Sommario
Il lavoro analizza in dettaglio un fenomeno di biforcazione dinamica degenere che si presenta in sistemi autonomi. La situazione sorge quando un coefficiente si annulla in un punto critico dove viene anche simultaneamente violata la condizione di trasversalità di Hopf. Si dimostra analiticamente che, a differenza del fenomeno di biforcazione alla Hopf, in questo caso l'esistenza di una famiglia ≪biforcante≫ di cicli-limite non può essere assicurata; infatti una condizione di esistenza emerge come parte integrante dell'analisi. Sotto tale condizione possono sorgere diversi fenomeni topologicamente distinti; si discutono le condizioni che danno luogo a questi casi. Le equazioni asintotiche degli itinerari di biforcazione, la famiglia di ciclilimite e le relazioni frequenza-ampiezza vengono fornite in forme generalied espliciteche possono essere usate direttamente nelle analisi di problemi specifici.
Summary
A degenerate dynamic bifurcation phenomenon exhibited by autonomous systems is analyzed in detail. The situation arises when a key coefficient vanishes at a critical point where Hopf's transversality condition is also violated simultaneously. It is demostrated analytically that, unlike the phenomenon of Hopf bifurcation, in this case the existence of bifurcating family of limit cycles cannot be guaranteed. Indeed, an existence condition emerges as an integral part of the analysis. Under this existence condition, several topologically distinct phenomena may arise, and the conditions giving rise to such cases are discussed. the asymptotic equations of the bifurcating paths, the family of limit cycles and frequency-amplitude relationships are given in general, explicitforms which can be used in the analyses of specific problems directly.
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Atadan, A.S., Huseyin, K. A double Hopf bifurcation phenomenon. Meccanica 21, 123–129 (1986). https://doi.org/10.1007/BF01556692
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DOI: https://doi.org/10.1007/BF01556692