoCM, 20 (2020), 681-752.  doi: 10.1007/s10208-019-09437-9.

[8]

G. Falqui and C.-M. Viallet, Singularity, complexity, and quasi-integrability of rational mappings, Commun. Math. Phys., 154 (1993), 111-125.  doi: 10.1007/BF02096835.

[9]

A. Gasull and V. Mañosa, A Darboux-type theory of integrability for discrete dynamical systems, Journal of Difference Equations and Applications, 8 (2010), 1171-1191.  doi: 10.1080/1023619021000054042.

[10]

F. Golse, A. Mahalov and B. Nicolaenko, Bursting dynamics of the 3D Euler equations in cylindrical domains, in Instability in Models Connected with Fluid Flows: 1, Int. Math. Ser. (N.Y.), vol 6 (2008), New York: Springer 300–338. doi: 10.1007/978-0-387-75217-4_7.

[11]

F. A. HaggarG. B. ByrnesG. R. W. Quispel and H. W. Capel, k-integrals and k-Lie symmetries in discrete dynamical systems, Physica A, 233 (1996), 379-394.  doi: 10.1016/S0378-4371(96)00142-2.

[12]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, Berlin, 2006.

[13]

J. Hietarinta, N. Joshi and F. W. Nijhoff, Discrete Systems and Integrability, CUP, 2016. doi: 10.1017/CBO9781107337411.

[14]

R. Hirota and K. Kimura, Discretization of the Euler top, J. Phys. Soc. Jap., 69 (2000), 627-630.  doi: 10.1143/JPSJ.69.627.

[15]

A. N. W. Hone and M. Petrera, Three dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras, Journal of Geometric Mechanics, 1 (2009), 55-85.  doi: 10.3934/jgm.2009.1.55.

[16]

A. N. W. Hone and G. R. W. Quispel, Analogues of Kahan's method for higher order equations of higher degree, in Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, (eds F. Nijhoff, Y. Shi, and D. Zhang), Springer (2020), 175–189. doi: 10.1007/978-3-030-57000-2_9.

[17]

W. Kahan, Unconventional numerical methods for trajectory calculations, Unpublished Lecture Notes, (1993).

[18]

K. Kimura and R. Hirota, Discretization of the Lagrange top, J. Phys. Soc. Jap., 69 (2000), 3193-3199.  doi: 10.1143/JPSJ.69.3193.

[19] E. M. McMillan, Topics in Modern Physics. A Tribute to E.U. Condon (eds E. Britton and H. Odabasi), Colorado University Press, Boulder, 1971. 
[20]

M. PetreraA. Pfadler and Y. B. Suris, On integrability of Hirota–Kimura type discretizations, Regular and Chaotic Dynamics, 16 (2011), 245-289.  doi: 10.1134/S1560354711030051.

[21]

G. R. W. QuispelH. W. CapelV. G. Papageorgiou and F. W. Nijhoff, Integrable mappings derived from soliton equations, Physica A, 173 (1991), 243-266.  doi: 10.1016/0378-4371(91)90258-E.

[22]

G. R. W. QuispelJ. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations, Physics Letters A, 126 (1988), 419-421.  doi: 10.1016/0375-9601(88)90803-1.

[23]

G. R. W. QuispelJ. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations. II, Physica D: Nonlinear Phenomena, 34 (1989), 183-192.  doi: 10.1016/0167-2789(89)90233-9.

[24]

J. A. G. Roberts and F. Vivaldi, Arithmetical method to detect integrability in maps, Phys Rev Lett, 90 (2003), 034102.  doi: 10.1103/PhysRevLett.90.034102.

[25]

P. H. van der KampE. CelledoniR. I. McLachlanD. I. McLarenB. Owren and G. R. W. Quispel, Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation, Journal of Physics A: Mathematical and Theoretical, 52 (2019), 045204.  doi: 10.1088/1751-8121/aaf51e.

[26]

P. H. van der KampT. E. KouloukasG. R. W. QuispelD. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multisums of products, Proc. Roy. Soc. A, 470 (2014), 20140481.  doi: 10.1098/rspa.2014.0481.

[27]

F. Wilczek, Fundamentals, Ten Keys to Reality, Penguin 2021. doi: 10.17104/9783406775536.

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishstadt, Mathematical Aspects of Classical and Celestial Mechanics, Encyclopedia of Mathematical Sciences, 3, Springer, Berlin, 1993. doi: 10.1007/978-3-642-61237-4.

[2]

E. Celledoni, C. Evripidou, D. I. McLaren, B. Owren, G. R. W. Quispel, B. K. Tapley and P. H. van der Kamp, Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps, J. Phys. A, 52 (2019), 31LT01, 11 pp. doi: 10.1088/1751-8121/ab294b.

[3]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202, 20 pp. doi: 10.1088/1751-8113/47/36/365202.

[4]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Discretization of polynomial vector fields by polarization, Proc. Roy. Soc. A, 471 (2014), 20150390, 10 pp. doi: 10.1098/rspa.2015.0390.

