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Journal of Applied Nonlinear Dynamics 1(2) (2012) 159--171 | DOI:10.5890/JAND.2012.05.004

Christoffer R. Heckman$^{1}$; Richard H. Rand$^{2}$

$^{1}$ Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA

$^{2}$ Dept. Mathematics, Dept. MAE, Cornell University, Ithaca, NY 14853, USA

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Abstract

A delay differential equation (DDE) which exhibits a double Hopf or Hopf-Hopf bifurcation [1] is studied using both Lindstedt’s method and center manifold reduction. Results are checked by comparison with a numerical continuation program (DDEBIFTOOL). This work has application to the dynamics of two interacting microbubbles.

References

1. [1]	Guckenheimer, J. and Kuznetsov, Y.A. (2008), Hopf-Hopf bifurcation, Scholarpedia, 3(8), 1856-2008.

2. [2]	Heckman, C.R., Sah, S.M. and Rand, R.H. (2010), Dynamics of microbubble oscillators with delay coupling, Commun Nonlinear Sci Numer Simulat, 15, 2735-2743.

3. [3]	Heckman, C.R. and Rand, R.H. (2011), Dynamics of Coupled Microbubbles with Large Fluid Compressibility Delays, Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011), July 24-29, 2011, Rome, Italy.

4. [4]	Heckman, C.R., Kotas, J. and Rand, R.H. (2012), Center Manifold Reduction of the Hopf-Hopf bifurcation in a Time Delay System, The First International Conference on Structural Nonlinear Dynamics and Diagnosis, April 30-May 2, 2012, Marrakech, Morocco.

5. [5]	Heckman C.R., Kotas J. and Rand R.H.(to appear), Center Manifold Reduction of the Hopf-Hopf bifur cation in a Time Delay System, Euro. Soc. App. Ind. Math..

6. [6]	Rand, R.H. (2005), Lecture Notes in Nonlinear Vibrations, version 52, http://www.math.cornell.edu/~rand/randdocs/nlvibe52.pdf

7. [7]	Rand, R.H. (2011), Differential-Delay Equations, in Luo, A.C.J. and Sun, J-Q. (eds.), Complex Systems: Fractionality, Time-delay and Synchronization, 83-117, HEP-Springer, Beijing-Berlin

8. [8]	Xu, J., Chung, K.W. and Chan, C.L. (2007), An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks, SIAM J. Applied Dynamical Systems, 6 (1), 29-60.

9. [9]	Ma, S., Lu, Q. and Feng, Z. (2008), Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control, J. Analysis and Applications, 338, 993-1007.

10. [10]	Yu, P., Yuan, Y. and Xu, J. (2002), Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback, Commun. Nonlinear Sci. Numer. Simulat., 7, 69-91.

11. [11]	Chen, Z. and Yu, P. (2005), Double-Hopf Bifurcation in an Oscillator with External Forcing and Time- Delayed Feedback Control, Proceeding of DETC 2005.

12. [12]	Buono, P.L. and Bèlair, J. (2003), Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Diff. Eq., 189, 234-266.

13. [13]	Kalmár-Nagy, T., Stépán, G. and Moon, F.C. (2001), Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations, Nonlinear Dynamics, 26, 121-142.

14. [14]	Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York.

15. [15]	Engelborghs, K., Luzyanina, T., Samaey, G., Roose, D. and Verheyden, K., DDE-BIFTOOL, available from http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml