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The pathology of differential equations

Published online by Cambridge University Press:  09 April 2009

T. M. Cherry
T. M. Cherry
Affiliation:
University of Melbourne
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Abstract

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Type
Presidental Address
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 1 , August 1959 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 1959

References

[1]Poincaré, H., Acta Math. 13 (1890), 1.Google Scholar
Hadamard, J., J. de Math. [5] 4 (1898), 29. Considering a surface of negative curvature of genus exceeding 1, he shows that the set of those geodesics which do not run off ultimately to infinity is a pathological component of the set of all geodesics.Google Scholar
Morse, M., Trans. American Math. Soc. 22 (1921), 84. Shows that a closed surface of negative curvature contains geodesics that are recurrent and of discontinuous type.CrossRefGoogle Scholar
Birkhoff, G. D., Dynamical Systems, American Math. Soc. Colloq. Pub. 9 (1927). For general reference.Google Scholar
[2]Birkhoff, G. D., Bull. Soc. Math. France 40 (1912), 305, and Dynamical Systems, 197.CrossRefGoogle Scholar
[3]Lefschetz, S., Differential equations; geometric theory Interscience, New York, 1957.Google Scholar
Stoker, J. J., Non-linear Vibrations, Interscience, New York, 1950.Google Scholar
Andronow, A. A. and Chaikin, C. E., Theory of Oscillations, Princeton 1949.Google Scholar
[4]Denjoy, A., C.R. Acad. Sci. Paris 194 (1932), 830 and J. de Math. [911 (1932), 333.Google Scholar
[5]Cherry, T. M., Proc. London Math. Soc. [2] 44 (1938), 175.CrossRefGoogle Scholar
[6]Cherry, T. M., Proc. Cambridge Phil. Soc. 22 (1924), 325. The result was known in 1920 by Birkhoff and guessed, but not proved, by Whittaker E. T., Proc. Roy. Soc. Edinburgh 37 (1916), 95. Apparently, and surprisingly, it seems to have remained unnoticed by Poincaré.CrossRefGoogle Scholar
[7]The evidence as to essential divergence is usually indirect: that convergence would be inconsistent with properties otherwise established; for example, inconsistent with the pattern of periodic solutions near a given periodic solution.Google Scholar
[8]Poincaré, H., loc. cit. and Les Méthodes Nouvelles de la Mécanique Céleste, Chap. 7.Google Scholar
[9]Moser, J., Comm. Pure Appl. Math. 9 (1956), 673.CrossRefGoogle Scholar
[10]I believe that this phenomenon was discovered by Rayleigh, or Maxwell, , but the earliest reference I know is Stephenson,, A. Phil. Mag. [6] 15 (1908), 233. The device of taking ⋅ to be piecewise constant was used by van der Pol, B. Physica (1925), 157 in discussing the linear approximation for the case x ˜ 0. If the forced oscillation of A is horizontal instead of vertical, the downward position x = π of the pendulum becomes unstable when p >n, and there are then two inclined positions of stable equilibrium!Google Scholar
[11]Artin, E., Abh. Math. Sem. Hamburg 2 (1923), 170.Google Scholar
[12]Littlewood, J. E., Acta Math. 97 (1957), 267 and 98 (1957), 1.CrossRefGoogle Scholar
[13]Dulac, H., J. École Polytech. 9 (1904), 1.Google Scholar
[14]Siegel, C. L., Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. 2a (1952), 21.Google Scholar
[15]Moser, J., Comm. Pure Appl. Math. 11 (1959), 257.CrossRefGoogle Scholar
[16]See for example Fatou, P., Bull. Soc. Math. France 47 (1919), 161 at Sec. 8.CrossRefGoogle Scholar
[17]Cherry, T. M., Proc. Cambridge Phil. Soc. 22 (1925), 510.CrossRefGoogle Scholar