A New 6D Hyperchaotic System with Four Positive Lyapunov Exponents Coined

2015 ◽  
Vol 25 (04) ◽  
pp. 1550060 ◽  
Author(s):  
Qigui Yang ◽  
Waleed Mahgoub Osman ◽  
Chuntao Chen
Keyword(s):  
Lyapunov Exponents ◽  
Lorenz System ◽  
Feedback Controller ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Nonlinear Feedback ◽  
Dynamical Behaviors ◽  
Linear Feedback Controller ◽  
The Lorenz System ◽  
Hyperbolic Equilibrium

This paper reports the finding of a new six-dimensional (6D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 5D hyperchaotic system that is constructed by adding a linear feedback controller and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with four positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of the hyperchaotic and chaotic attractors. In addition, stability of the hyperbolic equilibrium is analyzed and two complete mathematical characterizations for 6D Hopf bifurcation are given.

A New 7D Hyperchaotic System with Five Positive Lyapunov Exponents Coined

2018 ◽  
Vol 28 (05) ◽  
pp. 1850057 ◽  
Author(s):  
Qigui Yang ◽  
Daoyu Zhu ◽  
Lingbing Yang
Keyword(s):  
Lyapunov Exponents ◽  
Lorenz System ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Nonlinear Feedback ◽  
Feedback Controllers ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
Hyperbolic Equilibrium ◽  
Good Agreement

This paper reports the finding of a new seven-dimensional (7D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 6D hyperchaotic system that is constructed by adding two linear feedback controllers and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with five positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of hyperchaotic and chaotic attractors. Moreover, stability of the hyperbolic equilibrium is analyzed and a complete mathematical characterization for 7D Hopf bifurcation is given. Finally, circuit experiment implements the hyperchaotic attractor of the 7D system, showing very good agreement with the simulation results.

Observer-Based Feedback Linearizing Control of an Electromagnetic Suspension

1996 ◽  
Vol 118 (3) ◽  
pp. 615-619 ◽  
Author(s):  
B. C. Fabien
Keyword(s):  
Feedback Controller ◽  
Superior Performance ◽  
Linear Feedback ◽  
Nonlinear Controller ◽  
Electromagnetic Suspension ◽  
Weakly Coupled ◽  
Coupled Riccati Equations ◽  
Linear Feedback Controller ◽  
Lyapunov’S Second Method ◽  
Feedback Linearizable Systems

This paper develops a stabilizing observer-based feedback linearizing controller for a single-axis electromagnetic suspension. The controller uses only the measured output of the system, and is shown to be robust with respect to parameter uncertainty. The controller differs from other robust feedback linearizing controllers that have appeared in recent literature, because it is continuous, and non-adaptive. Lyapunov’s second method is used to prove stability and robustness of the controller. The controller has a simple structure and its gains are determined by solving two weakly coupled Riccati equations. Numerical simulations are performed to compare a linear feedback controller and the observer-based feedback linearizing controller. Results obtained demonstrate that the nonlinear controller yields superior performance when compared with the linear feedback controller. The controller synthesis technique developed in this paper is applicable to other fully feedback linearizable systems, not just electromagnetic suspensions.

Three-Stage Feedback Controller Design With Applications to a Three Time-Scale System

2016 ◽  
Author(s):  
Verica Radisavljevic-Gajic ◽  
Milos Milanovic
Keyword(s):  
Time Scale ◽  
Controller Design ◽  
Feedback Controller ◽  
Linear Feedback ◽  
Feedback Controllers ◽  
Linear Feedback Controller ◽  
Full State ◽  
Full State Feedback ◽  
A New Technique ◽  
Natural Decomposition

A new technique was presented that facilitates design of independent full-state feedback controllers at the subsystem levels. Different types of local controllers, for example, eigenvalue assignment, robust, optimal (in some sense L1, H2, H∞, ...) may be used to control different subsystems. This feature has not been available for any known linear feedback controller design. In the second part of the paper, we specialize the results obtained to the three time-scale linear systems (singularly perturbed control systems) that have natural decomposition into slow, fast, and very fast subsystems. The proposed technique eliminates numerical ill-condition of the original three-time scale problems.

