Bifurcation analysis and dynamics of a Lorenz –type dynamical system 

Authors
Abstract
./files/site1/files/0Abstract1.pdfIn this paper we consider a continues Lorenz – type dynamical system. Dynamical behaviors of this system such as computing equilibrium points, different bifurcation curves and computation of normal form coefficient of each bifurcation point analytically and numerically. In particular we derived sufficient conditions for existence of Hopf and Pitchfork bifurcations and determined criticality of these bifurcations. By means of numerical simulations, we show that the system may have chaotic behavior under some conditions. By employing numerical continuation method, we first compute bifurcation curves and then compute all codimension 1 and 2 bifurcation along these curves with determined of the corresponding normal form coefficient.
Keywords

1. E .L‎. ‎Allgower and K. George‎," Numerical continuation methods: An introduction", ‎spring-verlag‎, ‎berlin‎, ‎(1990‎). 2. E. J. Doedel, W. Govaerts, and YU. A. Kuznetsov,” Computation of periodic solution bifurcations in ODE using bordered systems”, SIAM Journal on Numerical Analysis,1-33. 3. Q .Bi‎, ‎B. Jiang‎ and ‎X‎. ‎Han ,"Hopf Bifurcation analysis in T system‎, ‎nonlinear analysis: Real World Application", 11 (2010) 522-527‎. 4. F.R‎. ‎Gantmacher,"Theory of Matrices‎", ‎Volume 2‎, ‎Chelsea‎, ‎New York‎ (‎1989)‎. 5. C. Li , J. C. Sprott, W. Thio, Linearization of the Lorenz system, Physics Letters A, 379 (2015) 888-893. 6. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, New York, Springer-Verlag, Third edition (2004). 7. G‎. ‎Tigan‎, "Analysis of a 3D chaotic system", ‎Chaos Soliton Fractals 36 (2008) 1315-1319‎. 8. G‎. ‎Tigan‎, "Bifurcation and stability in a system derived from the Lorenz system"‎, ‎Sci. Bull. Politehnica Univ.Timisoara Tomal 50(64) (Fascicola 1) (2005) 61-72‎. 9. G‎. ‎Tigan, "Hopf Bifurcation in the T system"‎, ‎Mathematical Bulletin 30 (2006) 9-16‎. 10. T‎. ‎Yang‎, "A Survey of chaotic secure communication system‎, ‎International Journal of Computational Cognition", 2 (2004) 81-130‎. 11. Q. Yang, Y.Chen, Complex Dynamics in the Unified Lorenz-Type System, International Journal of Bifurcation and Chaos, 24(4) (2014) 1450055.