Abstract
Within twelve years, from Poincaré’s Mémoire sur les courbes définies par une équation différentielle (1881–1886) to Lyapunov’s thesis Obshcˇaya zadacˇa ob unstoicˇivosti dvizˇeniya (1892), the qualitative theory of differential equations emerged almost from scratch as the core of a new field in mathematics; both Poincaré and Lyapunov were motivated by problems in mechanics, celestial mechanics above all. Even if he did not match Poincaré’s prodigious creativity between 1880 and 1883, Lyapunov developed from 1888 to 1892 a theory of dynamical stability which makes his 1892 thesis both a pioneering piece of work and a classic; in particular he developed a general stability criterion which now bears his name: the Lyapunov function.
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Bibliography
Andronov, A.A., and L.S. Pontrjagin. 1937. Systèmes grossiers. Doklady Akademii Nauk 14: 247–251.
Arrow, K., and F. Hahn. 1971. General competitive analysis. San Francisco: Holden- Day.
Aubin, J.P., and A. Cellina. 1984. Differential inclusions. Berlin: Springer.
Benhabib, J., and K. Nishimura. 1979. The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth. Journal of Economic Theory 21: 421–444.
Brock, W.A., and J.A. Scheinkman. 1976. Global asymptotic stability of optimal control systems with applications to the theory of economic growth. Journal of Economic Theory 12: 164–190.
Cass, D., and K. Shell. 1976. The structure and stability of competitive dynamical systems. Journal of Economic Theory 12: 31–70.
Champsaur, P., J. Drèze, and C. Henry. 1977. Stability theorems with economic applications. Econometrica 45: 273–294.
Coddington, E.A., and N. Levinson. 1955. Theory of ordinary differential equations. New York: McGraw-Hill.
Drèze, J.H., and E. Sheshinski. 1984. On industry equilibrium under uncertainty. Journal of Economic Theory 33: 88–97.
Grandmont, J.M. 1985. On endogenous competitive business cycles. Econometrica 53: 995–1045.
Hahn, W. 1967. Stability of motion. Berlin: Springer.
Lyapunov, A. 1892. Obshcˇaya zadacˇa ob ustoicˇivosti dvizˇeniya (The general problem of the stability of motion). Kharkov Mathematical Society. The 1907 French translation has been reproduced in Annals of Mathematics Studies 17. Princeton: Princeton University Press, 1949.
Lyapunov, A. 1906–1912. Sur les figures d’équilibre peu différentes des ellipsoïdes d’une masse liquide homogène douée d’un mouvement de rotation. 3 vols. St. Petersburg: Académie impériale des Sciences.
Poincaré, H. 1881. Mémoire sur les courbes définies par une équation différentielle. Journal de Mathématiques Pures et Appliquées 7(3), 1881, 375–422; 8(3), 1882, 251–296; 1(4), 1885, 167–244; 2(4), 1886, 151–217.
Poincaré, H. 1882. Les formes d’équilibre d’une masse fluide en rotation. Revue Générale des Sciences 3: 809–815.
Rouche, N., and J. Mawhin. 1980. Ordinary differential, equations: Stability and periodic solutions, Surveys and reference works in mathematics. London: Pitman.
Rouche, N., Habets, P., Laloy, M. 1977. Stability theory by Lyapunov’s direct method, Applied mathematical sciences, vol. 22. Berlin: Springer.
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Henry, C. (2018). Lyapunov Functions. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_932
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DOI: https://doi.org/10.1057/978-1-349-95189-5_932
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