Darboux theory of integrability for polynomial vector fields on 𝕊ⁿ

References

• G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème série 2(60–96) (1878), pp. 123–144; 151–200.
   Google Scholar
• F. Cantrijn and W. Sarlet, Generalizations of Noether's theorem in classical mechanics, SIAM Rev. 23 (1981), pp. 467–494.
   Web of Science ®Google Scholar
• P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
   Google Scholar
• T.C. Bountis, A. Ramani, B. Grammaticos, and B. Dorizzi, On the complete and partial integrability of non-Hamiltonian systems, Phys. A 128 (1984), pp. 268–288.
   Google Scholar
• P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21 (1968), pp. 467–490.
   Web of Science ®Google Scholar
• H.J. Giacomini, C.E. Repetto, and O.P. Zandron, Integrals of motion of three-dimensional non-Hamiltonian dynamical systems, J. Phys. A 24 (1991), pp. 4567–4574.
   Google Scholar
• J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep. 147 (1987), pp. 87–154.
   Web of Science ®Google Scholar
• J. Chavarriga, J. Llibre, and J. Sotomayor, Algebraic solutions for polynomial systems with emphasis in the quadratic case, Expositiones Math. 15 (1997), pp. 161–173.
   Google Scholar
• C.J. Christopher, Invariant algebraic curves and conditions for a center, Proc. R. Soc. Edinburgh 124A (1994), pp. 1209–1229.
   Google Scholar
• C.J. Christopher and J. Llibre, Algebraic aspects of integrability for polynomial systems, Qualit. Theory Dyn. Syst. 1 (1999), pp. 71–95.
   Google Scholar
• C.J. Christopher and J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Diff. Equ. 16 (2000), pp. 5–19.
   Google Scholar
• J.P. Jouanolou, Equations de Pfaff algébriques, Lectures Notes in Mathematics, Vol. 708, Springer-Verlag, New York/Berlin, 1979.
   Google Scholar
• J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations, Ordinary Differential Equations, A. Cañada, P. Drabek, and A. Fonda, eds., Elsevier /North-Holland, Amsterdam, 2004, pp. 437–533.
   Google Scholar
• J. Llibre and X. Zhang, On the Darboux integrability of the polynomial differential systems, Qual. Theory Dyn. Syst. 11 (2012), pp. 129–144.
   Web of Science ®Google Scholar
• J.V. Pereira, Integrabilidade de equaçoes diferenciais no plano complexo, Vol. 25, Monografias del IMCA, Lima, 2002.
   Google Scholar
• H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II, Rend. Circ. Mat. Palermo. 5 (1891), pp. 161–191; 11 (1897), pp. 193–239.
   Google Scholar
• M.J. Prelle and M.F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), pp. 613–636.
   Web of Science ®Google Scholar
• D. Schlomiuk, Elementary first integrals and algebraic invariant curves of differential equations, Expositiones Math. 11 (1993), pp. 433–454.
   Google Scholar
• D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), pp. 799–841.
   Web of Science ®Google Scholar
• D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk, ed., Kluwer Academic Publishers, Dordrecht, 1993, pp. 429–467.
   Google Scholar
• M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333–688 (1992) pp. 673–688.
   Web of Science ®Google Scholar
• J. Llibre and Y. Bolaños, Rational first integrals for polynomial vector fields on algebraic hypersurfaces of , Int. J. Bifurc. Chaos 22 (2012), pp. 1250270–11.
   Web of Science ®Google Scholar
• J. Llibre and J.C. Medrado, On the invariant hyperplanes for d–dimensional polynomial vector fields, J. Phys. A 40 (2007), pp. 8385–8391.
   Google Scholar
• J. Llibre and X. Zhang, Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces, Rend. Circ. Mat. Palermo. 51 (2002), pp. 109–126.
   Google Scholar
• J. Llibre and X. Zhang, Darboux theory of integrability in taking into account the multiplicity, J. Differ. Equ. 246 (2009), pp. 541–551.
   Web of Science ®Google Scholar
• J. Llibre and X. Zhang, Rational first integrals in the Darboux theory of integrability in , Bull. Sci. Math. 134 (2010), pp. 189–195.
   Web of Science ®Google Scholar
• J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in taking into account the multiplicity at infinity, Bull. Sci. Math. 133 (2009), pp. 765–778.
   Web of Science ®Google Scholar
• C.J. Christopher, J. Llibre, and J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 (2007), pp. 63–117.
   Web of Science ®Google Scholar
• F. Dumortier, J. Llibre, and J.C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer–Verlag, New York, 2006.
   Google Scholar
• J.M. Strelcyn and S. Wojciechowski, A method of finding integrals for three–dimensional dynamical systems, Phys. Lett. A 133 (1988), pp. 207–212.
   Web of Science ®Google Scholar
• J.C. Artés, B. Grünbaum, and J. Llibre, On the number of invariant straight lines for polynomial differential systems, Pacific J. Math. 184 (1998), pp. 207–230.
   Web of Science ®Google Scholar