Ultimate bound estimation set and chaos synchronization for a financial risk system

2018 ◽  
Vol 154 ◽  
pp. 19-33 ◽  
Author(s):  
Wei Gao ◽  
Li Yan ◽  
Mohammadhossein Saeedi ◽  
Hassan Saberi Nik
Keyword(s):  
Chaos Synchronization ◽  
Financial Risk ◽  
Ultimate Bound ◽  
Bound Estimation

scholarly journals Qualitative Study of a 4D Chaos Financial System

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Fuchen Zhang ◽  
Gaoxiang Yang ◽  
Yong Zhang ◽  
Xiaofeng Liao ◽  
Guangyun Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Dynamical Behavior ◽  
Lyapunov Stability Theory ◽  
Attraction Domain ◽  
Ultimate Boundedness ◽  
Lyapunov Dimension ◽  
Simulation Results ◽  
Ultimate Bound

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.

scholarly journals Analysis of a Lorenz-Like Chaotic System by Lyapunov Functions

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Fuchen Zhang
Keyword(s):  
Chaotic System ◽  
Chaos Synchronization ◽  
Lorenz System ◽  
Chen System ◽  
Lyapunov Function Method ◽  
Exponential Attractivity ◽  
Positively Invariant Set ◽  
Ultimate Bound ◽  
Special Case ◽  
Theoretical Results

In this paper, we investigate the ultimate bound set and positively invariant set of a 3D Lorenz-like chaotic system, which is different from the well-known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family. Furthermore, we investigate the global exponential attractive set of this system via the Lyapunov function method. The rate of the trajectories going from the exterior of the globally exponential attractive set to the interior of the globally exponential attractive set is also obtained for all the positive parameters values a,b,c. The innovation of this paper is that our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters (λ and m); that is, for the fixed parameters of the system, we have a series of sets depending on λ and m. The results contain the known result as a special case for the fixed λ and m. The efficiency of the scheme is shown numerically. The theoretical results may find wide applications in chaos control and chaos synchronization.

Dynamics of a New 5D Hyperchaotic System of Lorenz Type

2018 ◽  
Vol 28 (03) ◽  
pp. 1850036 ◽  
Author(s):  
Fuchen Zhang ◽  
Rui Chen ◽  
Xingyuan Wang ◽  
Xiusu Chen ◽  
Chunlai Mu ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Optimization Theory ◽  
Lyapunov Stability Theory ◽  
Hyperchaotic System ◽  
Ultimate Boundedness ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Hyperchaotic Systems ◽  
Ultimate Bound

Ultimate boundedness of chaotic dynamical systems is one of the fundamental concepts in dynamical systems, which plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors and the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, chaos synchronization. However, it is often difficult to obtain the bounds of the hyperchaotic systems due to the complex algebraic structure of the hyperchaotic systems. This paper has investigated the boundedness of solutions of a nonlinear hyperchaotic system. We have obtained the global exponential attractive set and the ultimate bound set for this system. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lyapunov stability theory and optimization theory. To show the ultimate bound region, numerical simulations are provided.

ULTIMATE BOUND ESTIMATION OF A CLASS OF HIGH DIMENSIONAL QUADRATIC AUTONOMOUS DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (09) ◽  
pp. 2679-2694 ◽  
Author(s):  
PEI WANG ◽  
DAMEI LI ◽  
XIAOQUN WU ◽  
JINHU LÜ ◽  
XINGHUO YU
Keyword(s):  
Dynamical Systems ◽  
Chaotic System ◽  
Lorenz System ◽  
High Dimensional ◽  
Hyperchaotic System ◽  
Sufficient Condition ◽  
System A ◽  
Ultimate Bound ◽  
Autonomous Dynamical Systems ◽  
Bound Estimation

This paper aims to propose a unified approach for the ultimate bound estimation of a class of High Dimensional Quadratic Autonomous Dynamical Systems (HDQADS). Using the proposed method and the optimization idea, a sufficient condition is then given for estimating the ultimate bounds of a class of HDQADS. To validate the above sufficient condition, this paper further investigates the ultimate bound estimation of a hyperchaotic system, a 6D and a 9D chaotic system, separately. Moreover, the ultimate bounds for a general Lorenz system, a low-order atmospheric circulation model, and a new 3D chaotic system are also discussed in detail. In particular, it should be pointed out that a unified and accurate ultimate bound estimation is attained for the generalized Lorenz system and it includes several well-known results as its special cases. Some numerical simulations are also given to verify and visualize the corresponding theoretical results.

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