Invariant Algebraic Surfaces and Impasses

• da Silva, Paulo Ricardo ^[1] ; Perez, Otavio Henrique ^[1]
  1. [1] Universidade Estadual Paulista

     Universidade Estadual Paulista

     Brasil

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00465-x
• Enlaces
  • Texto completo
• Resumen

  • Polynomial vector fields X : R3 → R3 that have invariant algebraic surfaces of the form M = { f (x, y)z − g(x, y) = 0} are considered. We prove that trajectories of X on M are solutions of a constrained differential system having I = { f (x, y) = 0} as impasse curve. The main goal of the paper is to study the flow on M near points that are projected on typical impasse singularities. The Falkner–Skan equation (Llibre and Valls in Comput Fluids 86:71– 76, 2013), the Lorenz system (Llibre and Zhang in J Math Phys 43:1622–1645, 2002) and the Chen system (Lu and Zhang in Int J Bifurc Chaos 17–8:2739–2748, 2007) are some of the well-known polynomial systems that fit our hypotheses.

• Referencias bibliográficas
  • Cao, J., Chen, C., Zhang, X.: The Chen system having an invariant algebraic surface. Int. J. Bifurc. Chaos 1(8), 3753–3758 (2008).
  • Cao, J., Zhang, X.: Dynamics of the Lorenz system having an invariant algebraic surface. J. Math. Phys. 48, 1–13 (2007)
  • Cardin, P.T., Silva, P.R., Teixeira, M.A.: Implicit differential equations with impasse singularities and singular perturbation problems....
  • Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)
  • Chua, L.O., Oka, H.: Normal forms for constrained nonlinear differential equations. I. Theory. IEEE Trans. Circuits Syst. 35(7), 881–901 (1988)
  • Chua, L.O., Oka, H.: Normal forms for constrained nonlinear differential equations. II. Bifurcation. IEEE Trans. Circuits Syst. 36(1), 71–88...
  • Cima, A., Llibre, J.: Bounded polynomial vector fields. Trans. Am. Math. Soc. 318, 557–579 (1990)
  • Dee, G.T., Saarloos, W.: Bistable systems with propagating fronts leading to patternformation. Phys. Rev. Lett. 60, 2641–2644 (1988)
  • Falkner, G., Skan, S.W.: Solutions of the boundary layer equations. Philos. Mag. 12, 865–896 (1931)
  • Ferragut, A., Gasull, A.: Seeking Darboux polynomials. Acta Appl. Math. 139, 167–186 (2015)
  • Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7–4, 353–369 (1937)
  • Llibre, J., Messias, M., Silva, P.R.: Global dynamics of the Lorenz system with invariant algebraic surfaces. Int. J. Bifurc. Chaos 20–10,...
  • Llibre, J., Messias, M., Silva, P.R.: Global dynamics of stationary solutions of the extended FisherKolmogorov equation. J. Math. Phys. 52,...
  • Llibre, J., Messias, M., Silva, P.R.: Global dynamics in the Poincaré ball of the Chen system having invariant algebraic surfaces. Int. J....
  • Llibre, J., Sotomayor, J.: Structural stability of constrained polynomial systems. Bull. Lond. Math. Soc. 30–6, 589–595 (1998)
  • Llibre, J., Valls, C.: On the Darboux integrability of Blasius and Falkner–Skan equation. Comput. Fluids 86, 71–76 (2013)
  • Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Lorenz System. J. Math. Phys. 43, 1622–1645 (2002)
  • Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)
  • Lu, T., Zhang, X.: Darboux polynomials and algebraic integrability of the Chen system. Int. J. Bifurc. Chaos 17–8, 2739–2748 (2007)
  • Smale, S.: On the mathematical foundation of electrical networks. J. Differ. Geom. 7, 193–210 (1972)
  • Sotomayor, J., Zhitomirskii, M.: Impasse singularities of differential systems of the form A(x)x = F(x). J. Differ. Equ. 169, 567–587...
  • Zhitomirskii, M.: Local normal forms for constrained systems on 2-manifolds. B. da Soc. Bras. de Mat. 24, 211–232 (1993)