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Homogeneous darboux polynomials and generalising integrable ODE systems
Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia |
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.
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. |
[2] |
O. I. Bogoyavlenskij,
Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.
doi: 10.1134/S1560354708060051. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, 2001.
doi: 10.1142/9789812811943. |
[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. |
[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. |
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. |
[2] |
O. I. Bogoyavlenskij,
Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.
doi: 10.1134/S1560354708060051. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, 2001.
doi: 10.1142/9789812811943. |
[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. |
[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. |
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