Periodic Solutions Bifurcating From a Curve of Singularity of the Jerk System

  • Niazy Hady Hussein 1Department of Mathematics, Faculty of Science, Soran University-Erbil, Kurdistan Region, Iraq 2Department of Mathematics, College of Basic Education, University of Raparin, Ranya, Kurdistan Region, Iraq
Keywords: Jerk system, periodic orbit, zero-Hopf singularity, the method of averaging

Abstract

We investigate a periodic solution which bifurcates from a curve of the singularity of the jerk system in .More precisely, we give the explicit states for the existence of a periodic solution of the jerk system with a nonisolated singular point, where for each singular point has a simple pair of purely imaginary and one zero eigenvalues. We recall for this point of singularity as a zero-Hopf (z-H) singular point. The coefficients in the jerk system are described for which the z-H singularity occur at each point of that curve of singularity. We show that for each point at that curve of singularity there is only one family of parameters which exhibits such type of singular points. The method of averaging in the second order is utilized to determine one periodic solution which bifurcates from any point of that curve of singularity. As far as, we realize that this investigation is the study on bifurcations from a curve of nonisolated z-H singularity to provide a periodic solution via the method of averaging. Under a generic small perturbation at the parameters, we prove that a periodic solution will be bifurcated at any point that located on a curve of a singularity of the jerk system.

References

References

BALDOM'A, I. & SEARA, T. 2006. Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity, 16(6), 543-582.
BROER, H., W. & VEGTER, G. 2006. Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory and Dynamical Systems. 4(4), 509-525.
BUICA, A., FRAN,COISE, J., P. & LLIBRA, J. 2007. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure and Applied Analysis, 6 (1), 103.
C^ANDIDO, M. R. & LLIBRA, L. 2018. Periodic Orbits Bifurcating from a Nonisolated Zero-Hopf Equilibrium of Three-Dimensional Differential Systems Revisited. International Journal of Bifurcation and Chaos. 28(05), 1850058.
CASTELLANOS, V., LLIBRA, L. & QUILANTAN, I. 2003. Simultaneous periodic orbits bifurcating from two z-H equilibria in a tritrophic food chain model. Journal of Applied Mathematics and Physics, 1 (07), 31.
CHOW, S.-N & HALE, J., K. 2012 Methods of bifurcation theory Vol. 251. Springer Science and Business Media.
EUZ'EBIO, R., D. & LLIBRA, L. 2017. Zero-Hopf bifurcation in a chua system. Nonlinear Analysis: Real World Applications, 37, 31–40.
EUZ'EBIO, R., D., LLIBRA, L. & VIDAL, C. 2015. Zero-Hopf bifurcation in the fitzhughnagumo system. Mathematical Methods in the Applied Sciences, 38 (17), 4289–4299.
GARC'IA, I., LLIBRA, L. & 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.
GOTTLIEB,H. P. 1990. Simple nonlinear jerk functions with periodic solutions.American Journal of Physics. 66(10), 903-906.
JOHN, G. 1981. On a codimension two bifurcation. In Dynamical systems and turbulence, Warwick 1980 (pp. 99–142). Springer.
JOHN, G. & PHILIP, H. 2013. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Vol. 42). Springer Science and Business Media.
KUZNETSOV, Y. A. 2013. Elements of applied bifurcation theory (Vol. 112). Springer Science and Business Media.
LLIBRA, L. 2014. Periodic orbits in the zero-hopf bifurcation of the r¨ossler system’. Romanian Astronomical Journal, 24 (1), 49–60.
LLIBRA, L., MAKHOUF, A. & BADI, S. 2009. 3-dimensional hopf bifurcation via averaging theory of second order. Discrete and Continuous Dynamical Systems A, 25 (4), 1287–1295.
LLIBRA, L., OLIVERIRA, R. D. & VALLS, C. 2015. On the integrability and the zero-hopf bifurcation of a chen–wang differential system. Nonlinear Dynamics, 80 (1-2), 353–361.
LLIBRA, L., & P´EREZ-CHAVELA, E. 2014. Zero-hopf bifurcation for a class of Lorenz type systems. Discrete and Continuous Dynamical Systems-Series B, 19(6), 1731–1736.
LLIBRA, L. & XIAO, D. 2014. 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.
MARSDEN, J. E. & McCRACKEN, M. 2012. The hopf bifurcation and its applications (Vol. 19). Springer Science and Business Media.
MOLAIE, M., JAFARI, S., SPROTT, J. C. & GOLPAYEGANI, S. M. 2013. Simple chaotic flows with one stable equilibrium. International Journal of Bifurcation and Chaos, 23 (11), 1350188.
RIZGAR, H. 2017. Zero-Hopf Bifurcation in the Rössler’s Second System. ZANCO Journal of Pure and Applied Sciences, 29(5), 66-75.
O’MALLEY, Jr. R. 1987. Averaging methods in nonlinear dynamical systems. (jasanders and f. verhulst). Society for Industrial and Applied Mathematics.
PI, D., H. & and ZHANG, X. 2009. Limit cycles of differential systems via the averaging methods. Canadian Appl. Math. Quarterly. 17, 243-269.
SANDERS, J. A., VERHULST, F. & MURDOCK, J. 2007. Averaging methods in nonlinear dynamical systems. Society for Industrial and Applied Mathematics. (Vol. 59). Springer
SPROTT, J. C. 1997. Simplest dissipative chaotic low. Physics letters A. 228(4-5), 271-274.
WEI, Z., SPROTT, J. & CHEN, H. 2015. Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium. Physics Letters A, 379 (37), 2184 - 2187.
Published
2020-04-22
How to Cite
Hussein, N. (2020) “Periodic Solutions Bifurcating From a Curve of Singularity of the Jerk System”, Zanco Journal of Pure and Applied Sciences, 32(2), pp. 55-61. doi: 10.21271/ZJPAS.32.2.6.
Section
Mathematics ,Physics and Engineering Researches