scholarly journals A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Chunmei Wang ◽  
Chunhua Hu ◽  
Jingwei Han ◽  
Shijian Cang
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Poincaré Map ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Poincare Map ◽  
Topological Horseshoe ◽  
Simulation Techniques

A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

HYPERCHAOS IN A SPACECRAFT POWER SYSTEM

2011 ◽  
Vol 21 (06) ◽  
pp. 1719-1726 ◽  
Author(s):  
QINGDU LI ◽  
XIAO-SONG YANG ◽  
SHU CHEN
Keyword(s):  
Power System ◽  
Numerical Computation ◽  
Lyapunov Exponents ◽  
Cross Section ◽  
Strange Attractor ◽  
Poincaré Map ◽  
Poincare Map ◽  
Topological Horseshoe

This paper presents some rigorous arguments on a chaotic strange attractor in a spacecraft power system, which is a 3D system with hysteresis switching. By carefully picking a suitable cross-section with respect to the attractor, we find a topological horseshoe of the corresponding third-returned Poincaré map, thus giving a rigorous verification of the existence of chaos in this system. Numerical computation shows that the two Lyapunov exponents of the Poincaré map are both positive and that there exists a two-directional expansion in this horseshoe, suggesting that this attractor is hyperchaotic in nature.

scholarly journals Dynamical analysis and adaptive fuzzy control for the fractional-order financial risk chaotic system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sukono ◽  
Aceng Sambas ◽  
Shaobo He ◽  
Heng Liu ◽  
Sundarapandian Vaidyanathan ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Fuzzy Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Financial Risk ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Adaptive Fuzzy Control ◽  
Dynamical Analysis ◽  
Adaptive Fuzzy

AbstractIn this paper, a fractional-order model of a financial risk dynamical system is proposed and the complex behavior of such a system is presented. The basic dynamical behavior of this financial risk dynamic system, such as chaotic attractor, Lyapunov exponents, and bifurcation analysis, is investigated. We find that numerical results display periodic behavior and chaotic behavior of the system. The results of theoretical models and numerical simulation are helpful for better understanding of other similar nonlinear financial risk dynamic systems. Furthermore, the adaptive fuzzy control for the fractional-order financial risk chaotic system is investigated on the fractional Lyapunov stability criterion. Finally, numerical simulation is given to confirm the effectiveness of the proposed method.

HORSESHOES IN A CHAOTIC SYSTEM WITH ONLY ONE STABLE EQUILIBRIUM

2013 ◽  
Vol 23 (01) ◽  
pp. 1350002 ◽  
Author(s):  
SONGMEI HUAN ◽  
QINGDU LI ◽  
XIAO-SONG YANG
Keyword(s):  
Chaotic System ◽  
Poincaré Map ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Computer Assisted ◽  
Poincare Map ◽  
Topological Horseshoes

To confirm the numerically demonstrated chaotic behavior in a chaotic system with only one stable equilibrium reported by Wang and Chen, we resort to Poincaré map technique and present a rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoes theory.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

2007 ◽  
Vol 17 (03) ◽  
pp. 837-850 ◽  
Author(s):  
SHIGEKI TSUJI ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Coupled Systems ◽  
Poincare Map ◽  
Circuit Implementation ◽  
Coupling Structure ◽  
Bifurcation Phenomena ◽  
Junctional Coupling ◽  
Current Coupling

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

CHAOTIC COEXISTENCE IN A RESOURCE–CONSUMER MODEL

2019 ◽  
Vol 27 (02) ◽  
pp. 167-184
Author(s):  
DENIS G. LADEIRA ◽  
MARCELO M. de OLIVEIRA
Keyword(s):  
Lyapunov Exponents ◽  
Limit Cycle ◽  
Food Supply ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Dynamic Variable ◽  
Interspecies Competition ◽  
Successive Period ◽  
Consumer Model ◽  
Rate Of Consumption

We study the interspecies competition in a simple resource–consumer model which includes the resource supply as a dynamic variable. In the model, an organism of each species needs to consume a certain minimum rate of resource (food) to survive; a higher rate of consumption is required for reproduction. We analyze the orbit diagrams and Lyapunov exponents in order to characterize the system dynamical behavior. We observe that under a periodic food supply, the system can exhibit coexistence in the form of limit cycle oscillations. For a certain parameter range, we observe chaotic behavior emerging from successive period doublings and quasi-periodicity.

scholarly journals S-Box Based Image Encryption Application Using a Chaotic System without Equilibrium

2019 ◽  
Vol 9 (4) ◽  
pp. 781 ◽  
Author(s):  
Xiong Wang ◽  
Ünal Çavuşoğlu ◽  
Sezgin Kacar ◽  
Akif Akgul ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Behavior ◽  
Encryption Algorithm ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Image Encryption Algorithm

Chaotic systems without equilibrium are of interest because they are the systems with hidden attractors. A nonequilibrium system with chaos is introduced in this work. Chaotic behavior of the system is verified by phase portraits, Lyapunov exponents, and entropy. We have implemented a real electronic circuit of the system and reported experimental results. By using this new chaotic system, we have constructed S-boxes which are applied to propose a novel image encryption algorithm. In the designed encryption algorithm, three S-boxes with strong cryptographic properties are used for the sub-byte operation. Particularly, the S-box for the sub-byte process is selected randomly. In addition, performance analyses of S-boxes and security analyses of the encryption processes have been presented.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THE SMOOTH CHUA'S EQUATION

2008 ◽  
Vol 18 (08) ◽  
pp. 2391-2396 ◽  
Author(s):  
QUAN YUAN ◽  
XIAO-SONG YANG
Keyword(s):  
Poincaré Map ◽  
Chaotic Behavior ◽  
Computer Assisted ◽  
Poincaré Section ◽  
Poincare Map ◽  
Poincare Section ◽  
Numerical Studies ◽  
Topological Horseshoes

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

GENERATION OF AN EIGHT-WING CHAOTIC ATTRACTOR FROM QI 3-D FOUR-WING CHAOTIC SYSTEM

2012 ◽  
Vol 22 (12) ◽  
pp. 1250287 ◽  
Author(s):  
GUOYUAN QI ◽  
ZHONGLIN WANG ◽  
YANLING GUO
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Poincaré Map ◽  
Electronic Circuit ◽  
Constant Parameter ◽  
Poincare Map ◽  
Compound Structure ◽  
Topological Structures ◽  
Dynamical Behaviors ◽  
Map Analysis

This paper presents an eight-wing chaotic attractor by replacing a constant parameter with a switch function in Qi four-wing 3-D chaotic system. The eight-wing chaotic attractor has more complicated topological structures and dynamics than the original one. Some basic dynamical behaviors and the compound structure of the proposed 3-D system are investigated. Poincaré-map analysis shows that the system has an extremely rich dynamics. The physical existence of the eight-wing chaotic attractor is verified by an electronic circuit FPGA.

scholarly journals Horseshoe Chaos in a Simple Memristive Circuit

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lei Wang ◽  
XiaoSong Yang ◽  
WenJie Hu ◽  
Quan Yuan
Keyword(s):  
Stability Analysis ◽  
Poincaré Map ◽  
Circuit Model ◽  
Computer Assisted ◽  
Poincaré Section ◽  
Poincare Map ◽  
Poincare Section ◽  
Topological Horseshoe ◽  
Memristive Circuit ◽  
The Stability

A simple memristive circuit model is revisited and the stability analysis is to be given. Furthermore, we resort to Poincaré section and Poincaré map technique and present rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoe theory.

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