Zero-Hopf Bifurcation in the Generalized Stretch-Twist-Fold Flow

Niazy H. Hussein, Azad I. Amen

Abstract


A zero-Hopf equilibrium in a three-dimensional system is an isolated equilibrium point which has a zero eigenvalue and a simple pair of purely imaginary eigenvalues. In general, for such an equilibrium, there is no theory for finding when some periodic solutions are bifurcated by perturbing the parameters of the system. In this work, we describe the values of the parameters for which a zero-Hopf equilibrium occurs at the equilibrium points in the generalized stretch-twist-fold flow. Thus, only one condition for parameters of the generalized stretch-twist-fold flow introduced in a system (Eq. 1) is found for which the equilibrium point is a zero-Hopf equilibrium. For this condition, we use the averaging method to provide the existence of a periodic solution, which bifurcates from the zero-Hopf equilibrium point. The main result in this paper is Theorem 1, which gives a periodic solution of the generalized stretch-twist-fold flow.


Keywords


Stretch-twist-fold flow; Zero-Hopf equilibrium point; Zero-Hopf bifurcation; Periodic solution; The averaging method.

Full Text:

PDF

References


Vainshtein, S.I. and Zel’dovich, Y.A. Origin of magnetic fields in astrophysics (turbulent “dynamo” mechanisms). Soviet Physics Uspekhi, 1972, 15(2), 159-172.

Moffatt, H.K. Stretch, twist and fold. Nature, 1989, 341(6240), 285-286.

Childress, S. and Gilbert, A.D. Stretch, Twist, Fold: The Fast Dynamo. Springer, Berlin/Heidelberg, 1995.

Bajer, K. and Moffatt, H. On a class of steady confined stokes flows with chaotic streamlines. Journal of Fluid Mechanics, 1990, 212, 337-363.

Samuel, I.V., Roald, Z.S., Rosner, R. and Kim, E. Fractal properties of the stretch-twist-fold magnetic dynamo. Physical Review E, 1996, 53(5), 4729-4744.

Jianghong, B. and Qigui, Y. Complex dynamics in the stretch-twist-fold flow. Nonlinear Dynamics, 2010, 61(4), 773-781.

Jianghong, B. and Qigui, Y. A new method to find homoclinic and heteroclinic orbits. Applied Mathematics and Computation, 2011, 217(14), 6526-6540.

Jianghong, B. and Qigui, Y. Bifurcation analysis of the generalized stretch-twist-fold flow. Applied Mathematics and Computation, 2014, 229, 16-26.

Jianghong, B. and Qigui, Y. Darboux integrability of the stretch-twist-fold flow. Nonlinear Dynamics, 2014, 76(1), 797-807.

Buzzi, C., Llibre, J. and Medrado, J. Hopf and zero-Hopf bifurcations in the Hindmarsh–Rose system. Nonlinear Dynamics. 2016, 83(3), 1549-1556.

Junze, L., Yebei L. and Zhouchao W. Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua’s system. Advances in Difference Equations, 2018, 1, 141-158.

Llibre, J. Periodic orbits in the zero-hopf bifurcation of the Rossler system. Roman. Astron. J., 2014, 24(1), 49-60.

Llibre, J., Oliveira, R.D.S. and Valls, C. On the integrability and the zero-Hopf bifurcation of a Chen-Wang differential system. Nonlinear Dynamics, 2015, 80(1-2), 353-361.

Barreira, L., Llibre, J. and Valls, C. Limit cycles bifurcating from a zero–Hopf singularity in arbitrary dimension, Nonlinear Dynamics. 2018, 92, 1159-1166.

Llibre, J. and Amar M. Zero-Hopf bifurcation in the generalized Michelson system. Chaos, Solutions and Fractals, 2016, 89, 228-231

Llibre, J. and Perez-Chavela, E. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete Continuous Dynamical Systems-Series B, 2014, 19(6), 1731-1736. Llibre, J. and Rodrigo D. Zero-Hopf bifurcation in a Chua system. Nonlinear Analysis: Real World Applications, 2017, 37, 31-40.

Guckenheimer, J. On a Codimension two bifurcation, in: Dynamical Systems and Turbulence. Warwick, 1980, Springer, 1981, 99-142.

Guckenheimer, J. and Holmes, P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, 2013, 42. Springer Science and Business Media, 2013.

Scheurle, J. and Marsden, J. Bifurcation to quasi-periodic tori in the interaction of steady state and hopf bifurcations, SIAM Journal on Mathematical Analysis, 1984, 15(6), 1055-1074

Kuznetsov, Y.A. Elements of applied bifurcation theory, Vol. 112, Springer Science and Business Media, 2013.

Baldom´a, I. and Seara, T. Brakdown of heteroclinic orbits for some analytic unfoldings of the Zero-Hopf singularity, Journal of Nonlinear Science., 2006, 16, 543-582.

Broer, H. and Vegter, G., Subordinate Silnikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynamic Systems, 1984, 4, 509-525.

Fatou, P. On the movement of a system subject to short-period forces. Bulletin of the Mathematical Society of France. 1928, 56, 98-139.

Bogoliubov, N. and Krylov, N. The application of methods of nonlinear mechanics in the theory of stationary oscillations, Publication 8 of Ukrainian Academy of Sciences, Kiev, 1934.

Bogolyubov, N. On some statistical methods in mathematical physics, Izdat. Akad. Nauk Ukr. SSR, Kiev. 1945.

Sanders, J.A. and Verhulst, F. Averaging Methods in Nonlinear Dynamical Systems, in: Applied Mathematics Science, Vol. 59, Springer, 1985.




DOI: http://dx.doi.org/10.24200/squjs.vol24iss2pp122-128

Refbacks

  • There are currently no refbacks.


Copyright (c) 2020 Niazy H. Hussein, Azad I. Amen

Creative Commons License
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

SQUJS 2017-CC BY-ND

This journal and its content is licensed under a Attribution-NoDerivatives 4.0 International.

Flag Counter