a b s t r a c tDynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of a child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with a numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically.

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The paper investigates an age-structured infinite-horizon optimal control model of harvesting a biological resource, interpreted as fish. Time and age are considered as continuum variables. The main result shows that in case of selective fishing, where only fish of prescribed sizes is harvested, it may be advantageous in the log run to implement a periodic fishing effort, rather than constant (the latter suggested by single-fish models that disregard the age-heterogeneity). Thus taking into account the age-structure of the fish may qualitatively change the theoretically optimal fishing mode. This result is obtained by developing a technique for reliable numerical verification of second order necessary optimality conditions for the considered problem. This technique could be useful for other optimal control problems of periodic age-structured systems."Aside from the fact that understanding optimal harvesting of age-structured fish populations is a theoretically fascinating open question, there is no doubt that the pressure to include the age-structure in economically relevant fishery models is steadily increasing."We partly fulfill this gap in the existing literature. The model we consider is continuous in age and time and has a similar first order PDE form as the one considered in [3,4]. However, the model in [3,4] assumes fishing in all ages (sizes), while for superiority of periodic fishing (in comparison with asymptotically constant one) it is of key importance that the fishing is selective: say, only fish of prescribed sizes is harvested, as it happens in practice. The results in [3,4] are not applicable in this case.This paper takes the averaged net revenue of fishing in the long run as a performance criterion to be maximized. Alternatively, one can consider the usual maximization of the total discounted net revenue, which is also economically meaningful. The technique presented below may be useful also for this problem.Our approach consists of the following steps. First we characterize the optimal constant fishing effort. Second, using ideas from [7], we develop a procedure (called in [7] properness test) for checking if the optimal constant control dominates all its additive (small) periodic variations of sinusoidal type. This properness test is based on second order variational analysis of the problem at hand. A remarkable feature is that it requires only solving ODE systems which are easily constructed from the original problems data, and are independent from the size of the variation. Third, for particular data specifications we perform the test numerically in a reliable way, which is possible, thanks to the abovementioned feature of the test. Cases where proper periodic fishing strictly dominates the optimal constant fishing are presented.On the other hand, if the fishing is not selective the picture changes. We prove that optimal fishing which is constant always exists in this case. This shows that the selectivity of fishing is the reason for which non-constant fishing efforts may be more profitable t...