A novel chaotic system and its topological horseshoe

Li, Chunlai; Wu, Lei; Li, Hongmin; Tong, Yaonan

doi:10.15388/na.18.1.14032

Cited by 37 publications

(40 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“… In Section 3.2, pp. 70–71 , when a = 40, b = 5, c = 30, and e gradually changes from 0 to 10, numerical simulations show that the Lyapunov exponent spectrums keep invariable and that the system is always chaotic. In accordance with the statements in p.72 and Theorem , in this paper, the dynamics of E 0 and E ± is independent of the parameter e .…”

Section: Some Explanations For the Results Inmentioning

confidence: 99%

“…Especially, do there exist fold bifurcation, Hopf bifurcation, zero‐Hopf bifurcation, and so on in the system ? Such problems are not answered in either. How about the dynamical behavior of the system at infinity? This has not been discussed yet at all.…”

Section: Introductionmentioning

confidence: 97%

“…Recently, Li et al proposed the following chaotic system in

({, \begin{matrix} \dot{x} & = a (y - x) + yz, \\ \dot{y} & = (c - a) x - xz + cy, \\ ż & = e y^{2} - bz, \end{matrix})

where a , b , c , and e are positive real parameters (For the convenience of writing, the state variables x 1 , x 2 , and x 3 in are written as x , y , and z in the present paper). Combining theoretical analysis and numerical simulation, they mainly analyzed the stability of equilibrium point of system as well as topological horseshoe; for readers' convenience, the main results obtained in are abstracted as follows. Result In order to obtain the equilibria of system , let us suppose

x_{*} = \sqrt{\frac{b [(c - 2 a) + {(c^{2} + 4 ac)}^{0.5}]}{2 e}}, y_{*} = \sqrt{\frac{b [(c - 2 a) - {(c^{2} + 4 ac)}^{0.5}]}{2 e}} .

Then, if 2 c − a ≥0, we obtain three equilibria of system :

P_{0} = (0, 0, 0), P_{1} (x_{*} + \frac{e x_{*}^{3}}{ab}, x_{*}, \frac{e x_{*}^{2}}{b}), P_{2} (- x_{*} - \frac{e x_{*}^{3}}{ab}, - x_{*}, e)

…”

Section: Introductionmentioning

confidence: 99%

“…Although there is some good work for the system in , some problems still obviously exist in that paper as follows. Result is not precise or even wrong.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some new insights into a known Chen‐like system

Wang

2015

Math Methods in App Sciences

View full text Add to dashboard Cite

In the paper entitled ‘A novel chaotic system and its topological horseshoe’ in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66–77], proposed the 3D chaotic system, trueẋ=a(y−x)+yz,trueẏ=(c−a)x−xz+cy,ż=ey2−bz, and discussed some of its dynamics according to theoretical and numerical analysis of its parameters (a,b,c,e)∈double-struckR+4. The present work is devoted to giving some new insights into the system for b≥0. Combining theoretical analysis and numerical simulations, some new results are formulated. On the one hand, after some known errors, mainly the distribution of its equilibrium point which is pointed out, correct results are formulated. On the other hand, some of its more rich dynamical properties hiding and not found previously, such as the stability, fold bifurcation, pitchfork bifurcation, degenerated pitchfork bifurcation, and Hopf bifurcation of its isolated equilibria, the dynamics of non‐isolated equilibria, the singularly degenerate heteroclinic cycle, the heteroclinic orbit, and the dynamics at infinity are clearly revealed. Using these results, one can easily explain those interesting phenomena for invariant Lyapunov exponent spectrum and amplitude control that are presented in the known literature. What is more important, we probably demonstrate a new route to chaos. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

Section: Some Explanations For the Results Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

“…Recently, Li et al proposed the following chaotic system in

({, \begin{matrix} \dot{x} & = a (y - x) + yz, \\ \dot{y} & = (c - a) x - xz + cy, \\ ż & = e y^{2} - bz, \end{matrix})

x_{*} = \sqrt{\frac{b [(c - 2 a) + {(c^{2} + 4 ac)}^{0.5}]}{2 e}}, y_{*} = \sqrt{\frac{b [(c - 2 a) - {(c^{2} + 4 ac)}^{0.5}]}{2 e}} .

Then, if 2 c − a ≥0, we obtain three equilibria of system :

P_{0} = (0, 0, 0), P_{1} (x_{*} + \frac{e x_{*}^{3}}{ab}, x_{*}, \frac{e x_{*}^{2}}{b}), P_{2} (- x_{*} - \frac{e x_{*}^{3}}{ab}, - x_{*}, e)

…”

Section: Introductionmentioning

confidence: 99%

“…Although there is some good work for the system in , some problems still obviously exist in that paper as follows. Result is not precise or even wrong.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Some new insights into a known Chen‐like system

Wang

2015

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…5 Afterwards, Lü et al described a new chaotic system which has represented the transition between Lorenz and Chen systems. 6 Recently, several new chaotic attractors have been discovered, [7][8][9] and many more will be revealed due to their potential applications especially in secure communication. [10][11][12] Chua's attractor is the most famous chaos circuit; the others are Colpitts and Wien-bridge 13 attractors.…”

Section: Introductionmentioning

confidence: 99%

Synchronization of Vilnius Chaotic Oscillators With Active and Passive Control

Kocamaz

Uyaroğlu

2014

J CIRCUIT SYST COMP

View full text Add to dashboard Cite

In this study, active and passive control techniques are applied for the synchronization of two identical Vilnius chaotic oscillators. The di®erential equations of Vilnius oscillator are described according to its circuit model. Based on Lyapunov function, the active and passive controllers are used to realize the synchronization of Vilnius chaotic systems. Numerical simulations are presented to verify and compare the e®ectiveness of proposed control techniques.

show abstract

Global chaos synchronization of new chaotic system using linear active control

Ahmad

Saaban

Ibrahim³

et al. 2014

Complexity

View full text Add to dashboard Cite

Chaos synchronization is a procedure where one chaotic oscillator is forced to adjust the properties of another chaotic oscillator for all future states. This research paper studies and investigates the global chaos synchronization problem of two identical chaotic systems and two non‐identical chaotic systems using the linear active control technique. Based on the Lyapunov stability theory and using the linear active control technique, the stabilizing controllers are designed for asymptotically global stability of the closed‐loop system for both identical and non‐identical synchronization. Numerical simulations and graphs are imparted to justify the efficiency and effectiveness of the proposed scheme. All simulations have been done by using mathematica 9. © 2014 Wiley Periodicals, Inc. Complexity 21: 379–386, 2015

show abstract

A novel chaotic system and its topological horseshoe

Cited by 37 publications

References 17 publications

Some new insights into a known Chen‐like system

Some new insights into a known Chen‐like system

Synchronization of Vilnius Chaotic Oscillators With Active and Passive Control

Global chaos synchronization of new chaotic system using linear active control

Contact Info

Product

Resources

About