Geodesic
Generic Profit Singularities in Arnold's Model of Cyclic Processes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 250 (2005), pp. 79-94.

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Generic singularities of the maximum time-averaged profit are classified for a one-parameter smooth pair of families of profit densities and control systems with positive velocities on the circle. For example, a typical passage of this profit through a local nondegenerate extremum of the profit density under the change of the parameter yields the singularity $(|p|^{3/2}\pm \nobreak p^2)(1+\mathop {\mathrm {sgn}} p)$ of the profit up to a diffeomorphism of its graph space fibered over the parameter. 
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A. A. Davydov. Generic Profit Singularities in Arnold's Model of Cyclic Processes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 250 (2005), pp. 79-94. http://geodesic.mathdoc.fr/item/TM_2005_250_a3/

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