Geodesic
On the First Conjugate Point Function for Nonlinear Differential Equations
Canadian mathematical bulletin, Tome 18 (1975) no. 4, pp. 577-585

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We are concerned with the nth order differential equation y(n) = (x, y, y′, ...,y(n-1)), where it is assumed throughout that f is continuous on [α,β) × Rn, α < β≤∞, and that solutions of initial value problems are unique and exist on [α, β). The definition of the first conjugate point function η1(t) for linear homogeneous equations is extended to this nonlinear case. Our main concern is what properties of this conjugacy function are valid in the nonlinear case. 
Peterson, Allan C.; Sukup, Dwight V. On the First Conjugate Point Function for Nonlinear Differential Equations. Canadian mathematical bulletin, Tome 18 (1975) no. 4, pp. 577-585. doi: 10.4153/CMB-1975-102-1
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     author = {Peterson, Allan C. and Sukup, Dwight V.},
     title = {On the {First} {Conjugate} {Point} {Function} for {Nonlinear} {Differential} {Equations}},
     journal = {Canadian mathematical bulletin},
     pages = {577--585},
     year = {1975},
     volume = {18},
     number = {4},
     doi = {10.4153/CMB-1975-102-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1975-102-1/}
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