Qualitative theory of differential equations

In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively few differential equations that can be solved explicitly, but using tools from analysis and topology, one can "solve" them in the qualitative sense, obtaining information about their properties.[1]

It was used by Benjamin Kuipers in the book Qualitative reasoning: modeling and simulation with incomplete knowledge to demonstrate how the theory of PDEs can be applied even in situations where only qualitative knowledge is available.

## References

1. ↑ Hazewinkel, Michiel, ed. (2001), "Qualitative theory of differential equations", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Qualitative_theory_of_differential_equations

## Further reading

* Kuipers, Benjamin. Qualitative reasoning: modeling and simulation with incomplete knowledge. MIT press, 1994.

* Viktor Vladimirovich Nemytskii, Vyacheslav Stepanov, Qualitative theory of differential equations, Princeton University Press, Princeton, 1960.
* Brauer, Fred (1969). The qualitative theory of ordinary differential equations: an introduction. https://archive.org/details/qualitativetheor0000brau.

### Original references

* Henri Poincaré, "Mémoire sur les courbes définies par une équation différentielle", Journal de Mathématiques Pures et Appliquées (1881, in French)
* Lyapunov, Aleksandr M. (1992). "The general problem of the stability of motion". International Journal of Control 55 (3): 531–534. doi:10.1080/00207179208934253. ISSN 0020-7179. (it was translated from the original Russian into French and then into this English version, the original is from the year 1892)

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