Abstract
In this paper, we consider dissipative second order differential equations. Given a finite number of subharmonic solutions, we show that the number of subharmonic solutions of any order is bounded from below by a number determined by the waveforms of the given solutions. As a consequence, we obtain a sufficient condition on the waveforms for the existence of infinitely many subharmonic solutions.
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References
J. S. Birman, Braids, Links, and Mapping Class Groups. Ann. of Math. Studies82, Princeton Univ. Press, Princeton, 1974.
R. Bowen, Entropy and the fundamental group. Lecture Notes in Math.668, Springer, 1978, 21–29.
M. L. Cartwright, Forced oscillations in nonlinear systems. Contributions to the Theory of Nonlinear Oscillations, Ann. of Math. Studies20, Princeton Univ. Press, Princeton, 1950.
P. Hartman, Ordinary Differential Equations. Wiley, New York, 1964.
N. Levinson, Transformation theory of non-linear differential equations of the second order. Ann. of Math.,45 (1944), 723–737.
T. Matsuoka, The number and linking of periodic solutions of periodic systems. Invent. Math.,70 (1983), 319–340.
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Dedicated to Professor Minoru Nakaoka on his 60th birthday
Partially supported by Grant-in-Aid for Scientific Research.
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Matsuoka, T. Waveform in the dynamical study of ordinary differential equations. Japan J. Appl. Math. 1, 417–434 (1984). https://doi.org/10.1007/BF03167067
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DOI: https://doi.org/10.1007/BF03167067