» Download Article

Authors
Griselda Rut Itovich; Jorge Luis Moiola; Hernan Cendra

Digital Object Identifier (DOI)
10.3182/20060628-3-FR-3903.00042

Page Numbers:
233-238

Index Terms
differential equations,frequency methods,harmonic balance techniques,stability analysis,Nyquist diagrams

Abstract
The interactions between two frequencies in a nonlinear model of resonant coupled electrical circuits are analyzed. This situation is related with the existence of one Hopf degeneracy known as non resonant double Hopf bifurcation. The dynamics of its unfolding is evaluated through the determination of the Neimark-Sacker bifurcation curves, which involve the appearance of quasiperiodic solutions. Moreover, fold and flip bifurcations of cycles have also been detected nearby the singularity using a frequency domain approach.

References

[1] Balanov, A. G., N. B. Janson and E. Schöll (2005).
    Delayed feedback control of chaos: Bifurcation
     analysis. Physical Review E 71, 016222.
[2] Chamara, P. A. and B. D. Coller (2004). A study
    of double flutter. Journal of Fluids and Structures
     19, 863-879.
[3] Coller, B.D. (2003). Intriguing nonlinear dynamics
     of a controller with a sluggish actuator.
    Automatica 39, 2049-2058.
[4] Gattulli, V., F. Di Fabio and A. Luongo (2003).
    One to one resonant double Hopf bifurcation
     in aeroelastic oscillators with tuned mass
    dampers. Journal of Sound and Vibration
    262(2), 201-217.
[5] Golubitsky, M. and W. F. Langford (1981). Classification
     and unfoldings of degenerate Hopf bifurcations.
     Journal of Differential Equations
    41, 375-415.
[6] Guckenheimer, J. and P. Holmes (1993). Nonlinear
     Oscillations, Dynamical Systems, and Bifurcations
     of Vector Fields. 4th ed.. Springer-Verlag.
     New York.
[7] Itovich, G. R. and J. L. Moiola (2002). Characterization
     of dynamic bifurcations in the frequency
     domain. International Journal of Bifurcation
     and Chaos 12(1), 87-101.
[8] Itovich, G. R. and J. L. Moiola (2005). Double Hopf
    bifurcations analysis using frequency domain
    methods. Nonlinear Dynamics 39, 235-258.
[9] Khibnik, A. I., Y. A. Kuznetsov, V. V. Levitin and
    E. V. Nikolaev (1993). Continuation techniques
     and interactive software for bifurcation
    analysis of ODE's and iterated maps. Physica
    D 62, 360-371.
[10] Kuznetsov, Y. A. (1998). Elements of Applied Bifurcation
      Theory. Vol. 34. 2nd ed.. Springer-Verlag.
      New York.
[11] Mees, A. I. and L. Chua (1979). The Hopf bifurcation
      theorem and its applications to non-linear
      oscillations in circuits and systems.
     IEEE Transactions on Circuits and Systems
     26(4), 235-254.
[12] Moiola, J. L. and G. Chen (1996). Hopf Bifurcation
     Analysis - A Frequency Domain Approach.
     World Scientific. Singapore.
[13] Yu, P. (2002). Analysis on double Hopf bifurcation
      using computer algebra with the aid of
     multiple scales. Nonlinear Dynamics 27,19-
     53.
[14] Zhang, Q. C. and A. Y. T. Leung (2000). Normal
      form of double Hopf in forced oscillators.
      Journal of Sound and Vibration
     231(4), 1057-1069.