scholarly journals Research on Hopf Bifurcation of a 4D Hyperchaotic System

2017 ◽  
Vol 06 (04) ◽  
pp. 474-480
Author(s):  
玉明 陈
Keyword(s):  
Hopf Bifurcation ◽  
Hyperchaotic System

scholarly journals Dynamics of a New Hyperchaotic System with Only One Equilibrium Point

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xiang Li ◽  
Ranchao Wu
Keyword(s):  
Hopf Bifurcation ◽  
Hyperchaotic System ◽  
Compound Structure ◽  
Normal Form Theory ◽  
Controlled System ◽  
Forming Mechanism ◽  
Form Theory ◽  
The Lorenz System ◽  
The Stability ◽  
New Hyperchaotic System

A new 4D hyperchaotic system is constructed based on the Lorenz system. The compound structure and forming mechanism of the new hyperchaotic attractor are studied via a controlled system with constant controllers. Furthermore, it is found that the Hopf bifurcation occurs in this hyperchaotic system when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation as well as the stability of bifurcating periodic solutions is presented in detail by virtue of the normal form theory. Numerical simulations are given to illustrate and verify the results.

scholarly journals Hopf Bifurcation Analysis for a Stochastic Discrete-Time Hyperchaotic System

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Jie Ran ◽  
Yu-Qin Li ◽  
Shao-Juan Ma ◽  
Juan Wu
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Discrete Time ◽  
Control Method ◽  
Deterministic System ◽  
Hyperchaotic System ◽  
Discrete Time System ◽  
Amplitude Control ◽  
Random Disturbances ◽  
Dynamical Features

The dynamics of a discrete-time hyperchaotic system and the amplitude control of Hopf bifurcation for a stochastic discrete-time hyperchaotic system are investigated in this paper. Numerical simulations are presented to exhibit the complex dynamical behaviors in the discrete-time hyperchaotic system. Furthermore, the stochastic discrete-time hyperchaotic system with random parameters is transformed into its equivalent deterministic system with the orthogonal polynomial theory of discrete random function. In addition, the dynamical features of the discrete-time hyperchaotic system with random disturbances are obtained through its equivalent deterministic system. By using the Hopf bifurcation conditions of the deterministic discrete-time system, the specific conditions for the existence of Hopf bifurcation in the equivalent deterministic system are derived. And the amplitude control with random intensity is discussed in detail. Finally, the feasibility of the control method is demonstrated by numerical simulations.

A 5D HYPERCHAOTIC SYSTEM WITH THREE POSITIVE LYAPUNOV EXPONENTS COINED

2013 ◽  
Vol 23 (06) ◽  
pp. 1350109 ◽  
Author(s):  
QIGUI YANG ◽  
CHUNTAO CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Power Spectrum ◽  
Lyapunov Exponents ◽  
Hyperchaotic System ◽  
Unique Equilibrium ◽  
Nonlinear Controller ◽  
Dynamical Behaviors ◽  
The Stability ◽  
New Hyperchaotic System ◽  
New System

This paper reports the finding of a five-dimensional (5D) new hyperchaotic system with three positive Lyapunov exponents, which is obtained by adding a nonlinear controller to the first equation of a 4D hyperchaotic system. The algebraical form of the hyperchaotic system is very similar to the 5D controlled Lorenz-like systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under unique equilibrium or three equilibria. To further analyze the new system, the corresponding hyperchaotic and chaotic attractor are firstly numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation, analysis of power spectrum and Poincaré projections. Moreover, some complex dynamical behaviors such as the stability of hyperbolic or nonhyperbolic equilibrium and two complete mathematical characterizations for 5D Hopf bifurcation are rigorously derived and studied.

Hopf Bifurcation Analysis for a Novel Hyperchaotic System

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Kejun Zhuang
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Local Stability ◽  
Center Manifold ◽  
Hyperchaotic System ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability

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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