Non-formally integrable centers admitting an algebraic inverse integrating factor

Antonio Algaba; Natalia Fuentes; Cristóbal García; Manuel Reyes

doi:10.3934/dcds.2018041

Article Contents

2018, Volume 38, Issue 3: 967-988. Doi: 10.3934/dcds.2018041

This issue Previous Article Next Article Exponential multiple mixing for some partially hyperbolic flows on products of

$ {\rm{PSL}}(2, \mathbb{R})$

Non-formally integrable centers admitting an algebraic inverse integrating factor

Department of Mathematics, Faculty of Experimental Sciences, Avda. Tres de Marzo s/n, 21071 Huelva, Spain

* Corresponding author: Manuel Reyes
* Corresponding author: Manuel Reyes

Received: March 31, 2016

Revised: September 30, 2017

Published: June 2018

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Abstract

We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system $ -y^3\partial_x+x^3\partial_y$ having an algebraic inverse integrating factor.

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Mathematics Subject Classification: 34C05, 34C14, 34C20.

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Table 1. Range and co-range of operator $\ell_{j} $ for the system (8).

Range($\ell_{2}$)=span{$-bx^2-3y^2,3ax^2+2bxy$}.
If $a\ne 0,$ Cor($\ell_{2}$)=span{$xy$}. If $a=0$, Cor($\ell_{2}$)=span{$x^2$}.
Range($\ell_{3}$)=span{$-2bx^3-6xy^2,6ax^3+4bx^2y-3h,6ax^2y+4bxy^2$}.
Cor($\ell_{3}$)=span{$h$}.
Range($\ell_{4}$)=span{$3bx^4+9x^2y^2,-9ax^4-6bx^3y+6xh,$ $ -9ax^3y-6bx^2y^2+3yh$}.
Cor($\ell_{4}$)=span{$xh,yh$}.

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Table 2. Range and co-range of operator $\ell_{j} $ for the system (15)

Range($\ell_{6}$)=span{$0$},	Cor($\ell_{6}$)=span{$x^2$}
Range($\ell_{7}$)=span{$0$},	Cor($\ell_{7}$)=span{$xy$}
Range($\ell_{8}$)=span{$y^2$},	Cor($\ell_{8}$)=span{$0$}
Range($\ell_{9}$)=span{$x^3$},	Cor($\ell_{9}$)=span{$0$}
Range($\ell_{10}$)=span{$0$},	Cor($\ell_{10}$)=span{$x^2y$}
Range($\ell_{11}$)=span{$xy^2$},	Cor($\ell_{11}$)=span{$0$}
Range($\ell_{12}$)=span{$7x^4-12h$},	Cor($\ell_{12}$)=span{$h$}
Range($\ell_{13}$)=span{$x^3y$},	Cor($\ell_{13}$)={$0$}
Range($\ell_{14}$)=span{$x^2y^2$},	Cor($\ell_{14}$)={$0$}
Range($\ell_{15}$)=span{$x^3-6xh$},	Cor($\ell_{15}$)=span{$xh$}
Range($\ell_{16}$)=span{$11x^4y-12yh$},	Cor($\ell_{16}$)=span{$yh$}
Range($\ell_{17}$)=span{$x^3y^2$},	Cor($\ell_{17}$)={$0$}
Range($\ell_{18}$)=span{$13x^6-36x^2h$},	Cor($\ell_{18}$)=span{$x^2h$}
Range($\ell_{19}$)=span{$7x^5-12xyh$},	Cor($\ell_{19}$)=span{$xyh$}
Range($\ell_{22}$)=span{$17x^6y-9x^2yh$},	Cor($\ell_{22}$)=span{$x^2yh$}

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Table 3. Range and co-range of operator $\ell_{j} $ for the system (21).

Range($\ell_{3}$)=span{$x^3,y^3$}.
Cor($\ell_{3}$)=span{$x^2y,xy^2$}.
Range($\ell_{4}$)=span{$xy^3,x^4+2h,x^3y$}.
Cor($\ell_{4}$)=span{$x^2y^2,h$}.
Range($\ell_{5}$)=span{$x^2y^3,3x^5+8xh,3x^4y+4yh,x^3y^2$}.
Cor($\ell_{5}$)=span{$xh,yh$}.
Range($\ell_{6}$)=span{$x^3y^3,x^6+3x^2h,x^5y+2xyh, x^4y^2+y^2h$}.
Cor($\ell_{6}$)=span{$x^2h,xyh,y^2h$}.

