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A Geometric Approach to Dynamical Systems in ℝN

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Applied Nonlinear Analysis

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Abstract

In 2-dimensional dynamical systems defined by some differential equation \( \dot x = f(x) \) , global existence and asymptotics of the solutions follow from a geometric condition concerning the characteristics, on which one component of the vector function f is vanishing. By extending this geometric condition to n dimensions we find 2 classes of differential equations which have global solutions for all positive times. Additional monotonicity of the characteristics implies the existence of a unique stationary point which is asymptotically stable and globally attractive.

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References

  1. Amann, H. (1990). Ordinary differential equations, de Gruyter, Berlin.

    Google Scholar 

  2. Berman, A. and Plemmons, R.J. (1979). Nonnegative matrices in the mathematical sciences, Academic Press New York.

    Google Scholar 

  3. Bony, J.M. (1969). Principe du maximum, inégalité de Harnack et unicité du probléme de Cauchy pour les opérators elliptiques dégénérés, Ann. Inst. Fourier, 19:277–304.

    MathSciNet  MATH  Google Scholar 

  4. Brezis, H. (1970). On a characterisation of flow-invariant sets, Comm. Pure Appl. Math., 23:261–263.

    MathSciNet  MATH  Google Scholar 

  5. Gasull, A., Llibre, J. and Satomayor, J. (1991). Global asymptotic stability of differential equations in the plane, J. Differential Equations, 91:327–335.

    Article  MathSciNet  Google Scholar 

  6. Hartman, Ph. (1964). Ordinary differential equations, Wiley New York.

    Google Scholar 

  7. Hartman, Ph. and Olech, C. (1962). On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104:154–178.

    MathSciNet  Google Scholar 

  8. Hirsch, M. (1982). Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal., 13:167–179.

    Article  MathSciNet  MATH  Google Scholar 

  9. Hirsch, M. (1988b). Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383:1–53.

    MathSciNet  MATH  Google Scholar 

  10. Hirsch, M. and Smale, St. (1974). Differential equations, dynamical systems, and linear algebra, Academic Press New York.

    Google Scholar 

  11. Jiang, J.F. (1998). The complete classification of asymptotic behaviour for bounded cooperative Lotka-Volterra systems with assumption (SM), Quart. Appl. Math., LVI:37–53.

    Google Scholar 

  12. Ladde, G.S. and Lakshmikantham, V. (1974). On flow-invariant sets, Pacific J. Math., 51:215–220.

    MathSciNet  Google Scholar 

  13. Lakshmikantham, V., Leela, S. and Martynyuk, A.A. (1989). Stability analysis of nonlinear systems, Decker New York.

    Google Scholar 

  14. Lu, Zh. and Takeuchi, Y. (1994). Qualitative stability and global stability for Lotka-Volterra systems, J. Math. Anal. Appl., 182:260–268.

    Article  MathSciNet  Google Scholar 

  15. Meisters, G. and Olech, C. (1988). Solution of the global asymptotic stability Jacobian conjecture for the polynomial case, in: Analyse Mathàmatique et Applications, Gauthier-Villars Paris, 373–381.

    Google Scholar 

  16. Pao, C.V. (1992). Nonlinear parabolic and elliptic equations, Plenum Press New York.

    Google Scholar 

  17. Rautmann, R. (1972). A criterion for global existence in case of ordinary differential equations, Applicable Anal., 2:187–194.

    MathSciNet  MATH  Google Scholar 

  18. Rautmann, R. (1985). Eine Klasse asymptotisch stabiler autonomer Systeme und Normschranken fur spezielle Reaktions-Diffusions-Prozesse, Z. Angew. Math. Mech., 65:207–217.

    MathSciNet  MATH  Google Scholar 

  19. Rautmann, R. (1987). Global bounds for some reaction diffusion processes, Methoden Verfahren Math. Phys., 33:87–98.

    MathSciNet  MATH  Google Scholar 

  20. Rautmann, R. (1991). On tests for stability, Methoden Verfahren Math. Phys., 37:201–212.

    MathSciNet  MATH  Google Scholar 

  21. Redheffer, R.M. (1972). The theorems of Bony and Brezis on flow-invariant sets, Amer. Math. Monthly, 79:740–747.

    MathSciNet  MATH  Google Scholar 

  22. Redheffer, R.M. and Walter, W. (1984). Solution of the stability problem for a class of generalized Volterra prey-predator systems, J. Differential Equations, 52:245–263.

    Article  MathSciNet  Google Scholar 

  23. Smith, H.L. (1986). On the asymptotic behaviour of a class of deterministic models of cooperative species, SIAM J. Appl. Math., 3:368–375.

    Google Scholar 

  24. Smith, H.L. (1995). Monotone dynamical systems, AMS Mathematical surveys and monographs 41, Providence R.I.

    Google Scholar 

  25. Volkmann, P. (1975). über die Invarianz-Sätze von Bony und Brezis in normierten Räumen, Arch. Math., XXVI:89–93.

    MathSciNet  Google Scholar 

  26. Walter, W. (1970).Differential and integral inequalities, Springer Berlin.

    Google Scholar 

  27. Wintner, A. (1945). The non-local existence problem of ordinary differential equations, Amer. J. Math., 67:277–284.

    MathSciNet  MATH  Google Scholar 

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Authors
  1. Reimund Rautmann
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Editor information

Editors and Affiliations

  1. I.S.T. Technical University, Lisbon, Portugal

    Adélia Sequeira  & Juha Hans Videman  & 

  2. University of Pisa, Pisa, Italy

    Hugo Beirão da Veiga

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© 2002 Kluwer Academic Publishers

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Rautmann, R. (2002). A Geometric Approach to Dynamical Systems in ℝN. In: Sequeira, A., da Veiga, H.B., Videman, J.H. (eds) Applied Nonlinear Analysis. Springer, Boston, MA. https://doi.org/10.1007/0-306-47096-9_30

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  • DOI: https://doi.org/10.1007/0-306-47096-9_30

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-306-46303-7

  • Online ISBN: 978-0-306-47096-7

  • eBook Packages: Springer Book Archive

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