doi: 10.3934/dcdss.2022104
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Integrability and linearizability of symmetric three-dimensional quadratic systems

1. 

Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, Maribor, 2000, Slovenia

2. 

Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, Maribor, 2000, Slovenia

3. 

Faculty of Natural Science and Mathematics, University of Maribor, Koroška cesta 160, Maribor, 2000, Slovenia

The paper is dedicated to prof. Jibin Li on his 80th birthday

Received  December 2021 Revised  April 2022 Early access April 2022

Fund Project: The authors acknowledge the support of the work by the Slovenian Research Agency (core research program P1-0306)

We study local integrability and linearizability of polynomial and analytic systems of ODEs. It is proven that in the case of non-degenerate singularity if an analytic system is completely integrable and one equation is linearizable, then the system is linearizable in a neighborhood of the singularity. Some integrable and linearizable systems in a family of three-dimensional quadratic autonomous systems of ODEs depending on ten parameters are found.

Citation: Barbara Arcet, Valery G. Romanovski. Integrability and linearizability of symmetric three-dimensional quadratic systems. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022104
References:
[1]

V. AntonovW. FernandesV. Romanovski and N. Shcheglova, First integrals of the May-Leonard asymmetric system, Mathematics, 7 (2019), 292.  doi: 10.3390/math7030292.

[2]

B. Arcet, J. Giné and V. Romanovski, Linearizability of planar polynomial Hamiltonian systems, Nonlinear Anal. Real World Appl., 63 (2022), 103422, 19 pp. doi: 10.1016/j.nonrwa.2021.103422.

[3]

W. Aziz, Integrability and linearizability of three dimensional vector fields, Qual. Theory Dyn. Syst., 13 (2014), 197-213.  doi: 10.1007/s12346-014-0113-0.

[4]

W. Aziz, Integrability and linearizability problems of three dimensional Lotka-Volterra equations of rank-2, Qual. Theory Dyn. Syst., 18 (2019), 1113-1134.  doi: 10.1007/s12346-019-00329-5.

[5]

W. Aziz and C. Christopher, Local integrability and linearizability of three-dimensional Lotka-Volterra systems, Appl. Math. Comput., 219 (2011), 4067-4081.  doi: 10.1016/j.amc.2012.10.051.

[6]

L. R. Berrone and H. J. Giacomini, Inverse Jacobian multipliers, Rend. Circ. Mat. Palermo, 52 (2003), 77-130.  doi: 10.1007/BF02871926.

[7]

Y. N. Bibikov, Local Theory on Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702. Springer-Verlag, Berlin-New York, 1979.

[8]

L. Cairó and J. Llibre, Darboux integrability for 3D Lotka-Volterra systems, J. Phys. A, 33 (2000), 2395-2406.  doi: 10.1088/0305-4470/33/12/307.

[9]

X. ChenJ. GinéV. G. Romanovski and D. S. Shafer, The $1:-q$ resonant center problem for certain cubic Lotka-Volterra systems, Appl. Math. Comput., 218 (2012), 11620-11633.  doi: 10.1016/j.amc.2012.05.045.

[10]

Y. T. Christodoulides and P. A. Damianou, Darboux polynomials for Lotka-Volterra Systems in three dimensions, J. Nonlinear Math. Phys., 16 (2009), 339-354.  doi: 10.1142/S1402925109000261.

[11]

C. ChristopherP. Mardesic and C. Rousseau, Normalizable, integrable and linearizable saddle points for complex quadratic systems in $\mathbb{C}^2$, J. Dyn. Control. Syst., 9 (2003), 311-363.  doi: 10.1023/A:1024643521094.

[12]

C. Christopher and C. Rousseau, Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbb{C}^2$, Publ. Mat., 45 (2001), 95-123.  doi: 10.5565/PUBLMAT_45101_04.

[13]

W. Decker, G. M. Greuel, G. Pfister and H. Shönemann, Singular (4-1-2-A Computer Algebra System for Polynomial Computations, 2019, Available from: https://www.singular.uni-kl.de.

[14]

W. Decker, S. Laplagne, G. Pfister and H. Shönemann, SINGULAR (3- 1 Library for Computing the Prime Decomposition and Radical of Ideals, Primdec.lib), 2010.

[15]

M. DukarićR. Oliveira and V. G. Romanovski, Local integrability and linearizability of a $(1:-1:-1)$ resonant quadratic system, J. Dyn. Differ. Equ., 29 (2011), 597-613.  doi: 10.1007/s10884-015-9486-2.

[16]

H. Dulac, Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Sci. Math., 32 (1908), 230-252. 

[17]

A. FronvilleA. P. Sadovski and H. Żołądek, Solution of the $1:-2$ resonant center problem in the quadratic case, Fundam. Math., 157 (1998), 191-207.  doi: 10.4064/fm-157-2-3-191-207.

[18]

Z. HuM. AldazharovaT. M. Aldibekov and V. G. Romanovski, Integrability of $3$-dim polynomial systems with three invariant planes, Nonlinear Dyn., 74 (2013), 1077-1092.  doi: 10.1007/s11071-013-1025-2.

