Qualitative theory of differential equations

  • Martin Braun
Part of the Texts in Applied Mathematics book series (TAM, volume 11)


In this chapter we consider the differential equation
$$\dot x = f\left( {t,x} \right)$$
$$ x = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)} \\ \vdots\\ {{x_n}\left( t \right)} \end{array}} \right], $$
$$ f\left( {t,x} \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \\ \vdots\\ {{f_n}\left( {t,{x_1}, \ldots ,{x_n}} \right)} \end{array}} \right] $$
is a nonlinear function of x1, ..., xn. Unfortunately, there are no known methods of solving Equation. This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of explicitly. For example, let x1(t) and x2(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 x1(t) and x2(t) are governed by the differential equation. In this case, we are not really interested in the values of x1(t) and x2(t) at every time t. Rather, we are interested in the qualitative properties of x1(t) and x2(t). Specically, we wish to answer the following questions.


Equilibrium Point Phase Portrait Bifurcation Point Equilibrium Solution Future Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Richardson, L. F. 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.Google Scholar
  4. 3.
    Howes, D. R., and Thrall, R. M., A theory of ideal linear weights for heteroge?neous 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.zbMATHGoogle 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: “Leons sur la theorie mathematique de la lutte pour la vie.” Paris, 1931.Google Scholar
  9. Gause, G. F. 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 therory of epidemics, Proceedings Roy. Stat. Soc., A, 115, 700 - 721, 1927.zbMATHGoogle Scholar
  12. Waltman, P, P., ‘Deterministic threshold models in the theory of epidemics,’ Springer-Verlag, New York, 1974.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

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

Personalised recommendations