Polynomial maps with hidden complex dynamics

Zhang, Xu; Chen, Guanrong

doi:10.3934/dcdsb.2018293

Cited by 9 publications

(10 citation statements)

References 25 publications

(30 reference statements)

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“…Consider the generalized Hénon map [26]: The bifurcation calculation is performed with a as the variable parameter and d as the incremental parameter. In addition, following theorems 4.1 and 4.4 in Ref.…”

Section: Bifurcation Analysis Of Generalized Hénon Mapsmentioning

confidence: 99%

“…In addition, following theorems 4.1 and 4.4 in Ref. [26], the range of the bifurcation calculation is ensured to be a > 0, 0 < d ≤ 1.…”

Section: Bifurcation Analysis Of Generalized Hénon Mapsmentioning

confidence: 99%

“…Jiang et al [25] studied a class of two-dimensional quadratic maps with hidden attractors. Zhang and Chen [26] studied a class of generalized Hénon maps and showed the coexistence of an attracting fixed point and a hidden attractor, and the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.…”

Section: Introductionmentioning

confidence: 99%

“…It is noted that, in Ref. [26], only part of the parameter region was analyzed, leaving many interesting problems for further studies. In this paper, we carry out more detailed analysis of the generalized Hénon map.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation analysis of a class of generalized Hénon maps with hidden dynamics

Amoh

Zhang

Chen

et al. 2021

IEEJ Transactions Elec Engng

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Although continuous systems such as the Chua circuit are known as systems with hidden attractors, hidden attractors also exist in classical discrete maps, such as a generalized Hénon map. A hidden attractor is an attractor that does not overlap with its own attracting region in its vicinity, which makes it difficult to visualize. In this paper, a local bifurcation analysis method for discrete maps is described, and the bifurcation analysis of the generalized Hénon map is performed using the method. The bifurcation structure, as the parameters are changed, shows a certain law, and the interesting Neimark-Sacker bifurcation and period-doubling bifurcation are confirmed to occur simultaneously. It was also found that the hidden attractors exist in the rectangular characteristic chaotic regions, and they appear relatively frequently near the window of chaos.

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Section: Bifurcation Analysis Of Generalized Hénon Mapsmentioning

confidence: 99%

“…In addition, following theorems 4.1 and 4.4 in Ref. [26], the range of the bifurcation calculation is ensured to be a > 0, 0 < d ≤ 1.…”

Section: Bifurcation Analysis Of Generalized Hénon Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bifurcation analysis of a class of generalized Hénon maps with hidden dynamics

Amoh

Zhang

Chen

et al. 2021

IEEJ Transactions Elec Engng

Self Cite

View full text Add to dashboard Cite

show abstract

“…In fact, discrete iterative map is a specific dynamical system with instant states described by continuous-time variables, which can seize the essential behavior of the dynamical flow. Despite the greater simplicity in their mathematical models, discrete iterative maps can also display chaotic behaviors [4]− [6], which are attracting much attention due to their engineering application merits [7]− [9].…”

Section: Introductionmentioning

confidence: 99%

Extreme Multistability in Simple Area-Preserving Map

Bao

Zhu

et al. 2020

IEEE Access

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Initial condition-relied extreme multistability has been recently found in many continuous dynamical systems. However, such a specific phenomenon has not yet been discovered in a discrete iterative map. To investigate this phenomenon, this paper proposes a two-dimensional conservative map only with one sine nonlinearity. The proposed simple discrete map is area-preserving in the phase space and displays the coexistence of infinite chaotic and quasi-periodic orbits caused by infinite fixed points. Multiple numerical results indicate that the area-preserving chaotic and quasi-periodic orbits have the initial condition-relied quasi-periodic route to chaos and initial condition-boosting bifurcation dynamics, which allow the simple area-preserving map to emerge the complex phenomenon of extreme multistability. Furthermore, a microcontroller-based hardware platform is developed to implement the initial conditionboosting chaotic signals.

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Initial-switched boosting bifurcations in 2D hyperchaotic map

Bao

Zhu

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Recently, the coexistence of initial-boosting attractors in continuous-time systems has been attracting more attention. How do you implement the coexistence of initial-boosting attractors in a discrete-time map? To address this issue, this paper proposes a novel two-dimensional (2D) hyperchaotic map with a simple algebraic structure. The 2D hyperchaotic map has two special cases of line and no fixed points. The parameter-dependent and initial-boosting bifurcations for these two cases of line and no fixed points are investigated by employing several numerical methods. The simulated results indicate that complex dynamical behaviors including hyperchaos, chaos, and period are closely related to the control parameter and initial conditions. Particularly, the boosting bifurcations of the 2D hyperchaotic map are switched by one of its initial conditions. The distinct property allows the dynamic amplitudes of hyperchaotic/chaotic sequences to be controlled by switching the initial condition, which is especially suitable for chaos-based engineering applications. Besides, a microcontroller-based hardware platform is developed to confirm the generation of initial-switched boosting hyperchaos/chaos.

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Polynomial maps with hidden complex dynamics

Cited by 9 publications

References 25 publications

Bifurcation analysis of a class of generalized Hénon maps with hidden dynamics

Bifurcation analysis of a class of generalized Hénon maps with hidden dynamics

Extreme Multistability in Simple Area-Preserving Map

Initial-switched boosting bifurcations in 2D hyperchaotic map

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