Zero-Hopf Bifurcation in the Rössler’s Second System

Rizgar Haji Salih

Abstract


     This paper is devoted to study the zero-Hopf bifurcation of the Rössler's second system. We characterize the parameters for which a zero-Hopf equilibrium point takes place at each point. We prove that there are three one-parameter families exhibiting such equilibria. The averaging theory of the first order is also applied to prove the existence of one periodic orbit bifurcating from the zero-Hopf equilibrium at the origin. Here, to visualize this, FireFlies software is used.

Keywords


R¨ossler’s second system, periodic orbit, averaging theory, zero Hopf bifurcation.

Full Text:

PDF

References


Amen, A., Salih, R. and Aziz, W. (2009), 'Hopf Bifurcation Analysis for Stability Nontrivial Critical Points of the Rössler's Second System', Journal of Koya University 12, 77-95.

Baldomá, I. and Seara, T. M. (2006), 'Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity', Journal of Nonlinear Science 16(6), 543-582.

Baldomă, I. and Seara, T. M. (2008), 'The inner equation for generic analytic unfoldings of the Hopf-zero singularity', Discrete Contin. Dyn. Syst. Ser. B 10(2-3), 323--347.

Broer, H. W. and Vegter, G. (1984), 'Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension', Ergodic Theory and Dynamical Systems 4(04), 509--525.

Buică, A. and Llibre, J. (2004), 'Averaging methods for finding periodic orbits via Brouwer degree', Bulletin des Sciences Mathématiques 128(1), 7--22.

Buică, A., Francoise, J. and Llibre, J. (2007), 'Periodic solutions of nonlinear periodic differential systems with a small parameter', Communications on Pure and Applied Analysis 6(1), 103-111.

Castellanos, V., Llibre, J. and Quilantan, I. (2013), 'Simultaneous Periodic Orbits Bifurcating from Two Zero-Hopf Equilibria in a Tritrophic Food Chain Model', Journal of Applied Mathematics and Physics 1(7), 31-38.

Champneys, A. R. and Kirk, V. (2004), 'The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities', Physica D: Nonlinear Phenomena 195(1-2), 77-105.

Chow, S. N. and Hale, J. K. (1982), Methods of bifurcation theory, Vol. 251, Springer-Verlag, New York-Berlin.

Euzébio, R. D., Llibre, J. and Vidal, C. (2015), 'Zero-Hopf bifurcation in the FitzHugh-Nagumo system', Mathematical Methods in the Applied Sciences 38(17), 4289--4299.

Euzébio, R. D. and Llibre, J. (2017), 'Zero-Hopf bifurcation in a Chua system ', Nonlinear Analysis: Real World Applications 37, 31 - 40.

García, I. A., Llibre, J. & Maza, S. (2014), 'On the periodic orbit bifurcating from a zero Hopf bifurcation in systems with two slow and one fast variables', Applied Mathematics and Computation 232, 84-90.

Guckenheimer, J. (1981), On a codimension two bifurcation, in David Rand and Lai-Sang Young, ed. ' Dynamical Systems and Turbulence, Warwick 1980', Springer, Berlin, 99-142.

Guckenheimer, J. and Holmes, P. (2013), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer New York.

Han, M. (1998), 'Existence of Periodic Orbits and Invariant Tori in Codimension Two Bifurcations of Three-Dimensional Systems', Journal of Systems Science and Mathematical Sciences 18(4), 403-409.

Kokubu, H. and Roussarie, R. (2004), 'Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part I', Journal of Dynamics and Differential Equations 16(2), 513--557.

Kuznetsov, Y. A. (2004), Elements of applied bifurcation theory, Vol. 112, Springer-Verlag, New York.

Llibre, J., Makhlouf, A. and Badi, S. (2009), '3-dimensional Hopf bifurcation via averaging theory of second order', Discrete and Continuous Dynamical Systems-Series A (DCDS-A) 25(4), 1287-1295.

Llibre, J. (2014a), 'Periodic Orbits in the Zero-Hopf Bifurcation of the Rössler System', Romanian Astronomical Journal 24(1), 49-60.

Llibre, J. and Pérez-Chavela, E. (2014b), 'Zero-Hopf bifurcation for a class of Lorenz-type systems', Discrete and Continuous Dynamical Systems - Series B 19(6), 1731-1736.

Llibre, J. and Xiao, D. (2014c), 'Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems', Proceedings of the American Mathematical Society 142(6), 2047-2062.

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

Marsden, J. E. and McCracken, M. (1976), The Hopf Bifurcation and Its Applications, Vol. 19, Springer New York.

Merrison-Hort, R. (2015), 'Fireflies: New Software for Interactively Exploring Dynamical Systems Using GPU Computing', International Journal of Bifurcation and Chaos 25(13), 1550181.

Messias, M., Nespoli, C. and Dalbelo, T. M. (2008), 'Mechanics for the creation of strange attractors in Rössler's second system', TEMA Tend. Mat. Apl. Comput. 9(2), 275--285.

Rössler, O. E. (1979), 'Continuous Chaos-four Prototype Equations', Annals of the New York Academy of Sciences 316(1), 376--392.

Salih, R. (2009), 'Studying the Stability of the Origin for the Rössler's Second System', Journal of Koya University 10, 29-44.

Sanders, J. A.; Verhulst, F. and Murdock, J. (2007), Averaging Methods in Nonlinear Dynamical Systems, Vol. 59, Springer, New York.

Scheurle, J. and Marsden, J. (1984), 'Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations', SIAM Journal on Mathematical Analysis 15(6), 1055-1074.




Copyright (c) 2017 ZANCO Journal of Pure and Applied Sciences

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

ZANCO Journal of Pure and Applied Sciences (print version: ISSN 2218-0230 online version: ISSN 2412-3986, DOI: 10.21271) is published by Salahaddin University-Erbil / Department of Scientific Publications. Responsibility for the contents rests upon the authors and not upon  Salahaddin University-Erbil or the Journal Editorial and Advisory Boards. 

Department of Scientific Publication Office: The Central Library of Salahaddin University-Erbil, Kirkuk Road, Erbil, Kurdistan, Iraq. Cell Phone: +964 (0)750 7761675, email: zanco.scientific@su.edu.krd. www.su.edu.krd, www.zancojournals.su.edu.krd

Copyright and Reprint Permission: It is the policy of ZANCO to own the copyright to the technical contributions it publishes and to facilitate the appropriate reuse of this material by others. Photocopying is permitted with credit to the source for individuals for individual use.

Copyright © 2017. All Rights Reserved. Salahaddin University-Erbil