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References

	A. Algaba , E. Freire , E. Gamero and C. García , Quasihomogeneous normal forms, J. Comput. Appl. Math., 150 (2003) , 193-216. doi: 10.1016/S0377-0427(02)00660-X.
	A. Algaba , N. Fuentes , C. García and M. Reyes , A class of non-integrable systems admitting an inverse integrating factor, J. Math. Anal. Appl., 420 (2014) , 1439-1454. doi: 10.1016/j.jmaa.2014.06.047.
	A. Algaba , E. Gamero and C. García , The integrability problem for a class of planar systems, Nonlinearity, 22 (2009) , 395-420. doi: 10.1088/0951-7715/22/2/009.
	A. Algaba , C. García and J. Giné , Analytic integrability for some degenerate planar vector fields, J. Differential Equations, 257 (2014) , 549-565. doi: 10.1016/j.jde.2014.04.010.
	A. Algaba , C. García and M. Reyes , Nilpotent systems admitting an algebraic inverse integrating factor over $ \mathbb{C}((x,y))$, Qualitative Theory of Dynamical Systems, 10 (2011) , 303-316. doi: 10.1007/s12346-011-0046-9.
	A. Algaba , C. García and M. Reyes , Characterization of a monodromic singular point of a planar vector field, Nonlinear Analysis, 74 (2011) , 5402-5414. doi: 10.1016/j.na.2011.05.023.
	A. Algaba , C. García and M. Reyes , Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems, Chaos Solitons & Fractals, 45 (2012) , 869-878. doi: 10.1016/j.chaos.2012.02.016.
	A. García , C. Algaba and M. Reyes , Like-linearizations of vector fields, Bulletin des Sciences Mathématiques, 133 (2009) , 806-816. doi: 10.1016/j.bulsci.2009.09.006.
	M. Berthier and R. Moussu , Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier, 44 (1994) , 465-494. doi: 10.5802/aif.1406.
	A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems; Geometry, Topology, Classification, Chapman and Hall, 2004.
	J. Chavarriga , H. Giacomini , J. Giné and J. Llibre , On the integrability of two-dimensional flows, J. Differential Equations, 157 (1999) , 163-182. doi: 10.1006/jdeq.1998.3621.
	J. Chavarriga , H. Giacomini , J. Giné and J. Llibre , Darboux integrability and the inverse integrating factor, J. Differential Equations, 194 (2003) , 116-139. doi: 10.1016/S0022-0396(03)00190-6.
	C. J. Christopher and J. Llibre , Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations, 16 (2000) , 5-19.
	C. Christopher , P. Mardesic and C. Rousseau , Normalizable, integrable, and linealizable saddle points for complex quadratic systems in $ \mathbb{C}^2$, J. Dyn. Control Syst., 9 (2003) , 311-363. doi: 10.1023/A:1024643521094.
	H. R. Dullin and A. Pelayo , Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Science, 26 (2016) , 787-811. doi: 10.1007/s00332-016-9290-0.
	H. Eliasson , Normal forms for Hamiltonian systems with Poisson commuting, integrals -elliptic case, Comm. Math. Helv., 65 (1990) , 4-35. doi: 10.1007/BF02566590.
	A. Enciso and D. Peralta-Salas , Existence and vanishing set of inverse integrating factors for analytic vector fields, Bull. London Math. Soc., 41 (2009) , 1112-1124. doi: 10.1112/blms/bdp090.
	I. García , H. Giacomini and M. Grau , The inverse integrating factor and the Poincaré map, Trans. Amer. Math. Soc., 362 (2010) , 3591-3612. doi: 10.1090/S0002-9947-10-05014-2.
	I. García , H. Giacomini and M. Grau , Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor, J. Dyn. Differ. Equat., 23 (2011) , 251-281. doi: 10.1007/s10884-011-9209-2.
	I. García and M. Grau , A survey on the inverse integrating factor, Qual. Theory Dyn. Sist., 9 (2010) , 115-166. doi: 10.1007/s12346-010-0023-8.
	I. García and D. Shafer , Integral invariants and limit sets of planar vector fields, J. Differential Equations, 217 (2005) , 363-376. doi: 10.1016/j.jde.2005.06.022.
	A. Gasull and J. Torregrosa , Center problem for several differential equations via Cherkas' method, J. Math. Anal. Appl., 228 (1998) , 322-343. doi: 10.1006/jmaa.1998.6112.
	H. Giacomini , J. Llibre and M. Viano , On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996) , 501-516. doi: 10.1088/0951-7715/9/2/013.
	J. Giné and D. Peralta-Salas , Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012) , 344-357. doi: 10.1016/j.jde.2011.08.044.
	R. E. Kooij and C. J. Christopher , Algebraic invariant curves and the integrability of polynomial systems, Appl. Math. Lett., 6 (1993) , 51-53. doi: 10.1016/0893-9659(93)90123-5.
	L. Mazzi and M. Sabatini , A characterization of centers via first integrals, J. Differential Equations, 76 (1988) , 222-237. doi: 10.1016/0022-0396(88)90072-1.
	R. Moussu , Symétrie et forme normaledes centres et foyers dégénérés, Ergodic Theory Dynam. Sys., 2 (1982) , 241-251.
	H. Poincaré , Mémoire sur les courbes définies par les équations différentielles, J. Math., 37 (1881) , 375-422.
	M. J. Prelle and M. F. Singer , Elementary first integrals of of differential equations, Trans. Amer. Math. Soc., 279 (1983) , 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.
	A. P. Sadovskii , Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Diff. Urav., 12 (1976) , 1238-1246 (in Russian).
	S. Walcher , On the Poincaré problem, J. Differential Equations, 166 (2000) , 51-78. doi: 10.1006/jdeq.2000.3801.
	S. Walcher , Local integrating factors, J. Lie Theory, 13 (2003) , 279-289.