On hidden twin attractors and bifurcation in the Chua’s circuit

Li, Qingdu; Zeng, Hongzheng; Yang, Xiao‐Song

doi:10.1007/s11071-014-1290-8

Cited by 90 publications

(52 citation statements)

References 27 publications

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“…This type of attractor is completely different from the self-excited 2-scroll chaotic attractor reported in [1] and also different from the hidden chaotic spiral attractor reported in [20,26].…”

Section: -Scroll Chua's Attractor With Two Stable Node-focicontrasting

confidence: 85%

“…However, it is found that a hidden Chua's spiral attractor with one stable zero node-focus exists when the parameters = 8.8, = 12.0732, = 0.0052, = −0.1768, and = −1.1468, and the initial conditions are (9.4287, −0.5945, and −13.4705) [20,21], as shown in Figure 2 Chua's spiral attractors with one stable zero node-focus are observed [26,27], as shown in Figure 2(b). Note that Chua's nonlinearity is modified for the generations of these hidden Chua's attractors, and there exist the relations of < 0, < 0, | | < 1, and | | > 1 and Chua's nonlinearity is different from that in Figure 1(a).…”

Section: Remarks For Chua's Nonlinearities and The Generating Chaoticmentioning

confidence: 92%

“…Different from self-excited attractor, hidden attractor, whose attraction basin does not intersect with small neighborhoods of the equilibria of the system [19][20][21], is sensitive to the initial conditions and special analytical-numerical procedure should be adopted to locate its attractive basin [20]. Recently, by developing this procedure, hidden chaotic spiral attractor has been numerically observed in the classic Chua's system with one stable zero equilibrium point [20,26]. Furthermore, two coexisting hidden attractors have been numerically revealed and a dynamical route from one hidden limit cycle to hidden twin chaotic attractors by period-doubling bifurcations has been reported in the classic Chua's system [26,27].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, by developing this procedure, hidden chaotic spiral attractor has been numerically observed in the classic Chua's system with one stable zero equilibrium point [20,26]. Furthermore, two coexisting hidden attractors have been numerically revealed and a dynamical route from one hidden limit cycle to hidden twin chaotic attractors by period-doubling bifurcations has been reported in the classic Chua's system [26,27].…”

Section: Introductionmentioning

confidence: 99%

“…The mechanism to have different types of attractors in the classic Chua's system is due to different Chua's nonlinearities [1,[26][27][28]. Different Chua's nonlinearities formulate the three equilibrium points with different characteristics, leading to the generations of different types of Chua's attractors.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

Chu

Zhu

Qian

et al. 2018

Mathematical Problems in Engineering

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With new three-segment piecewise-linearity in the classic Chua's system, two new types of 2-scroll and 3-scroll Chua's attractors are found in this paper. By changing the outer segment slope of the three-segment piecewise-linearity as positive, the new 2-scroll Chua's attractor has emerged from one zero index-1 saddle-focus and two symmetric stable nonzero node-foci. In particular, by newly introducing a piecewise-linear control function, an improved Chua's system only with one zero index-2 saddle-focus and two stable nonzero node-foci is constructed, from which a 3-scroll Chua's attractor is converged. Some remarks for Chua's nonlinearities and the generating chaotic attractors are discussed, and the stabilities at the three equilibrium points are then analyzed, upon which the emerging mechanisms of the novel 2-scroll and 3-scroll Chua's attractors are explored in depth. Furthermore, an analog electronic circuit built with operational amplifier and analog multiplier is designed and hardware circuit experiments are measured to verify the numerical simulations. These novel 2-scroll and 3-scroll Chua's attractors reported in this paper are completely different from the classic Chua's attractors, which will enrich the dynamics of the classic Chua's system.

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Section: -Scroll Chua's Attractor With Two Stable Node-focicontrasting

confidence: 85%

Section: Remarks For Chua's Nonlinearities and The Generating Chaoticmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

Chu

Zhu

Qian

et al. 2018

Mathematical Problems in Engineering

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Creation and circuit implementation of two‐to‐eight–wing chaotic attractors using a 3D memristor‐based system

Zhong

Peng

Shahidehpour

2019

Circuit Theory & Apps

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This paper constructs a three-dimensional (3D) memristor-based system for creating multiwing chaotic attractors. A second-degree polynomial memristance function and a sixth-order exponent internal state memristor function with one parameter are employed, and the complexity of attractors is increased. A detailed analysis on dynamical behaviors of the proposed system are described, such as the bifurcation diagrams, finite-time local Lyapunov exponents, time series, phase portraits, and Poincaré maps. By adjusting the design parameters, the system displays two-to-eight-wing chaotic attractors, especially the five-wing and seven-wing attractors, which have never been found in the known systems. Further, we provide the calculation formula of the number of wings in the system, discuss the distribution of the involving inner holes on the plane, and design an electronic circuit to realize the proposed system. The experimental results of the circuit implementation agree with the numerical simulations on Matlab well. It indicates the potential engineering applications for various chaos-based information systems. KEYWORDSchaotic system, circuit implementation, Lyapunov exponents, memristor, the number of the wings INTRODUCTIONAfter the first solid state realization of a memristor was successfully fabricated in 2008 by researchers at HP, 1 memristor, the new two-terminal circuit element postulated by Chua 2 in 1971 and classified by Chua and Kang 3 in 1976, has been well known and studied thanks to its potential applications in neural networks, novel storage medium, artificial intelligence, and a wide range of other chaotic circuit. 4-12 Since the mathematical model involving chaos was first suggested by Lorenz 13 in 1963 and the term chaos was first used by Li and Yorke 14 in 1975, the design of new chaotic systems with complex topological structures are attracting more and more interest in the research. [15][16][17][18][19][20][21][22][23][24][25][26] In 1993, Stone and Miranda observed first multiwing chaotic attractor from a proto-Lorenz system. 27 Since then, people has found that a wide variety of memristor-based systems can exhibit the complicated dynamics, particularly in birth of multiwing attractors. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] This problem is interesting, especially in the case of a simple, low-dimensional system with less equilibrium points, more wings, and more coexisting attractors. The multiwing or multiscroll 45-53 chaotic attractors with more complex is of the meaningful applications in many fields, particularly, including chaotic broad band radio, encryption, and secure communication. 54 In the reported multiwing systems, most memristor models are based on 686

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Some new insights into a known Chen‐like system

Wang

2015

Math Methods in App Sciences

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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.

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On hidden twin attractors and bifurcation in the Chua’s circuit

Cited by 90 publications

References 27 publications

Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

Creation and circuit implementation of two‐to‐eight–wing chaotic attractors using a 3D memristor‐based system

Some new insights into a known Chen‐like system

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