[19]

Z. HuM. Han and V. G. Romanovski, Local integrability of a family of three-dimensional quadratic systems, Phys. D, 265 (2013), 78-86.  doi: 10.1016/j.physd.2013.09.001.

[20]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.  doi: 10.1142/S0218127403006352.

[21]

J. Li and Y. Lin, Normal form and critical points of the period of closed orbits for planar autonomous systems, Acta Math. Sin., 34 (1991), 490-501. 

[22]

Y. Liu and J. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1990), 10-23. 

[23]

Y. Liu, J. Li and W. Huang, Planar Dynamical Systems: Selected Classical Problems, De Gruyter, Berlin, Science Press New York, Ltd., New York, 2014. doi: 10.1515/9783110298369.

[24]

Y. Liu and P. Xiao, Critical point quantities and integrability conditions for a class of quintic systems, J. Cent. South Univ. Technol, 11 (2004), 109-112.  doi: 10.1007/s11771-004-0023-4.

[25]

J. LlibreC. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359.  doi: 10.1016/j.bulsci.2011.10.003.

[26]

J. Llibre and G. Rodríguez, Invariant hyperplanes and Darboux integrability for d-dimensional polynomial differential systems, Bull. Sci. Math., 124 (2000), 599-619.  doi: 10.1016/S0007-4497(00)01061-7.

[27]

J. Llibre and X. Zhang, On the Darboux integrability of polynomial differential systems, Qual. Theory Dyn. Syst., 11 (2012), 129-144.  doi: 10.1007/s12346-011-0053-x.

[28]

V. A. Pliss, On the reduction of an analytic system of differential equations to linear form, Differ. Equ., 5 (1969), 796-802. 

[29]

V. G. Romanovski and M. Prešern, An approach to solving systems of polynomials via modular arithmetics with applications, J. Comput. Appl. Math., 236 (2011), 196-208.  doi: 10.1016/j.cam.2011.06.018.

[30]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems. A Computational Algebra Approach, Birkhäuser, Boston-Basel-Berlin, 2009. doi: 10.1007/978-0-8176-4727-8.

[31]

V. G. RomanovskiY. Xia and X. Zhang, Varieties of local integrability of analytic differential systems and their applications, J. Differ. Equ., 257 (2014), 3079-3101.  doi: 10.1016/j.jde.2014.06.007.

[32]

P. Wang, M. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, ACM SIGSAM Bull., 16 (1982), 23 pp.

[33]

K. Wu and X. Zhang, Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces, Phys. D, 244 (2013), 25-35.  doi: 10.1016/j.physd.2012.10.011.

[34]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics, 47. Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.

[35]

X. Zhang, Analytic normalization of analytic integrable systems and the embedding flows, J. Differ. Equ., 250 (2008), 1080-1092.  doi: 10.1016/j.jde.2008.01.001.

[36]

H. Żołądek, The problem of center for resonant singular points of polynomial vector fields, J. Differ. Equ., 137 (1997), 94-118.  doi: 10.1006/jdeq.1997.3260.

show all references

References:
[1]

V. AntonovW. FernandesV. Romanovski and N. Shcheglova, First integrals of the May-Leonard asymmetric system, Mathematics, 7 (2019), 292.  doi: 10.3390/math7030292.

[2]

B. Arcet, J. Giné and V. Romanovski, Linearizability of planar polynomial Hamiltonian systems, Nonlinear Anal. Real World Appl., 63 (2022), 103422, 19 pp. doi: 10.1016/j.nonrwa.2021.103422.

[3]

W. Aziz, Integrability and linearizability of three dimensional vector fields, Qual. Theory Dyn. Syst., 13 (2014), 197-213.  doi: 10.1007/s12346-014-0113-0.

[4]

W. Aziz, Integrability and linearizability problems of three dimensional Lotka-Volterra equations of rank-2, Qual. Theory Dyn. Syst., 18 (2019), 1113-1134.  doi: 10.1007/s12346-019-00329-5.

[5]

W. Aziz and C. Christopher, Local integrability and linearizability of three-dimensional Lotka-Volterra systems, Appl. Math. Comput., 219 (2011), 4067-4081.  doi: 10.1016/j.amc.2012.10.051.

[6]

L. R. Berrone and H. J. Giacomini, Inverse Jacobian multipliers, Rend. Circ. Mat. Palermo, 52 (2003), 77-130.  doi: 10.1007/BF02871926.

[7]

Y. N. Bibikov, Local Theory on Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702. Springer-Verlag, Berlin-New York, 1979.

[8]

L. Cairó and J. Llibre, Darboux integrability for 3D Lotka-Volterra systems, J. Phys. A, 33 (2000), 2395-2406.  doi: 10.1088/0305-4470/33/12/307.

[9]

X. ChenJ. GinéV. G. Romanovski and D. S. Shafer, The $1:-q$ resonant center problem for certain cubic Lotka-Volterra systems, Appl. Math. Comput., 218 (2012), 11620-11633.  doi: 10.1016/j.amc.2012.05.045.

