Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 18
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Burkin, I. M. 2015. Method of “transition into space of derivatives”: 40 years of evolution. Differential Equations, Vol. 51, Issue. 13, p. 1717.

    Dirr, Gunther Ito, Hiroshi Rantzer, Anders and Rüffer, Björn S. 2015. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete and Continuous Dynamical Systems - Series B, Vol. 20, Issue. 8, p. 2497.

    Jiang, Jifa 2013. Convergence on cooperative cascade systems with length one. Journal of Differential Equations, Vol. 255, Issue. 11, p. 4081.

    Liao, Shu and Wang, Jin 2012. Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices. Chaos, Solitons & Fractals, Vol. 45, Issue. 7, p. 966.

    Yang, Maobin Zhang, Zhufan Li, Qiang and Zhang, Gang 2012. An SLBRS Model with Vertical Transmission of Computer Virus over the Internet. Discrete Dynamics in Nature and Society, Vol. 2012, p. 1.

    Roques, Lionel and Chekroun, Mickaël D. 2011. Probing chaos and biodiversity in a simple competition model. Ecological Complexity, Vol. 8, Issue. 1, p. 98.

    Tian, Jianjun Paul and Wang, Jin 2011. Global stability for cholera epidemic models. Mathematical Biosciences, Vol. 232, Issue. 1, p. 31.

    Zhou, Xueyong and Cui, Jingan 2011. Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate. Nonlinear Dynamics, Vol. 63, Issue. 4, p. 639.

    Li, Xue-Zhi and Zhou, Lin-Lin 2009. Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate. Chaos, Solitons & Fractals, Vol. 40, Issue. 2, p. 874.

    Sanchez, Luis A. 2009. Cones of rank 2 and the Poincaré–Bendixson property for a new class of monotone systems. Journal of Differential Equations, Vol. 246, Issue. 5, p. 1978.

    Hirsch, M.W. and Smith, Hal 2006.

    Vano, J A Wildenberg, J C Anderson, M B Noel, J K and Sprott, J C 2006. Chaos in low-dimensional Lotka–Volterra models of competition. Nonlinearity, Vol. 19, Issue. 10, p. 2391.

    Wang, Kaifa Wang, Wendi and Liu, Xianning 2006. Global stability in a viral infection model with lytic and nonlytic immune responses. Computers & Mathematics with Applications, Vol. 51, Issue. 9-10, p. 1593.

    Hui, Jing and Zhu, Deming 2005. Global stability and periodicity on SIS epidemic models with backward bifurcation. Computers & Mathematics with Applications, Vol. 50, Issue. 8-9, p. 1271.

    Arino, Julien McCluskey, C. Connell and van den Driessche, P. 2003. Global Results for an Epidemic Model with Vaccination that Exhibits Backward Bifurcation. SIAM Journal on Applied Mathematics, Vol. 64, Issue. 1, p. 260.

    Smith, Hal L. Wang, Liancheng and Li, Michael Y. 2001. Global Dynamics of an SEIR Epidemic Model with Vertical Transmission. SIAM Journal on Applied Mathematics, Vol. 62, Issue. 1, p. 58.

    Li, Michael Y. and Muldowney, James S. 2000. Dynamics of Differential Equations on Invariant Manifolds. Journal of Differential Equations, Vol. 168, Issue. 2, p. 295.

    Li, Michael Y. and Muldowney, James S. 1996. A Geometric Approach to Global-Stability Problems. SIAM Journal on Mathematical Analysis, Vol. 27, Issue. 4, p. 1070.

  • Ergodic Theory and Dynamical Systems, Volume 11, Issue 3
  • September 1991, pp. 443-454

Systems of differential equations that are competitive or cooperative. VI: A local Cr Closing Lemma for 3-dimensional systems

  • Morris W. Hirsch (a1)
  • DOI:
  • Published online: 01 September 2008

For certain Cr 3-dimensional cooperative or competitive vector fields F, where r is any positive integer, it is shown that for any nonwandering point p, every neighborhood of F in the Cr topology contains a vector field for which p is periodic, and which agrees with F outside a given neighborhood of p. The proof is based on the existence of invariant planar surfaces through p.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. W. Hirsch , (1982a), Systems of differential equations that are competitive or cooperative. I: Limit sets. SIAM J. Math. Anal. 13 167179.

M. W. Hirsch , (1983), Differential equations and convergence almost everywhere in strongly monotone semiflows. Contemp. Math. 17 267285.

M. W. Hirsch , (1984), The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. 11 164.

M. W. Hirsch , (1985), Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere. SIAM J. Math. Anal. 16 423439.

M. W. Hirsch , (1988b), Systems of differential equations that are competitive or cooperative. III: Competing species. Nonlinearity 1 5171.

M. W. Hirsch , (1989b), Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems. J. Diff. Eq. 80 94106.

M. W. Hirsch , C. C. Pugh , & M. Shub , (1977), Invariant Manifolds. Springer Lecture Notes in Mathematics 583. New York: Springer-Verlag.

E. Kamke , (1932), Zur Theorie der Systeme gewöhnlicher differential Gleichungen II. Acta Math. 58 5785.

R. Mañé , (1982), An ergodic closing lemma. Ann. Math. 116 503541.

M. Müller , (1926), Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen. Math. Zeitschr. 26 619645.

J. Palis , (1970), A note on Ω-stability. In Proc. Symp. Pure Math. 14, Global Analysis. S.-S. Chern and S. Smale , eds. Providence: American Mathematics Society.

M. L. Peixoto , (1988), The closing lemma for generalized recurrence in the plane. Trans. Amer. Math. Soc. 308 143158.

M. M. Peixoto , (1962), Structural stability on two-dimensional manifolds. Topology 1 101120.

D. Pixton , (1982), Planar homoclinic points. J. Diff. Eq. 44 365382.

C. Pugh , (1967a), The closing lemma. Amer. J. Math. 89 9561009.

C. Pugh , (1967b), An improved closing lemma and the general density theorem. Amer. J. Math. 89 10101021.

J. Selgrade , (1980), Asymptotic behavior of solutions to single loop positive feedback systems. J. Diff. Eq. 38 80103.

S. Smale , (1976), On the differential equations of species in competition. J. Math. Biol. 3 57.

H. L. Smith , (1986), Periodic orbits of competitive and cooperative systems. J. Diff. Eq. 65 361373.

H. L. Smith , (1986a), On the asymptotic behavior of a class of deterministic models of cooperating species. SIAM J. Appl. Math. 46 368375.

H. L. Smith , (1986b), Periodic solutions of periodic competitive and cooperative systems. SIAM J. Math. Anal. 17 12891318.

H. L. Smith , (1986c), Periodic competitive differential equations and the discrete dynamics of competitive maps. J. Diff. Eq. 64 165194.

H. L. Smith , (1986d), Competing subcommunities of mutualists and a generalized Kamke Theorem. SIAM J. Appl. Math. 46 856874.

H. L. Smith , (1988), Systems of differential equations which generate a monotone flow. A survey of results. SIAM Review 30 87113.

H. L. Smith and P. Waltman , (1987), A classification theorem for three-dimensional competitive systems. J. Diff. Equat. 70 325332

F. Takens , (1972), Homoclinic points in conservative systems. Invent. Math. 18 267292.

W. Wilson , (1969), Smoothing derivatives of functions and applications. Trans. Amer. Math. Soc. 139 413428.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *