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

2019 ◽  
Vol 29 (01) ◽  
pp. 1930001 ◽  
Author(s):  
Marius-F. Danca ◽  
Paul Bourke ◽  
Nikolay Kuznetsov
Keyword(s):  
Computer Graphic ◽  
Initial Conditions ◽  
Three Dimensions ◽  
Hidden Attractors ◽  
Attraction Basin ◽  
Graphic Analysis ◽  
Stable Equilibria ◽  
Attraction Basins ◽  
Graphical Structure ◽  
Advanced Computer

The attraction basin of hidden attractors does not intersect with small neighborhoods of any equilibrium point. 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 equilibria are attracted either by the stable equilibria, or are divergent.

scholarly journals Bifurcations, Hidden Chaos and Control in Fractional Maps

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 879 ◽  
Author(s):  
Adel Ouannas ◽  
Othman Abdullah Almatroud ◽  
Amina Aicha Khennaoui ◽  
Mohammad Mossa Alsawalha ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Discrete Fractional Calculus ◽  
Hidden Attractors ◽  
Nonlinear Dynamical ◽  
Control Schemes ◽  
Stable Equilibria ◽  
And Control

Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems.

Hidden Attractors and Dynamical Behaviors in an Extended Rikitake System

2015 ◽  
Vol 25 (02) ◽  
pp. 1550028 ◽  
Author(s):  
Zhouchao Wei ◽  
Wei Zhang ◽  
Zhen Wang ◽  
Minghui Yao
Keyword(s):  
Hopf Bifurcation ◽  
Initial Conditions ◽  
Dynamics Analysis ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
System Parameters ◽  
Dynamical Behaviors ◽  
Rikitake System ◽  
Wide Range ◽  
Stable Equilibria

In this paper, an extended Rikitake system is studied. Several issues, such as Hopf bifurcation, coexistence of stable equilibria and hidden attractor, and dynamics analysis at infinity are investigated either analytically or numerically. Especially, by a simple linear transformation, the wide range of hidden attractors is noticed, and the Lyapunov exponents diagram is given. The obtained results show that the unstable periodic solution generated by Hopf bifurcation leads to the hidden attractor. The existence of hidden attractors that may render the system's behavior unpredictable not only depends on the value of system parameters but also on the value of initial conditions. The phenomena are important and potentially problematic in engineering applications.

Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium

2018 ◽  
Vol 28 (13) ◽  
pp. 1850167 ◽  
Author(s):  
Sen Zhang ◽  
Yicheng Zeng ◽  
Zhijun Li ◽  
Chengyi Zhou
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
State Feedback Control ◽  
Hyperchaotic System ◽  
Hidden Attractors ◽  
System Parameters ◽  
Offset Boosting ◽  
Extreme Multistability ◽  
Hidden Extreme Multistability ◽  
New System

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.

scholarly journals Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Qiang Lai ◽  
Paul Didier Kamdem Kuate ◽  
Huiqin Pei ◽  
Hilaire Fotsin
Keyword(s):  
Adaptive Control ◽  
Chaotic System ◽  
Physical Meaning ◽  
Control Method ◽  
Initial Conditions ◽  
Periodic Motions ◽  
Hidden Attractors ◽  
Coexisting Attractors ◽  
Dynamic Behaviors

This paper proposes a new no-equilibrium chaotic system that has the ability to yield infinitely many coexisting hidden attractors. Dynamic behaviors of the system with respect to the parameters and initial conditions are numerically studied. It shows that the system has chaotic, quasiperiodic, and periodic motions for different parameters and coexists with a large number of hidden attractors for different initial conditions. The circuit and microcontroller implementations of the system are given for illustrating its physical meaning. Also, the synchronization conditions of the system are established based on the adaptive control method.

Robust and Dynamically Consistent Reduced Order Models

2013 ◽  
Author(s):  
David B. Segala ◽  
David Chelidze
Keyword(s):  
Initial Conditions ◽  
Dimensional Space ◽  
Full Scale ◽  
Three Dimensions ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Random Forcing ◽  
Technological Advances ◽  
Model Complex ◽  
Dynamical Consistency