[10]

Y. T. Christodoulides and P. A. Damianou, Darboux polynomials for Lotka-Volterra Systems in three dimensions, J. Nonlinear Math. Phys., 16 (2009), 339-354.  doi: 10.1142/S1402925109000261.

[11]

C. ChristopherP. Mardesic and C. Rousseau, Normalizable, integrable and linearizable saddle points for complex quadratic systems in $\mathbb{C}^2$, J. Dyn. Control. Syst., 9 (2003), 311-363.  doi: 10.1023/A:1024643521094.

[12]

C. Christopher and C. Rousseau, Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbb{C}^2$, Publ. Mat., 45 (2001), 95-123.  doi: 10.5565/PUBLMAT_45101_04.

[13]

W. Decker, G. M. Greuel, G. Pfister and H. Shönemann, Singular (4-1-2-A Computer Algebra System for Polynomial Computations, 2019, Available from: https://www.singular.uni-kl.de.

[14]

W. Decker, S. Laplagne, G. Pfister and H. Shönemann, SINGULAR (3- 1 Library for Computing the Prime Decomposition and Radical of Ideals, Primdec.lib), 2010.

[15]

M. DukarićR. Oliveira and V. G. Romanovski, Local integrability and linearizability of a $(1:-1:-1)$ resonant quadratic system, J. Dyn. Differ. Equ., 29 (2011), 597-613.  doi: 10.1007/s10884-015-9486-2.

[16]

H. Dulac, Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Sci. Math., 32 (1908), 230-252. 

[17]

A. FronvilleA. P. Sadovski and H. Żołądek, Solution of the $1:-2$ resonant center problem in the quadratic case, Fundam. Math., 157 (1998), 191-207.  doi: 10.4064/fm-157-2-3-191-207.

[18]

Z. HuM. AldazharovaT. M. Aldibekov and V. G. Romanovski, Integrability of $3$-dim polynomial systems with three invariant planes, Nonlinear Dyn., 74 (2013), 1077-1092.  doi: 10.1007/s11071-013-1025-2.

[19]

Z. HuM. Han and V. G. Romanovski, Local integrability of a family of three-dimensional quadratic systems, Phys. D, 265 (2013), 78-86.  doi: 10.1016/j.physd.2013.09.001.

[20]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.  doi: 10.1142/S0218127403006352.

[21]

J. Li and Y. Lin, Normal form and critical points of the period of closed orbits for planar autonomous systems, Acta Math. Sin., 34 (1991), 490-501. 

[22]

Y. Liu and J. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1990), 10-23. 

[23]

Y. Liu, J. Li and W. Huang, Planar Dynamical Systems: Selected Classical Problems, De Gruyter, Berlin, Science Press New York, Ltd., New York, 2014. doi: 10.1515/9783110298369.

[24]

Y. Liu and P. Xiao, Critical point quantities and integrability conditions for a class of quintic systems, J. Cent. South Univ. Technol, 11 (2004), 109-112.  doi: 10.1007/s11771-004-0023-4.

[25]

J. LlibreC. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359.  doi: 10.1016/j.bulsci.2011.10.003.

[26]

J. Llibre and G. Rodríguez, Invariant hyperplanes and Darboux integrability for d-dimensional polynomial differential systems, Bull. Sci. Math., 124 (2000), 599-619.  doi: 10.1016/S0007-4497(00)01061-7.

[27]

J. Llibre and X. Zhang, On the Darboux integrability of polynomial differential systems, Qual. Theory Dyn. Syst., 11 (2012), 129-144.  doi: 10.1007/s12346-011-0053-x.

[28]

V. A. Pliss, On the reduction of an analytic system of differential equations to linear form, Differ. Equ., 5 (1969), 796-802. 

[29]

V. G. Romanovski and M. Prešern, An approach to solving systems of polynomials via modular arithmetics with applications, J. Comput. Appl. Math., 236 (2011), 196-208.  doi: 10.1016/j.cam.2011.06.018.

[30]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems. A Computational Algebra Approach, Birkhäuser, Boston-Basel-Berlin, 2009. doi: 10.1007/978-0-8176-4727-8.

[31]

V. G. RomanovskiY. Xia and X. Zhang, Varieties of local integrability of analytic differential systems and their applications, J. Differ. Equ., 257 (2014), 3079-3101.  doi: 10.1016/j.jde.2014.06.007.

[32]

P. Wang, M. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, ACM SIGSAM Bull., 16 (1982), 23 pp.

[33]

K. Wu and X. Zhang, Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces, Phys. D, 244 (2013), 25-35.  doi: 10.1016/j.physd.2012.10.011.

[34]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics, 47. Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.

[35]

X. Zhang, Analytic normalization of analytic integrable systems and the embedding flows, J. Differ. Equ., 250 (2008), 1080-1092.  doi: 10.1016/j.jde.2008.01.001.

[36]

H. Żołądek, The problem of center for resonant singular points of polynomial vector fields, J. Differ. Equ., 137 (1997), 94-118.  doi: 10.1006/jdeq.1997.3260.

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