Ricerche di Matematica

, Volume 68, Issue 2, pp 811–840 | Cite as

Hopf bifurcations in dynamical systems

  • Salvatore RioneroEmail author


The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.


Instability Bifurcations Hopf and others unsteady bifurcations 

Mathematics Subject Classification

76E25 76E06 35B35 



This work has been performed under the auspices of the G.N.F.M. of INdAM.


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Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applications “R. Caccioppoli”University of Naples Federico II, Complesso Universitario Monte S. AngeloNaplesItaly

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