Centre Bifurcations for a Three Dimensional System with Quadratic Terms

Rizgar H. Salih; Mohammad S. Hasso; Surma H. Ibrahim

doi:10.21271/ZJPAS.32.2.7

Authors

Rizgar H. Salih Department of Mathematics, College of Basic Education, University of Raparin, Kurdistan Region-Iraq
Mohammad S. Hasso Department of Mathematics, Faculty of Science and Health, Koya University, Kurdistan Region-Iraq
Surma H. Ibrahim Department of Mathematics, Koya University, Faculty of Science and Health, Kurdistan Region-Iraq

DOI:

https://doi.org/10.21271/ZJPAS.32.2.7

Keywords:

Hopf and Centre Bifurcation; Periodic Solutions; Inverse Jacobi Multiplier.

Abstract

This article is devoted to study the bifurcated periodic orbits from centre for a differential equation of third order. Sufficient conditions for the existence of a centre are obtained by using inverse Jacobi multiplier. As a result, we found four sets of centre conditions on the centre manifold. For a given centre, it is shown that three periodic orbits can be bifurcated from the origin under two sets of condition and four periodic orbits under the other sets of condition. The cyclicityes are obtained by considering the linear parts of the corresponding Liapunov quantities of the perturbed system.

References

Ameen, A. I., 2015. Computing Isochronous Center Conditions for Polynomial Differential Systems. ZANCO Journal of Pure and Applied Sciences, 27(1), pp. 41--50.

Ameen, A. I., Salih, R. H. & Aziz, W., 2009. Hopf Bifurcation Analysis for Stablity Nontrivial Critical Points of the Rössler's Second System. Journal of Koya University, Volume 12, pp. 77-95.

Berrone, L. R. & Giacomini, H., 2003. Inverse Jacobi multipliers. Rendiconti del Circolo Matematico di Palermo, 52(1), pp. 77-130.

Bibikov, Y. N., 1979. Local Theory of Nonlinear Analytic Ordinary Differential Equations. s.l.:Springer.

Buică, A., Garc´ıa, I. & Maza, S., 2012. Existence of inverse Jacobi multipliers around Hopf points in Emphasis on the center problem. Journal of Differential Equation, pp. 6324- 6336.

Christopher, C., 2005. Estimating Limit Cycle Bifurcations from Centers. Differential Equation with symbolic computation, pp. 23-35.

Mahdi, A., 2013. Center problem for third-order ODEs. International Journal of Bifurcation and Chaos, Volume 23, p. 1350078.

Mahdi, A., Pessoa, C. & Hauenstein, . J., 2017. A hybrid symbolic-numerical approach to the center-focus problem. Journal of Symbolic Computation, Volume 82, pp. 57--73.

Salih, R. H., 2009. Studying the Stability of Origin Point for the Rossler's Second System. Journal of Koya University, Volume 10, pp. 29-44.

Salih, R. H., 2015. HOPF BIFURCATION AND CENTRE BIFURCATION IN THREE DIMENSIONAL LOTKA-VOLTERRA SYSTEMS. s.l.:PhD thesis, Plymouth University.

Salih, R. H., 2017. Zero-Hopf Bifurcation in the Rössler’s Second System. ZANCO Journal of Pure and Applied Sciences, 29(5), pp. 66-75.

Salih, R. H. & Ameen, A. I., 2008. Limit Cycles of Lorenz System With Hopf Bifurcation. AL-Rafidain Journal of Computer Sciences and Mathematics, pp. 81-99.

Salih, R. H. & Hasso, M. S., 2017. Centre bifurcations of periodic orbits for some special three dimensional systems. Electronic Journal of Qualitative Theory of Differential Equations, Issue 19, pp. 1-10.