Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

Zhang, Liping; Jiang, Haibo; Liu, Yang; Wei, Zhouchao; Bi, Qinsheng

doi:10.1142/s0218127421500474

Cited by 13 publications

(7 citation statements)

References 62 publications

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“…In this section, the dynamical behaviors of the rational map (1) will be investigated in five cases in accordance with different types of fixed points. In each case, the random bifurcation diagram [30] is drawn firstly to show the possible attractors of the rational map (1). If no multi-stability is observed, the forward (backward) bifurcation diagram will be presented to show the complex behaviors of the rational map (1).…”

Section: Dynamical Behaviors Of the Rational Mapmentioning

confidence: 99%

“…[21]. Great efforts [26][27][28][29][30] have also been made for searching and controlling hidden attractors. For example, Dudkowski et al [26,27] presented a new method to locate hidden and coexisting attractors in non-linear maps based on the concept of perpetual points.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [30], Zhang et al put forward a method to distinguish hidden and self-excited attractors by constructing random bifurcation diagrams. They employed the linear augmentation method to control hidden attractors and multi-stability in a class of two-dimensional maps.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A class of two-dimensional rational maps with self-excited and hidden attractors

Zhang¹,

Liu²,

Wei³

et al. 2022

Chinese Phys. B

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This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stabilities of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan-Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

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Section: Dynamical Behaviors Of the Rational Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A class of two-dimensional rational maps with self-excited and hidden attractors

Zhang¹,

Liu²,

Wei³

et al. 2022

Chinese Phys. B

Self Cite

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show abstract

“…Multi-stability and extreme multi-stability of dynamical systems have been found in many disciplines, including physics, chemistry, biology, and economics. [35,36] Very recently, extreme multi-stability of nonlinear maps has received much attention. [37][38][39][40] In Ref.…”

Section: Introductionmentioning

confidence: 99%

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

et al. 2022

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In this work, we present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are demonstrated and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map has been carried out to reveal the bifurcation mechanism of the dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors. They can exhibit extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, rarely shown in memristive maps. Potentially, this work can be used for secure communication, such as data and image encryption.

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“…It has been also shown that the existing LA control for flows can be extended to discrete dynamical systems as well. In this context, Zhang et al [41] applied this generalized control strategy to various classes of maps, for example, the one exhibiting hidden dynamics, no fixed points and the ones exhibiting one stable fixed point.…”

mentioning

confidence: 99%

Augmented dynamics of nonlinear systems: A review

Punetha,

Khatun,

Jafri

et al. 2024

EPL

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We discuss a simple yet powerful control technique called ‘Linear Augmentation’ (LA) for nonlinear dynamical systems. The linear augmentation can be perceived as a type of interaction that may occur naturally in dynamical systems as an environmental effect, or can be explicitly added to a system in order to control its collective dynamical behavior. LA has been known to effectively regulate resulting dynamics of various dynamical systems and can be used as a powerful control strategy in various applications. Examples include targeting attractor(s), regulating multistable dynamics, suppression of extreme events, and controlling chimera states in the nonlinear dynamical systems. In this review, we discuss augmented dynamics of nonlinear dynamical systems in this perspective with some of its interesting applications.

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Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

Cited by 13 publications

References 62 publications

A class of two-dimensional rational maps with self-excited and hidden attractors

A class of two-dimensional rational maps with self-excited and hidden attractors

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

Augmented dynamics of nonlinear systems: A review

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