Qualitative theory of differential equations

  • Martin Braun
Part of the Applied Mathematical Sciences book series (AMS, volume 15)


In this chapter we consider the differential equation
$$dot x = f(t,x)$$
$$x = \left[ \begin{array}{l} {x_1}(t) \\ \vdots \\ {x_n}(t) \\ \end{array} \right]$$
$$f(t,X) = \left[ \begin{array}{l} {f_1}(t,{x_1}, \ldots ,{x_n}) \\ \vdots \\ {f_n}(t,{x_1}, \ldots ,{x_n}) \\ \end{array} \right]$$
is a nonlinear function of x v ...,x n . Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let x 1(t) and x 2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x 1(t) and x 2(t) are governed by the differential equation (1). In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time t. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.


Equilibrium Point Phase Portrait Equilibrium Solution Future Time Negative Real Part 
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  1. Richardson, L. F., “Generalized foreign politics,” The British Journal of Psychology, monograph supplement #23, 1939.Google Scholar
  2. 1.
    Coleman, C. S., Combat Models, MAA Workshop on Modules in Applied Math, Cornell University, Aug. 1976.Google Scholar
  3. 2.
    Engel, J. H., A verification of Lanchester’s law, Operations Research, 2, (1954), 163–171.CrossRefGoogle Scholar
  4. 3.
    Howes, D. R., and Thrall, R. M., A theory of ideal linear weights for heterogeneous combat forces, Naval Research Logistics Quarterly, vol. 20, 1973, pp. 645–659.CrossRefGoogle Scholar
  5. 4.
    Lanchester, F. W., Aircraft in Warfare, the Dawn of the Fourth Arm. Tiptree, Constable and Co., Ltd., 1916.Google Scholar
  6. 5.
    Morehouse, C. P., The Iwo Jima Operation, USMCR, Historical Division, Hdqr. USMC, 1946.Google Scholar
  7. 6.
    Newcomb, R. F., Iwo Jima. New York: Holt, Rinehart, and Winston, 1965.Google Scholar
  8. Volterra, V: “Leçons sur la théorie mathématique de la lutte pour la vie.” Paris, 1931.Google Scholar
  9. Gause, G. F., ‘The Struggle for Existence,’ Dover Publications, New York, 1964.Google Scholar
  10. Bailey, N. T. J., “The mathematical theory of epidemics,’ 1957, New York.Google Scholar
  11. Kermack, W. O. and McKendrick, A. G., Contributions to the mathematical theory of epidemics, Proceedings Roy. Stat. Soc., A, 115, 700–721, 1927.zbMATHCrossRefGoogle Scholar
  12. Waltman, P., ‘Deterministic threshold models in the theory of epidemics,’ Springer-Verlag, New York, 1974.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag, New York, Inc. 1978

Authors and Affiliations

  • Martin Braun
    • 1
  1. 1.Department of Mathematics, Queens CollegeCity University of New YorkFlushingUSA

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