Skip to main content
    • Aa
    • Aa

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

  • Morris W. Hirsch (a1)

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? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 53 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 25th May 2017. This data will be updated every 24 hours.