Abstract
In this paper, we find the normal forms of polynomial differential systems in \({\mathbb{R}}^3\) which have at least three invariant algebraic surfaces. Also, we deduce the normal forms of polynomial differential systems in \({\mathbb{R}}^3\) having a parabolic cylinder with the equation \({\mathcal{P}} : y^2-z\), or having a hyperbolic parabolic with the equation \({\mathcal{H}} : x^2-y^2-z\) as invariant objects. The conditions to find a lower bound for the number of invariant algebraic curves for the deduced systems are obtained.
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References
Cima, A., Llibre, J.: Bounded polynomial vector fields. Trans. Am. Math. Soc. 318, 557–579 (1990)
Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. World scientific publishing Co. Pte. Ltd, Singapore (2001)
Jouanolou, J.P.: Equations de plaff algébriques lecture notes in mathematics. Springer, New York (1979)
Llibre, J., Bolans, Y.: Rational first integrals for polynomial vector fields on algebraic hypersurfaces of ${\mathbb{R}}^{n+1}$. Int. J. Bifurc. Chaos 22, 1–11 (2012)
Llibre, J., Messias, M., Reinol, A.C.: Normal forms for polynomial differential systems in ${\mathbb{R}}^3$ having an invariant quadric and a Darboux invariant. Int. J. Bifurc. Chaos 25, 1–16 (2015)
Llibre, J., Ramirez, R., Ramirez, V., Sadovskaia, N.: The 16th Hilbert problem restricted to circular algebraic limit cycles. J. Differ. Equ. 260, 5726–5760 (2016)
Llibre, J., Zhang, X.: Rational first integrals in the Darboux theory of integrability in ${\mathbf{C}}^n$. Bull. Sci. Math. 134, 189–195 (2010)
Llibre, J., Zhang, X.: Darboux theory of integrability for polynomial vector fields in ${\mathbb{R}}^n$ taking into account the multiplicity at infinity. Bull. Sci. Math. 133, 765–778 (2009)
Messias, M., Reinol, A.C.: Integrability and dynamics of quadratic three-dimensional differential systems having an invariant paraboloid. Int. J. Bifurc. Chaos 26, 1–23 (2016)
Molaei, M.R.: Hyperbolic dynamics of discrete dynamical systems on pseudo-Riemannian manifolds. Electron. Res. Announc. Math. Sci. 25, 8–15 (2018)
Molaei, M.R.: Topological cocycles. Bol. Soc. Mat. Mex. 24, 257–267 (2018)
Sadovskaia, N., RamÃrez, R.: Inverse approach to study the planar polynomial vector field with algebraic solutions. J. Phys. A: Math. Gen. 37, 3847–3868 (2004)
Velasco, E.A.G.: Generic properties of polynomial vector fields at infinity. Trans. Am. Math. Soc. 143, 201–221 (1969)
Zhang, X.: Integrability of Dynamical Systems: Algebra and Analysis. Springer, Berlin (2016)
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Khajoei, N., Molaei, M.R. Normal forms of polynomial differential systems in \({\mathbb{R}}^3\) having at least three invariant algebraic surfaces. Rend. Circ. Mat. Palermo, II. Ser 70, 1023–1035 (2021). https://doi.org/10.1007/s12215-020-00537-y
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DOI: https://doi.org/10.1007/s12215-020-00537-y