Perpetual points and hidden attractors in dynamical systems

Dudkowski, Dawid; Prasad, Awadhesh; Kapitaniak, Tomasz

doi:10.1016/j.physleta.2015.06.002

Cited by 82 publications

(57 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, chaotic attractors have been classified as self-excited or hidden attractors [13,14]. An attractor is said to be self-excited provided its basin of attraction overlaps with neighborhood of a critical point.…”

Section: Introductionmentioning

confidence: 99%

A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot

Vaidyanathan¹,

Sambas²,

Mamat³

et al. 2017

Archives of Control Sciences

View full text Add to dashboard Cite

A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot SUNDARAPANDIAN VAIDYANATHAN, ACENG SAMBAS, MUSTAFA MAMAT and MADA SANJAYA WS This research work proposes a new three-dimensional chaotic system with a hidden attractor. The proposed chaotic system consists of only two quadratic nonlinearities and the system possesses no critical points. The phase portraits and basic qualitative properties of the new chaotic system such as Lyapunov exponents and Lyapunov dimension have been described in detail. Finally, we give some engineering applications of the new chaotic system like circuit simulation and control of wireless mobile robot.

show abstract

Section: Introductionmentioning

confidence: 99%

A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot

Vaidyanathan¹,

Sambas²,

Mamat³

et al. 2017

Archives of Control Sciences

View full text Add to dashboard Cite

show abstract

“…It is defined as a point for which all accelerations of the system become zero, while at least one of the velocities remains nonzero. Although the role of such points is still not fully understood, many interesting features of PPs were revealed [1][2][3]. Thus, PPs may become a useful tool for analysis of nonlinear systems and help to better understand some complex dynamical phenomena.…”

Section: Introductionmentioning

confidence: 99%

“…Some sample-based methods were proposed [10,11], but they do not reveal any infor-mation on the topology of the phase space nor indicate the mechanisms of creation of hidden or rare attractors. The lack of unstable fixed points in the neighborhood of hidden attractors led to the hypothesis that PPs could serve as a tool to locate hidden attractors [3,4]. This concept has been widely investigated recently, and despite some limitations [12], PPs prove to be a useful tool to localize hidden and rare attractors in multiple types of systems [3,[13][14][15].…”

Section: Introductionmentioning

confidence: 99%

“…The lack of unstable fixed points in the neighborhood of hidden attractors led to the hypothesis that PPs could serve as a tool to locate hidden attractors [3,4]. This concept has been widely investigated recently, and despite some limitations [12], PPs prove to be a useful tool to localize hidden and rare attractors in multiple types of systems [3,[13][14][15]. Especially in electrical systems there is a need to search for efficient tool to localize non-trivial solutions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Experimental investigation of perpetual points in mechanical systems

Brzeski

Virgin

2017

Nonlinear Dyn

View full text Add to dashboard Cite

In dissipative dynamical systems, equilibrium (stationary) points have a dominant organizing effect on transient motion in phase space, especially in nonlinear systems. These time-independent solutions are readily defined in the context of ordinary differential equations, that is, they occur when all the time derivatives are simultaneously zero. However, there has been some recent interest in perpetual points: points at which the higher time derivatives are zero, but not necessarily the first. Previous work has focused on analytic work (including simulation) and some experimental studies of electric circuits. This paper focuses attention on the occurrence of these points in a simple mechanical system, including experimental verification. Thus, points of zero acceleration can be found in which the corresponding velocity is a maximum or minimum, but not zero. Specifically, the rigid-arm pendulum is used to generate data for which acceleration (and its derivative) can be evaluated. In this paper an experimental (mechanical) setup is described, specifically designed to investigate perpetual points, including a description of the data analysis approaches developed to identify their location.

show abstract

“…In the meantime, the coexistence of hidden attractors has attracted great attention by many researchers, e.g. [15][16][17][18][19]. In [15,16], the rare and hidden attractors in the externally excited van der Pol-Duffing oscillator have been investigated by using the concept of perpetual points in [17].…”

Section: Introductionmentioning

confidence: 99%