We consider the optimization problem of maximizing the time-averaged profit for the motion of a smooth polydynamical system on the circle in the presence of a smooth profit density. If the problem depends on a k-dimensional parameter, then the optimal averaged profit is a function of the parameter. It is known from  that an optimal motion can always be selected among stationary strategies and a special type of periodic motions called level cycles. We present a classification of all generic singularities of the optimal averaged profit if k ¿ 2 and the maximum is provided by level cycles. © 2014 Springer Science+Business Media New York.
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