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.

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.

A Novel Simple Hyperchaotic System with Two Coexisting Attractors

2019 ◽  
Vol 29 (14) ◽  
pp. 1950203 ◽  
Author(s):  
Jiaopeng Yang ◽  
Zhengrong Liu
Keyword(s):  
Complex Dynamics ◽  
Chaotic Dynamic ◽  
Cross Product ◽  
Hyperchaotic System ◽  
Compound Structure ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Periodic Attractors ◽  
New Hyperchaotic System ◽  
New System

This article introduces a new hyperchaotic system of four-dimensional autonomous ordinary differential equations, with only cubic cross-product nonlinearities, which can respectively display two hyperchaotic attractors with only nonhyperbolic equilibria line. Several issues such as basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new hyperchaotic and chaotic system are investigated, either theoretically or numerically. Of particular interest is the fact that the two coexisting attractors of the new hyperchaotic system are symmetrical, and this hyperchaotic system can generate plenty of complex dynamics including two coexisting chaotic or periodic attractors. Moreover, some chaotic features of the attractor are justified numerically. Finally, 0-1 test is used to analyze and describe the complex chaotic dynamic behavior of the new system.

A Hyperchaotic System and its Synchronization via Scalar Controller

2011 ◽  
Vol 50-51 ◽  
pp. 254-257
Author(s):  
Wu Chun Dai ◽  
Zheng Fu Cheng
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Mathematical Proof ◽  
Hyperchaotic System ◽  
Scalar Control ◽  
Response System ◽  
Dynamical Behaviors ◽  
Synchronization Scheme ◽  
New Hyperchaotic System

In this paper, a 4D hyperchaotic system is proposed. Some basic dynamical behaviors are explored by calculating its Lyapunov exponents, Poincar´e mapping, etc.. Finally, synchronization for this new hyperchaotic system is achieved via scalar control. The nonlinear terms in the response system are not dropped. The proposed synchronization scheme is simple and theoretically rigorous. The mathematical proof of this method is provided. Some numerical simulations are obtained. The numerical simulations coincide with the theoretical analysis.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

2021 ◽  
Vol 31 (08) ◽  
pp. 2150143
Author(s):  
Zunxian Li ◽  
Chengyi Xia
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Cellular Neural Networks ◽  
Bifurcation Theorem ◽  
Decoupling Method ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Approximate Expressions ◽  
Zero Equilibrium

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

GENERATING HYPERCHAOTIC ATTRACTORS WITH THREE POSITIVE LYAPUNOV EXPONENTS VIA STATE FEEDBACK CONTROL

2009 ◽  
Vol 19 (02) ◽  
pp. 651-660 ◽  
Author(s):  
GUOSI HU
Keyword(s):  
Lyapunov Exponents ◽  
State Feedback ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
State Feedback Control ◽  
Hyperchaotic System ◽  
Lorenz Chaotic System ◽  
Simulation Results ◽  
New Hyperchaotic System ◽  
Good Agreement

This letter presents a new hyperchaotic system, which was obtained by adding a nonlinear quadratic controller to the first equation and a linear controller to the second equation of the three-dimensional autonomous modified Lorenz chaotic system. This system uses only two multipliers but can generate very complex strange attractors with three positive Lyapunov exponents. The system is not only demonstrated by numerical simulations but also implemented via an electronic circuit, showing very good agreement with the simulation results.

scholarly journals Theoretical Analysis and Circuit Verification for Fractional-Order Chaotic Behavior in a New Hyperchaotic System

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ling Liu ◽  
Chongxin Liu
Keyword(s):  
Fractional Order ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Circuit Diagram ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Dynamical Behaviors ◽  
Mapping Parameter ◽  
New Hyperchaotic System ◽  
Observation Results

A novel nonlinear four-dimensional hyperchaotic system and its fractional-order form are presented. Some dynamical behaviors of this system are further investigated, including Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations, and calculated Lyapunov exponents. A simple fourth-channel block circuit diagram is designed for generating strange attractors of this dynamical system. Specifically, a novel network module fractance is introduced to achieve fractional-order circuit diagram for hardware implementation of the fractional attractors of this nonlinear hyperchaotic system with order as low as 0.9. Observation results have been observed by using oscilloscope which demonstrate that the fractional-order nonlinear hyperchaotic attractors exist indeed in this new system.

Hopf Bifurcation of a Generalized Delay-Induced Predator–Prey System with Habitat Complexity

2020 ◽  
Vol 30 (06) ◽  
pp. 2050082
Author(s):  
Zhihui Ma
Keyword(s):  
Hopf Bifurcation ◽  
Habitat Complexity ◽  
Sufficient Conditions ◽  
Explicit Formulas ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
The Stability ◽  
Theoretical Results ◽  
Positivity And Boundedness

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.

A Novel Hyperchaotic System With Adaptive Control, Synchronization, and Circuit Simulation

2018 ◽  
pp. 382-419 ◽  
Author(s):  
Sundarapandian Vaidyanathan ◽  
Ahmad Taher Azar ◽  
Aceng Sambas ◽  
Shikha Singh ◽  
Kammogne Soup Tewa Alain ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Dissipative System ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Qualitative Properties ◽  
New Hyperchaotic System

This chapter announces a new four-dimensional hyperchaotic system having two positive Lyapunov exponents, a zero Lyapunov exponent, and a negative Lyapunov exponent. Since the sum of the Lyapunov exponents of the new hyperchaotic system is shown to be negative, it is a dissipative system. The phase portraits of the new hyperchaotic system are displayed with both two-dimensional and three-dimensional phase portraits. Next, the qualitative properties of the new hyperchaotic system are dealt with in detail. It is shown that the new hyperchaotic system has three unstable equilibrium points. Explicitly, it is shown that the equilibrium at the origin is a saddle-point, while the other two equilibrium points are saddle-focus equilibrium points. Thus, it is shown that all three equilibrium points of the new hyperchaotic system are unstable. Numerical simulations with MATLAB have been shown to validate and demonstrate all the new results derived in this chapter. Finally, a circuit design of the new hyperchaotic system is implemented in MultiSim to validate the theoretical model.

CONSTRUCTING MULTIWING HYPERCHAOTIC ATTRACTORS

2010 ◽  
Vol 20 (03) ◽  
pp. 727-734 ◽  
Author(s):  
BO YU ◽  
GUOSI HU
Keyword(s):  
Numerical Simulation ◽  
Chaotic System ◽  
Topological Structure ◽  
Construction Method ◽  
Chaotic Attractors ◽  
Hyperchaotic System ◽  
Simple Construction ◽  
System A ◽  
New Hyperchaotic System ◽  
New System

Few reports have introduced chaotic attractors with both multiwing topological structure and hyperchaotic dynamics. A simple construction method, for designing chaotic system with multiwing attractors, is presented in this paper. The number of wings in the attractor was doubled on applying this method to an arbitrary smooth chaotic system. Moreover, the hyperchaotic property is preserved in the new system. A new hyperchaotic system with 16-wing attractors is constructed; the result system is not only verified via numerical simulation but also confirmed by a DSP-based experiment.

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