On the behavior of dynamic systems in the vicinity of existence boundaries of periodic motions. (English. Russian original) Zbl 0387.70029
J. Appl. Math. Mech. 41(1977), 642-650 (1978); translation from Prikl. Mat. Mekh. 41, 628-636 (1977).
Summary: Violation of stability conditions of existence of periodic motions of a multidimensional system induced by transition through the discontinuity point of the characteristic of one of the nonlinear elements of the system is considered. It is shown that at stronger discontinuity a motion close to the disturbed one generally loses its stability, and that the related existence boundary of the stability region becomes dangerous, if no section of a sliding mode makes it appearance. A dangerous boundary always corresponds to a bifurcation associated with the occurrence of incompletely elastic collisions. The possibility of unlimited complication of the parameter space structure due to the effect of boundary “blurring” is established. An example of the analysis of that structure is presented, and an estimate is made of the width of the band of boundary blurring.
MSC:
| 70J25 | Stability for problems in linear vibration theory |
| 34C25 | Periodic solutions to ordinary differential equations |
References:
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| [2] | Bautin, N. N., Behavior of Dynamic Systems Close to Stability Region Boundaries (1949), Gostekhizdat: Gostekhizdat Leningrad-Moscow |
| [3] | Feigin, M. I., Doubling of the oscillation period with C-bifurcations in piece-wise continuous systems, PMM, Vol. 34, N≗5 (1970) · Zbl 0224.34021 |
| [4] | Fedosenko, Iu. S.; Feigin, M. I., On the theory of the slipping state in dynamical systems with collisions, PMM, Vol. 36, N≗5 (1972) · Zbl 0287.70009 |
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