Bifurcations and chaos in piecewise-smooth dynamical systems by Zhusubaliyev T., Mosekilde E.

By Zhusubaliyev T., Mosekilde E.

Technical difficulties frequently result in differential equations with piecewise-smooth right-hand facets. difficulties in mechanical engineering, for example, violate the necessities of smoothness in the event that they contain collisions, finite clearances, or stick-slip phenomena. structures of this kind can exhibit a wide number of complex bifurcation situations that also lack a close description. This e-book offers many of the attention-grabbing new phenomena that you may become aware of in piecewise-smooth dynamical platforms. the sensible importance of those phenomena is proven via a chain of well-documented and life like purposes to switching strength converters, relay platforms, and kinds of pulse-width modulated regulate structures. different examples are derived from mechanical engineering, electronic electronics, and fiscal business-cycle concept. the themes thought of within the ebook contain abrupt transitions linked to converted period-doubling, saddle-node and Hopf bifurcations, the interaction among classical bifurcations and border-collision bifurcations, truncated bifurcation eventualities, period-tripling and -quadrupling bifurcations, multiple-choice bifurcations, new sorts of direct transitions to chaos, and torus destruction in nonsmooth structures. despite its orientation in the direction of engineering difficulties, the ebook addresses theoretical and numerical difficulties in adequate aspect to be of curiosity to nonlinear scientists more often than not.

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Dabei ist 7 das spezifische Gewicht des Bojenkorpers, € ein Parameter (bei vorhandenem Bojendeckel e = 1, ohne Bojendeckel e = 0), d die Wandstarke der Boje im Mantel und Deckel, a der Neigungswinkel zur Vertikalen. Daraus resultiert h = [{21 + r + er)j - r - 2b]/{r/d + 2) . 26: Eindringtiefe, Luftpolsterhohe und mo iiber relativer Befullung Fiir stabile aufrechte Lage ist folgendes zu verlangen: Bezogen auf einen willkiirlich gewahlten Drehpunkt muss das Aufrichtmoment grofier sein als das Moment, das zufolge der Schwerkraft die Boje umzuwerfen droht.

Numerisch findet man eine reelle Losung des Nennerpolynoms 53 = —3, 795. Daraus resultieren die konjugiert komplexen Losungen 0,0223 ± j 1,815. 27) Dieses Resultat ist in Abb. 9 dargestellt, der Schleifenfrequenzgang Fo{juj) in Abb. 10. Den Wurzelort zeigt Abb. 11 fiir das ^0(5) aus Abb. 10, in einer BezifFerung nach einem zusatzlichen Faktor V im Zahler. Die Wurzelortskurve besitzt eine Mehrfachlosung bei 5 = —0, 528. 36 2 Analyse einfacher Regelkreise u Vref • ^ J " . 5] ; den=[l 3 . 7 5 . .

199) liegt dann ein Doppelpol vor, wenn 4kRTD = 2a = 4^/kR + 1 TD = , "^^ = ^ kR = 0,351 -^ K{s) = 9(1 + 0,351s) . 201) zeigt die Abb. 41b unter Ti = 0,03. VergroBert man Ti, so ruckt man den Pol bei - 3 3 an die Nullstelle heran und die Stabilisierung wird geschwacht, siehe Wurzelortskurve Abb. 41c fiir Ti = 0,25. Die Sprungantwort des Regelkreises fiir den idealen PD-Regler unter kR = 9 und TD = 0,351 zeigt die Abb. 42. 54 * Regelung mit AUpass als Strecke Angabe: Eine AUpass-Strecke soil fehlerfrei und unter einem Dampfungsgrad von D = 0,7 geregelt werden.

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