Geodesic
On Periodic Solutions of x′′′ + ax′′ + b′ + g(x) = 0
Canadian mathematical bulletin, Tome 10 (1967) no. 1, pp. 75-77

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In [1] J. O. C. Ezeilo asks whether the equation 1 has periodic solutions for a ≠ 0. Since (1) has a two-dimensional space of solutions of period 2π if sin x is approximated by x, it is plausible to conclude, by analogy with x′′ + sin x = 0, that (1) does have periodic solutions. However, when one applies the standard theory of perturbation of periodic solutions (treating a as small, see [2]), one finds that the only real periodic solutions obtainable in this manner are the trivial ones x(t, a) = nπ for some integer n. 
Kemp, R.R.D. On Periodic Solutions of x′′′ + ax′′ + b′ + g(x) = 0. Canadian mathematical bulletin, Tome 10 (1967) no. 1, pp. 75-77. doi: 10.4153/CMB-1967-008-6
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     title = {On {Periodic} {Solutions} of x''' + ax'' + b' + g(x) = 0},
     journal = {Canadian mathematical bulletin},
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     year = {1967},
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