A Note on Hidden Transient Chaos in the Lorenz System

2017 ◽  
Vol 18 (5) ◽  
pp. 427-434 ◽  
Author(s):  
Quan Yuan ◽  
Fang-Yan Yang ◽  
Lei Wang
Keyword(s):  
Rayleigh Number ◽  
Lorenz System ◽  
Critical Value ◽  
Transient Chaos ◽  
Topological Horseshoe ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
The One

AbstractIn this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number $\rho $ somewhat smaller than the critical value 24.06 by studying the basins characterization of attraction of attractors and tracing the one-dimensional unstable manifold of the origin, indicating some interesting clues for detecting the existence of hidden transient chaos. In addition, horseshoes chaos is verified in the famous system for some parameters corresponding to the hidden transient chaos by the topological horseshoe theory.

HOMOCLINIC AND HETEROCLINIC ORBITS AND BIFURCATIONS OF A NEW LORENZ-TYPE SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2695-2712 ◽  
Author(s):  
XIANYI LI ◽  
HAIJUN WANG
Keyword(s):  
Chaotic Attractor ◽  
Lorenz System ◽  
Type System ◽  
Pitchfork Bifurcation ◽  
Heteroclinic Orbits ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
The Stability

In this paper, a new Lorenz-type system with chaotic attractor is formulated. The structure of the chaotic attractor in this new system is found to be completely different from that in the Lorenz system or the Chen system or the Lü system, etc., which motivates us to further study in detail its complicated dynamical behaviors, such as the number of its equilibrium, the stability of the hyperbolic and nonhyperbolic equilibrium, the degenerate pitchfork bifurcation, the Hopf bifurcation and the local manifold character, etc., when its parameters vary in their space. The existence or nonexistence of homoclinic and heteroclinic orbits of this system is also rigorously proved. Numerical simulation evidences are also presented to examine the corresponding theoretical analytical results.

Chaotic Control in a Fractional-Order Modified Van Der Pol Oscillator

2011 ◽  
Vol 474-476 ◽  
pp. 83-88
Author(s):  
Xin Gao
Keyword(s):  
Fractional Order ◽  
Feedback Controller ◽  
Van Der Pol Oscillator ◽  
Linear Feedback ◽  
Fractional Order Systems ◽  
Van Der Pol ◽  
Chaotic Control ◽  
Linear Feedback Controller

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator. We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4. Finally, a simple, but effective, linear feedback controller is also designed to stabilize the fractional order chaotic van der Pol oscillator.

scholarly journals Hyperchaos Numerical Simulation and Control in a 4D Hyperchaotic System

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Junhai Ma ◽  
Yujing Yang
Keyword(s):  
Lyapunov Exponents ◽  
Equilibrium Points ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Bifurcation Diagrams ◽  
Theory Analysis ◽  
Feedback Controllers ◽  
Dynamic Simulations ◽  
Wide Range ◽  
And Control

A hyperchaotic system is introduced, and the complex dynamical behaviors of such system are investigated by means of numerical simulations. The bifurcation diagrams, Lyapunov exponents, hyperchaotic attractors, the power spectrums, and time charts are mapped out through the theory analysis and dynamic simulations. The chaotic and hyper-chaotic attractors exist and alter over a wide range of parameters according to the variety of Lyapunov exponents and bifurcation diagrams. Furthermore, linear feedback controllers are designed for stabilizing the hyperchaos to the unstable equilibrium points; thus, we achieve the goal of a second control which is more useful in application.

Optimal Linear Feedback Control for a Class of Nonlinear Nonquadratic Non-Gaussian Problems

1991 ◽  
Vol 113 (4) ◽  
pp. 568-574 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Control System ◽  
Stochastic Systems ◽  
Feedback Controller ◽  
Performance Criteria ◽  
Linear Feedback ◽  
Feedback Gain ◽  
Linear Feedback Controller ◽  
Gaussian Method ◽  
Optimal Linear ◽  
Non Gaussian

An optimal linear feedback controller designed for a class of nonlinear stochastic systems with nonquadratic performance criteria by a non-Gaussian approach is presented. The non-Gaussian method is developed through expressing the unknown stationary output density function as a weighted sum of the Gaussian densities with undetermined parameters. With the aid of a Gaussian-sum density, the optimal feedback gain for a control system with complete state information is derived. By assuming that the separation principle is valid for the class of stochastic systems, a nonlinear precomputed-gain filter is then implemented. The method is illustrated by a Duffing-type control system and the performance of a linear feedback controller designed through both quadratic and nonquadratic performance indices is compared.

Sign in / Sign up

Export Citation Format

Share Document