scholarly journals A New Series of Three-Dimensional Chaotic Systems with Cross-Product Nonlinearities and Their Switching

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Xinquan Zhao ◽  
Feng Jiang ◽  
Zhigang Zhang ◽  
Junhao Hu
Keyword(s):  
Chaotic Systems ◽  
Three Dimensional ◽  
Cross Product ◽  
Invariant Set ◽  
Switching System ◽  
Positive Invariant Set

This paper introduces a new series of three-dimensional chaotic systems with cross-product nonlinearities. Based on some conditions, we analyze the globally exponentially or globally conditional exponentially attractive set and positive invariant set of these chaotic systems. Moreover, we give some known examples to show our results, and the exponential estimation is explicitly derived. Finally, we construct some three-dimensional chaotic systems with cross-product nonlinearities and study the switching system between them.

GLOBALLY EXPONENTIALLY ATTRACTIVE SETS AND POSITIVE INVARIANT SETS OF THREE-DIMENSIONAL AUTONOMOUS SYSTEMS WITH ONLY CROSS-PRODUCT NONLINEARITIES

2013 ◽  
Vol 23 (01) ◽  
pp. 1350007 ◽  
Author(s):  
XINQUAN ZHAO ◽  
FENG JIANG ◽  
JUNHAO HU
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Cross Product ◽  
Invariant Set ◽  
Invariant Sets ◽  
Special Cases ◽  
Attractive Sets ◽  
Positive Invariant Set

In this paper, the existence of globally exponentially attractive sets and positive invariant sets of three-dimensional autonomous systems with only cross-product nonlinearities are considered. Sufficient conditions, which guarantee the existence of globally exponentially attractive set and positive invariant set of the system, are obtained. The results of this paper comprise some existing relative results as in special cases. The approach presented in this paper can be applied to study other chaotic systems.

NEW ESTIMATIONS FOR GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE FAMILY OF THE LORENZ SYSTEMS

2006 ◽  
Vol 16 (11) ◽  
pp. 3383-3390 ◽  
Author(s):  
PEI YU ◽  
XIAOXIN LIAO
Keyword(s):  
Lyapunov Functions ◽  
Chaotic Systems ◽  
Invariant Set ◽  
The Other ◽  
Qualitative Behavior ◽  
The Family ◽  
Infinite Intervals ◽  
Globally Attractive ◽  
Generalized Lyapunov Functions ◽  
Positive Invariant Set

In this paper, we employ generalized Lyapunov functions to derive new estimations of the ultimate boundary for the trajectories of two types of Lorenz systems, one with parameters in finite intervals and the other in infinite intervals. The new estimations improve the results reported so far in the literature. In particular, for the singular cases: b → 1+ and a → 0+, we have obtained the estimations independent of a. Moreover, our method using elementary algebra greatly simplifies the proofs in the literature. This is an interesting attempt in obtaining information of the attractors which is difficult when merely based on differential equations. It indicates that Lyapunov function is still a powerful tool in the study of qualitative behavior of chaotic systems.

scholarly journals A Mathematical Model Related To Chemostat With Variable Yields

2012 ◽  
Vol 60 (2) ◽  
pp. 147-152
Author(s):  
S. M. Sohel Rana ◽  
Ummi Kulsum
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Qualitative Analysis ◽  
Global Stability ◽  
Three Dimensional ◽  
Invariant Set ◽  
Chemostat Model ◽  
Local And Global Stability ◽  
Positive Invariant Set

In this paper, a three dimensional chemostat model with variable yields is studied. The properties of the steady state points, the local and global stability, the Hopf bifurcation and the positive invariant set for the system are investigated by qualitative analysis of differential equations.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11482 Dhaka Univ. J. Sci. 60(2): 147-152, 2012 (July)

A New Category of Three-Dimensional Chaotic Flows with Identical Eigenvalues

2020 ◽  
Vol 30 (02) ◽  
pp. 2050026 ◽  
Author(s):  
Zahra Faghani ◽  
Fahimeh Nazarimehr ◽  
Sajad Jafari ◽  
Julien C. Sprott
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Special Property ◽  
Computer Search ◽  
Chaotic Flows ◽  
Stable Equilibria ◽  
Dynamical Properties ◽  
Simple Systems ◽  
First Time

In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems with identical eigenvalues were found. We believe that systems with identical eigenvalues are described here for the first time. These simple systems are listed in this paper, and their dynamical properties are investigated.

scholarly journals Ultimate Bound of a 3D Chaotic System and Its Application in Chaos Synchronization

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jiezhi Wang ◽  
Qing Zhang ◽  
Zengqiang Chen ◽  
Hang Li
Keyword(s):  
Chaotic System ◽  
Chaos Synchronization ◽  
Three Dimensional ◽  
Boundary Region ◽  
Cross Product ◽  
State Equation ◽  
Linear Coefficient ◽  
Analytical Expression ◽  
Product Term ◽  
Cross Product Term

Two ellipsoidal ultimate boundary regions of a special three-dimensional (3D) chaotic system are proposed. To this chaotic system, the linear coefficient of theith state variable in theith state equation has the same sign; it also has two one-order terms and one quadratic cross-product term in each equation. A numerical solution and an analytical expression of the ultimate bounds are received. To get the analytical expression of the ultimate boundary region, a new result of one maximum optimization question is proved. The corresponding ultimate boundary regions are demonstrated through numerical simulations. Utilizing the bounds obtained, a linear controller is proposed to achieve the complete chaos synchronization. Numerical simulation exhibits the feasibility of the designed scheme.

scholarly journals Low Model Analysis and Synchronous Simulation of the Wave Mechanics

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wenyuan Duan ◽  
Heyuan Wang ◽  
Meng Kan
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Chaotic System ◽  
Invariant Set ◽  
Positive Definite ◽  
Chaotic Attractors ◽  
Wave Mechanics ◽  
Wave Dynamics ◽  
Simulation Results ◽  
Positive Invariant Set

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

ROTATING SYNCHRONIZATION IN THREE-DIMENSIONAL CHAOTIC SYSTEMS

2012 ◽  
Vol 26 (20) ◽  
pp. 1250120 ◽  
Author(s):  
FUZHONG NIAN ◽  
XINGYUAN WANG
Keyword(s):  
Phase Space ◽  
Chaotic Systems ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Phase Space Reconstruction ◽  
Projective Synchronization ◽  
Reconstruction Method ◽  
Self Similarity ◽  
Application Fields ◽  
Three Dimensional Space

Projective synchronization investigates the synchronization of systems evolve in same orientation, however, in practice, the situation of same orientation is only minority, and the majority is different orientation. This paper investigates the latter, proposes the concept of rotating synchronization, and verifies its necessity and feasibility via theoretical analysis and numerical simulations. Three conclusions were elicited: first, in three-dimensional space, two arbitrary nonlinear chaotic systems who evolve in different orientation can realize synchronization at end; second, projective synchronization is a special case of rotating synchronization, so, the application fields of rotating synchronization is more broadly than that of the former; third, the overall evolving information can be reflected by single state variable's evolving, it has self-similarity, this is the same as the basic idea of phase space reconstruction method, it indicates that we got the same result from different approach, so, our method and the phase space reconstruction method are verified each other.

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