[5]

E. Celledoni, D. I. McLaren, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: The preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9 pp. doi: 10.1088/1751-8121/aafb1e.

[6]

E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2013), 025201, 12 pp. doi: 10.1088/1751-8113/46/2/025201.

[7]

G. Cheze and T. Combot, Symbolic Computations of first integrals for polynomial vector fields, JFoCM, 20 (2020), 681-752.  doi: 10.1007/s10208-019-09437-9.

[8]

G. Falqui and C.-M. Viallet, Singularity, complexity, and quasi-integrability of rational mappings, Commun. Math. Phys., 154 (1993), 111-125.  doi: 10.1007/BF02096835.

[9]

A. Gasull and V. Mañosa, A Darboux-type theory of integrability for discrete dynamical systems, Journal of Difference Equations and Applications, 8 (2010), 1171-1191.  doi: 10.1080/1023619021000054042.

[10]

F. Golse, A. Mahalov and B. Nicolaenko, Bursting dynamics of the 3D Euler equations in cylindrical domains, in Instability in Models Connected with Fluid Flows: 1, Int. Math. Ser. (N.Y.), vol 6 (2008), New York: Springer 300–338. doi: 10.1007/978-0-387-75217-4_7.

[11]

F. A. HaggarG. B. ByrnesG. R. W. Quispel and H. W. Capel, k-integrals and k-Lie symmetries in discrete dynamical systems, Physica A, 233 (1996), 379-394.  doi: 10.1016/S0378-4371(96)00142-2.

[12]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, Berlin, 2006.

[13]

J. Hietarinta, N. Joshi and F. W. Nijhoff, Discrete Systems and Integrability, CUP, 2016. doi: 10.1017/CBO9781107337411.

[14]

R. Hirota and K. Kimura, Discretization of the Euler top, J. Phys. Soc. Jap., 69 (2000), 627-630.  doi: 10.1143/JPSJ.69.627.

[15]

A. N. W. Hone and M. Petrera, Three dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras, Journal of Geometric Mechanics, 1 (2009), 55-85.  doi: 10.3934/jgm.2009.1.55.

[16]

A. N. W. Hone and G. R. W. Quispel, Analogues of Kahan's method for higher order equations of higher degree, in Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, (eds F. Nijhoff, Y. Shi, and D. Zhang), Springer (2020), 175–189. doi: 10.1007/978-3-030-57000-2_9.

[17]

W. Kahan, Unconventional numerical methods for trajectory calculations, Unpublished Lecture Notes, (1993).

[18]

K. Kimura and R. Hirota, Discretization of the Lagrange top, J. Phys. Soc. Jap., 69 (2000), 3193-3199.  doi: 10.1143/JPSJ.69.3193.

[19] E. M. McMillan, Topics in Modern Physics. A Tribute to E.U. Condon (eds E. Britton and H. Odabasi), Colorado University Press, Boulder, 1971. 
[20]

M. PetreraA. Pfadler and Y. B. Suris, On integrability of Hirota–Kimura type discretizations, Regular and Chaotic Dynamics, 16 (2011), 245-289.  doi: 10.1134/S1560354711030051.

[21]

G. R. W. QuispelH. W. CapelV. G. Papageorgiou and F. W. Nijhoff, Integrable mappings derived from soliton equations, Physica A, 173 (1991), 243-266.  doi: 10.1016/0378-4371(91)90258-E.

[22]

G. R. W. QuispelJ. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations, Physics Letters A, 126 (1988), 419-421.  doi: 10.1016/0375-9601(88)90803-1.

[23]

G. R. W. QuispelJ. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations. II, Physica D: Nonlinear Phenomena, 34 (1989), 183-192.  doi: 10.1016/0167-2789(89)90233-9.

[24]

J. A. G. Roberts and F. Vivaldi, Arithmetical method to detect integrability in maps, Phys Rev Lett, 90 (2003), 034102.  doi: 10.1103/PhysRevLett.90.034102.

[25]

P. H. van der KampE. CelledoniR. I. McLachlanD. I. McLarenB. Owren and G. R. W. Quispel, Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation, Journal of Physics A: Mathematical and Theoretical, 52 (2019), 045204.  doi: 10.1088/1751-8121/aaf51e.

[26]

P. H. van der KampT. E. KouloukasG. R. W. QuispelD. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multisums of products, Proc. Roy. Soc. A, 470 (2014), 20140481.  doi: 10.1098/rspa.2014.0481.

[27]

F. Wilczek, Fundamentals, Ten Keys to Reality, Penguin 2021. doi: 10.17104/9783406775536.

Figure 1.  Plots of the level sets $ p_{1,1} = 0 $ and $ p_{2,1} = 0 $ of the Kahan map in Example 1 (for $ h = \frac{1}{5} $), dotted red. Also shown are the corresponding second integrals of the ODE, $ x_1-2 $ and $ 1-x_1^2-x_2^2 $, solid blue
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