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# Hopf Bifurcation Analysis for a Novel Hyperchaotic System

[+] Author and Article Information
Kejun Zhuang

Institute of Applied Mathematics,
School of Statistics and Applied Mathematics,
Anhui University of Finance and Economics,
Bengbu 233030, China
e-mail: zhkj123@163.com

Contributed by the Design Engineering Division of ASME for publication in the Journalof Computationaland Nonlinear Dynamics. Manuscript received March 18, 2011; final manuscript received March 5, 2012; published online June 14, 2012. Assoc. Editor: Henryk Flashner.

J. Comput. Nonlinear Dynam 8(1), 014501 (Jun 14, 2012) (4 pages) Paper No: CND-11-1040; doi: 10.1115/1.4006327 History: Received March 18, 2011; Revised March 05, 2012

## Abstract

The paper mainly focuses on a novel hyperchaotic system. The local stability of equilibrium is analyzed and existence of Hopf bifurcation is established. Moreover, formulas for determining the stability and direction of bifurcating periodic solutions are derived by center manifold theorem and normal form theory. Finally, numerical simulation is given to illustrate the theoretical analysis.

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## Figures

Fig. 1

Phase portraits of hyperchaotic system (1) when (a,b,c,k) = (35,3,35,8). This is the projection on x-y-z plane.

Fig. 2

Phase portraits of hyperchaotic system (1) when (a,b,c,k) = (35,3,35,8). This is the projection on y-z-u plane.

Fig. 3

The equilibrium of (4) is stable when (a,b,c,k) = (5,3,−5,22). This is the projection on x-y-z plane.

Fig. 4

The equilibrium of (4) is stable when (a,b,c,k) = (5,3,−5,22). This is the projection on x-z-u plane.

Fig. 5

The equilibrium of (4) is stable when (a,b,c,k) = (5,3,−5,28). This is the projection on x-y-z plane.

Fig. 6

The equilibrium of (4) is stable when (a,b,c,k) = (5,3,−5,28). This is the projection on x-z-u plane.

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