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January  2021, 8(1): 1-8. doi: 10.3934/jcd.2021001

Homogeneous darboux polynomials and generalising integrable ODE systems

Peter H. van der Kamp , , D. I. McLaren and  G. R. W. Quispel

Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

* Corresponding author: Peter H. van der Kamp

Received  February 2020 Revised  May 2020 Published  August 2020

We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.

Keywords: Darboux polynomials, Lotka-Volterra systems, Liouville integrability, superintegrability, ordinary differential equations.
Mathematics Subject Classification: 34C14, 70H06.
Citation: Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001
References:
[1]

D.W. Albrecht, E.L. Mansfield and A.E. Milne, Algorithms for special integrals of ordinary differential equations, J. Phys. A: Math. Gen., 29 (1996), 973-991.  doi: 10.1088/0305-4470/29/5/013.  Google Scholar

[2]

O. I. Bogoyavlenskij, Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.  doi: 10.1134/S1560354708060051.  Google Scholar

[3]

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: Math. Theor., 52 (2019), 11 pp. doi: 10.1088/1751-8121/ab294b.  Google Scholar

[4]

H. Christodoulidi, A. N. W. Hone and T. E. Kouloukas, A new class of integrable Lotka-Volterra systems, J. Comput. Dyn., 6 (2019), 223-237.  doi: 10.3934/jcd.2019011.  Google Scholar

[5]

C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math. Anal. Appl., 195 (1995), 719-735.  doi: 10.1006/jmaa.1995.1385.  Google Scholar

[6]

C. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the dressing chain, J. Comput. Dyn., 6 (2019), 277-306.  doi: 10.3934/jcd.2019014.  Google Scholar

[7]

A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, 2001. doi: 10.1142/9789812811943.  Google Scholar

[8]

T. E. Kouloukas, G. R. W. Quispel and P. Vanhaecke, Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization, J. Phys. A, 49 (2016), 13 pp. doi: 10.1088/1751-8113/49/22/225201.  Google Scholar

[9]

D. T. Tran, Complete Integrability of Maps Obtained as Reductions of Integrable Lattice Equations, Ph.D thesis, La Trobe University, Australia, 2011. Google Scholar

[10]

P. H. van der Kamp, T. E. Kouloukas, G. R. W. Quispel, D. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140481. doi: 10.1098/rspa.2014.0481.  Google Scholar

show all references

References:
[1]

D.W. Albrecht, E.L. Mansfield and A.E. Milne, Algorithms for special integrals of ordinary differential equations, J. Phys. A: Math. Gen., 29 (1996), 973-991.  doi: 10.1088/0305-4470/29/5/013.  Google Scholar

[2]

O. I. Bogoyavlenskij, Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.  doi: 10.1134/S1560354708060051.  Google Scholar

[3]

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: Math. Theor., 52 (2019), 11 pp. doi: 10.1088/1751-8121/ab294b.  Google Scholar

[4]

H. Christodoulidi, A. N. W. Hone and T. E. Kouloukas, A new class of integrable Lotka-Volterra systems, J. Comput. Dyn., 6 (2019), 223-237.  doi: 10.3934/jcd.2019011.  Google Scholar

[5]

C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math. Anal. Appl., 195 (1995), 719-735.  doi: 10.1006/jmaa.1995.1385.  Google Scholar

[6]

C. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the dressing chain, J. Comput. Dyn., 6 (2019), 277-306.  doi: 10.3934/jcd.2019014.  Google Scholar

[7]

A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, 2001. doi: 10.1142/9789812811943.  Google Scholar

[8]

T. E. Kouloukas, G. R. W. Quispel and P. Vanhaecke, Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization, J. Phys. A, 49 (2016), 13 pp. doi: 10.1088/1751-8113/49/22/225201.  Google Scholar

[9]

D. T. Tran, Complete Integrability of Maps Obtained as Reductions of Integrable Lattice Equations, Ph.D thesis, La Trobe University, Australia, 2011. Google Scholar

[10]

P. H. van der Kamp, T. E. Kouloukas, G. R. W. Quispel, D. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140481. doi: 10.1098/rspa.2014.0481.  Google Scholar

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