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Some amazing phenomena in stability of nonlinear dynamical systems

Nonlinear dynamical systems possess some properties which at the first sight look unexpected and even surprising (see, e.g., [1,2]). In this paper a collec-tion of new qualitative results relating to stability of such systems is presented. In particular, for periodic oscillations of systems with non-monotonic elastic forces, new stability and instability effects are found. For some systems with uncertain bounded terms, nec-essary and sufficient stability conditions are obtained; the surprising feature is that they are independent upon arbitrary time-varying delays in the uncertain terms. It is shown that the known mathematical model of a swing – a pendulum with a periodic length – is incor-rect. An unknown feature relating to regions of para-metrical resonances is found.
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