The need for reduced order models (ROMs) has become considerable higher with the increasing technological advances that allows one to model complex dynamical systems. When using ROMs, the following two questions always arise: 1) “What is the lowest dimensional ROM?” and 2) “How well does the ROM capture the dynamics of the full scale system model?” This paper considers the newly developed concepts the authors refer to as subspace robustness — the ROM is valid over a range of initial conditions, forcing functions, and system parameters — and dynamical consistency — the ROM embeds the nonlinear manifold — which quanitatively answers each question. An eighteen degree-of-freedom pinned-pinned beam which is supported by two nonlinear springs is forced periodically and stochastically for building ROMs. Smooth and proper orthogonal decompositions (SOD and POD, respectively) based ROMs are dynamically consistent in four or greater dimensions. In the strictest sense POD-based ROMs are not considered coherent whereas, SOD-based ROMs are coherent in roughly five dimesions and greater. Is is shown that in the periodically forced case, the full scale dynamics are captured in a five-dimensional POD and SOD-based ROM. For the randomly forced case, POD and SOD-based ROMs need three dimensions but SOD captures the dynamics better in a lower-dimensional space. When the ROM is developed from a different set of initial conditions and forcing values, SOD outperforms POD in periodic forcing case and are equal in the random forcing case.

scholarly journals TIMELIKE GEODESIC MOTION IN HORAVA–LIFSHITZ SPACE–TIME

2010 ◽  
Vol 25 (07) ◽  
pp. 1439-1448 ◽  
Author(s):  
JUHUA CHEN ◽  
YONGJIU WANG
Keyword(s):  
Initial Conditions ◽  
Detailed Balance ◽  
Critical Value ◽  
Space Time ◽  
Three Dimensions ◽  
Geodesic Motion ◽  
Timelike Geodesic ◽  
Particle Orbit ◽  
Renormalizable Theory ◽  
Finite Distance

Recently a nonrelativistic renormalizable theory of gravitation has been proposed by P. Horava. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory is expected to flow to the relativistic value λ = 1, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. In this paper under allowing the lapse function to depend on the spatial coordinates xi as well as t, we obtain the spherically symmetric solutions. And then by analyzing the behavior of the effective potential for the particle, we investigate the timelike geodesic motion of particle in the Horava–Lifshitz space–time. We find that the nonradial particle falls from a finite distance to the center along the timelike geodesics when its energy is in an appropriate range. However, we find that it is complexity for radial particle along the timelike geodesics. There are three different cases due to the energy of radial particle: (i) when the energy of radial particle is higher than a critical value EC, the particle will fall directly from infinity to the singularity; (ii) when the energy of radial particle equals to the critical value EC, the particle orbit at r = rC is unstable, i.e. the particle will escape from r = rC to the infinity or to the singularity, depending on the initial conditions of the particle; (iii) when the energy of radial particle is in a proper range, the particle will rebound to the infinity or plunge to the singularity from a infinite distance, depending on the initial conditions of the particle.

Computer-graphic representation of mandibular movements in three dimensions. Part I. The horizontal plane

1978 ◽  
Vol 39 (4) ◽  
pp. 378-383 ◽  
Author(s):  
William H. Roedema ◽  
John G. Knapp ◽  
Judson Spencer ◽  
Michael K. Dever
Keyword(s):  
Horizontal Plane ◽  
Computer Graphic ◽  
Graphic Representation ◽  
Three Dimensions ◽  
Mandibular Movements

scholarly journals Anatomy of the Attraction Basins: Breaking with the Intuition

2019 ◽  
Vol 27 (3) ◽  
pp. 435-466 ◽  
Author(s):  
Leticia Hernando ◽  
Alexander Mendiburu ◽  
Jose A. Lozano
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Local Search Algorithms ◽  
Neighborhood Structure ◽  
Natural Landscape ◽  
Local Optima ◽  
Attraction Basin ◽  
Combinatorial Optimization Problems ◽  
New Findings ◽  
Attraction Basins

Solving combinatorial optimization problems efficiently requires the development of algorithms that consider the specific properties of the problems. In this sense, local search algorithms are designed over a neighborhood structure that partially accounts for these properties. Considering a neighborhood, the space is usually interpreted as a natural landscape, with valleys and mountains. Under this perception, it is commonly believed that, if maximizing, the solutions located in the slopes of the same mountain belong to the same attraction basin, with the peaks of the mountains being the local optima. Unfortunately, this is a widespread erroneous visualization of a combinatorial landscape. Thus, our aim is to clarify this aspect, providing a detailed analysis of, first, the existence of plateaus where the local optima are involved, and second, the properties that define the topology of the attraction basins, picturing a reliable visualization of the landscapes. Some of the features explored in this article have never been examined before. Hence, new findings about the structure of the attraction basins are shown. The study is focused on instances of permutation-based combinatorial optimization problems considering the 2-exchange and the insert neighborhoods. As a consequence of this work, we break away from the extended belief about the anatomy of attraction basins.

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