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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Analytical Prediction of Homoclinic Bifurcations Following a Supercritical Hopf Bifurcation

Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 209--222 | DOI:10.5890/DNC.2016.09.002

Tanushree Roy$^{1}$, Roy Choudhury$^{1}$, and Ugur Tanriver$^{2}$

$^{1}$ Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364 USA

$^{2}$ Department of Mathematics, Texas A&M University, Texarkana, TX 75505

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An analytical approach to homoclinic bifurcations at a saddle fixed point is developed in this paper based on high-order, high-accuracy approximations of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This orbit then expands as the Hopf bifurcation parameter(s) is(are) varied beyond the bifurcation value, with the analytical criterion proposed for homoclinic bifurcation being the merging of the periodic orbit with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. We apply our criterion to two systems here. Using approximations of the stable, post-Hopf periodic orbits to first, second, and third orders in a multiple-scales perturbation expansion, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely.


  1. [1]  Nayfeh, A. and Balachandran, B. (1995), Applied Nonlinear Dynamics,Wiley, New York.
  2. [2]  Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York.
  3. [3]  Afraimovich, V.S., Bykov, V.V., and Shilnikov, L.P. (1977), On the origin and structure of the Lorenz attractor, Sov. Phys. Doklady, 22 253.
  4. [4]  Sparrow, C.T. (1982), The Lorenz Equations: Bifurcations,Chaos, and Strange Atttractors, Springer-Verlag, New York.
  5. [5]  Wiggins, S. (1988), Global Bifurcations and Chaos, Springer-Verlag, New York.
  6. [6]  Shilnikov, L.P. (1965), A case of the existence of a denumerable set of periodic motions, Sov. Math. Doklady, 6, 163.
  7. [7]  Strogatz, S. (1994), Nonlinear Dynamics and Chaos, Addison-Wesley, Reading (Mass).
  8. [8]  Dominik Zobel (2013), Nonlinear Dynamics: Some exercises and solutions, Creative Commons License.
  9. [9]  Glendinning, P. (1994), Stability, Instability and Chaos, Cambridge U. Press, Cambridge.
  10. [10]  Krise, S. and Choudhury, S. Roy (2003), Bifurcattons and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons, and Fractals, 16, 59.
  11. [11]  Cheng, T. and Choudhury, S. Roy (2012), Bifurcations and chaos in a modified driven Chen’s system, Far East Journal of Dynamical Systems, 18 (2012), 1.