Memristive non-linear system and hidden attractor

Saha, Pramit; Saha, Dibakar; Ray, Apurba; Chowdhury, A. Roy

doi:10.1140/epjst/e2015-02480-1

Cited by 34 publications

(12 citation statements)

References 17 publications

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“…See also The European Physical Journal Special Topics: Multistability: Uncovering Hidden Attractors, 2015 (see [113][114][115][116][117][118][119][120][121][122][123][124]).…”

Section: Self-excited and Hidden Attractorsmentioning

confidence: 99%

Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey

Kuznetsov

2016

Lecture Notes in Electrical Engineering

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Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered. The material is mostly based on surveys [1-4].1 Analytical-numerical study of oscillations An oscillation dynamical system can generally be easily numerically localized if the initial data from its open neighborhood in the phase space (with the exception of a minor set of points) lead to a long-term behavior that approaches the oscillation. Such an oscillation (or a set of oscillations) is called an attractor, and its attracting set is called the basin of attraction (i.e., a set of initial data for which the trajectories tend to the attractor).When the theories of dynamical systems, oscillations, and chaos were first developed researchers mainly focused on analyzing equilibria stability, which can be easily done numerically or analytically, and on the birth of periodic or chaotic attractors from unstable equilibria. The structures of many physical dynamical International Conference on Advanced Engineering -Theory and Applications, 2015 (Ho Chi Minh City, Vietnam), http://icaeta.org/, plenary lecture arXiv:1510.04803v1 [nlin.CD] 16 Oct 2015 chaotic attractor in the Lorenz system with other parameters may be self-excited with respect to zero unstable equilibrium only, and the possible existence in the Lorenz system of a hidden chaotic attractor is an open problem.

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“…See also The European Physical Journal Special Topics: Multistability: Uncovering Hidden Attractors, 2015 (see [113][114][115][116][117][118][119][120][121][122][123][124]).…”

Section: Self-excited and Hidden Attractorsmentioning

confidence: 99%

Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey

Kuznetsov

2016

Lecture Notes in Electrical Engineering

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“…The classification of attractors as being hidden or self-excited was introduced in connection with the discovery of the first hidden Chua attractor [Kuznetsov et al, 2010;Bragin et al, 2011;Leonov et al, 2012;Kiseleva et al, 2017;Stankevich et al, 2017] and has captured much attention of scientists from around the world (see, e.g. [Burkin & Khien, 2014;Li & Sprott, 2014;Pham et al, 2014;Chen et al, 2015;Kuznetsov et al, 2015;Saha et al, 2015;Semenov et al, 2015;Sharma et al, 2015;Zhusubaliyev et al, 2015;Wi et al, 2015;Jafari et al, 2016;Menacer et al, 2016;Ojoniyi & Njah, 2016;Pham et al, 2016;Rocha & Medrano, 2016;Wei et al, 2016;Zelinka, 2016;Borah & Roy, 2017;Brzeski et al, 2017;Feng & Pan, 2017;Jiang , 2016;Kuznetsov et al, 2017;Ma et al, 2017;Messias & Reinol, 2017;Singh & Roy, 2017;Volos et al, 2017;Wei et al, 2017;Zhang et al, 2017]).…”

Section: Introductionmentioning

confidence: 99%

Graphical Structure of Attraction Basins of Hidden Chaotic Attractors: The Rabinovich–Fabrikant System

Danca

Bourke

Kuznetsov

2019

Int. J. Bifurcation Chaos

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For systems with hidden attractors and unstable equilibria, the property that hidden attractors are not connected with unstable equilibria is now accepted as one of their main characteristics. To the best of our knowledge this property has not been explored using realtime interactive three-dimensions graphics. Aided by advanced computer graphic analysis, in this paper, we explore this characteristic of a particular nonlinear system with very rich and unusual dynamics, the Rabinovich-Fabrikant system. It is shown that there exists a neighborhood of one of the unstable equilibria within which the initial conditions do not lead to the considered hidden chaotic attractor, but to one of the stable equilibria or are divergent. The trajectories starting from any neighborhood of the other unstable equlibria are attracted either by the stable equilibria, or are divergent.

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“…As a result, many novel memristive circuits have been constructed by integrating the memristors with versatile nonlinearities into some existing linear or nonlinear circuits [3][4][5][6][7][8][9][10][11]. In these memristive circuits, rich dynamical behaviors have been reported and tested by numerical simulations and hardware experiments, such as chaos and hyperchaos [12,13], hyperchaotic multiwing attractors [14,15], coexisting multiple attractors [16,17], hidden attractors [18,19], and complex transient chaos and hyperchaos [20]. It should be noted that the simplest chaotic circuit has been proposed based on a locally active nonlinear memristive element [4].…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamical Behaviors of a Fractional‐Order System Based on a Locally Active Memristor

Bao

Shi

et al. 2019

Complexity

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A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

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Memristive non-linear system and hidden attractor

Cited by 34 publications

References 17 publications

Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey

Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey

Graphical Structure of Attraction Basins of Hidden Chaotic Attractors: The Rabinovich–Fabrikant System

Complex Dynamical Behaviors of a Fractional‐Order System Based on a Locally Active Memristor

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