Abstract
Counter-examples to Aizerman's and Kalman's conjectures are considered. Investigation of the behaviour in the vicinity of the origin and at a distance from the origin is done. The simultaneous existence of both: a limit cycle and asymptotic or finite-time convergence is proved through the LPRS method and the Lyapunov method, respectively. Conclusions regarding the complex behaviour of these nonlinear dynamic systems are given.
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