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

Homogeneous darboux polynomials and generalising integrable ODE systems

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.

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. AlbrechtE.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. ChristodoulidiA. 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. EvripidouP. 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. AlbrechtE.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. ChristodoulidiA. 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. EvripidouP. 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

[1]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[3]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[4]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[5]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[6]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[7]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[8]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[9]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[10]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[11]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[12]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[13]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[14]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[15]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[16]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[17]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[18]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[19]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[20]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

 Impact Factor: 

Article outline

[Back to Top]