No abstract is provided for this article.
The main characteristic of a forced single-degree-of-freedom weakly nonlinear system is determined by its primary, super- and sub-harmonic resonances. A nonlinear parametric feedback control is proposed to modify the steady-state resonance responses, thus to reduce the amplitude of the response and to eliminate the saddle-node bifurcations that take place in the resonance responses. The nonlinear gain of the feedback control is determined by analyzing the bifurcation diagrams associated with the corresponding frequency-response equation, from the singularity theory approach. It is shown by illustrative examples that the proposed nonlinear feedback is effective for controlling three kinds of resonance responses.
Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying the onset of an inherent bifurcation, stabilizing a bifurcated solution or branch, changing the parameter value of an existing bifurcation point, modifying the shape or type of a bifurcation chain,introducing a new bifurcation at a preferable parameter value, monitoring the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, optimizing the system performance near a bifurcation point, or a combination of some of these objectives. This article offers an overview of this emerging, challenging, stimulating, and yet promising field of research,putting the main subject of bifurcation control into perspective.
In this paper, a new algorithm is proposed, which uses only local information to analyze community structures in complex networks. The algorithm is based on a table that describes a network and a virtual cache similar to the cache in the computer structure. When being tested on some typical computer-generated and real-world networks, this algorithm demonstrates excellent detection results and very fast processing performance, much faster than the existing comparable algorithms of the same kind.
Many complex networks possess vertex-degree distributions in a power-law form of <i>ck</i><sup>-<i>γ</i></sup>, where <i>k</i> is the degree variable and <i>c</i> and <i>γ</i> are constants. To better understand the mechanism of power-law formation in realworld networks, it is effective to analyze their degree variable sequences. We had shown before that, for a scale-free network of size N,if its vertex-degree sequence is <i>k</i><sub>1</sub> < <i>k</i><sub>2</sub> <…< <i>k</i><sub>l</sub>, where {<i>k</i><sub>1</sub>, <i>k</i><sub>2</sub>,…, <i>k</i><sub>l</sub>} is the set of all unequal vertex degrees in the network, and if its power exponent satisfies <i>γ</i>>1, then the length <i>l</i> of the vertex-degree sequence is of order <i>logN</i>. In the present paper, we further study complex networks with more general distributions and prove that the same conclusion holds even for non-network type of complex systems. In addition, we support the conclusion by verifying many real-world network and system examples. We finally discuss some potential applications of the new finding in various fields of science, technology and society.
No abstract is provided for this article.
In this paper, we present a method for transmitting ternary logic digital signals based on chaos masking, using a new unified chaotic system. The unified chaotic system is in the form of a general Lorenz system, which can generate the dynamics of Lorenz system, Chen system or Lü system, controlled by a simple analog switch. The ternary logic digital information signals can be encoded and then transmitted as a random-like chaotic signal. At the receiver end, this signal is synchronized with one of the subsystems and the ternary logic signals can be duly decoded. The overall communication system has been realized by electronic circuitry, and the design is explained in detail in this paper. Both simulation and experimental results show that the proposed system works well for electronic transmission of ternary logic digital signals.
In this paper, we study the scaling for the mean first-passage time (MFPT) of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.
This paper shows that a large class of chaotic systems, introduced in (Čelikovský and Vaněček, 1994), (Vaněček and Čelikovský, 1996) as the generalized Lorenz system, can be further generalized to the hyperbolic-type generalized Lorenz system. While the generalized Lorenz system unifies both the famous Lorenz system and new Chen's system (Ueta and Chen, 1999), (Chen and Ueta, 2000), the hyperbolic-type generalized Lorenz system introduced here is in some way complementary to it. Such a complementarity is especially clear when considering the canonical form of the generalized Lorenz system obtained in (Čelikovský and Chen, 2002), where the canonical form is characterized by the eigenvalues of the linearized part together with a key parameter τ ∈ (–1, ∞). The analogous canonical form of the hyperbolic-type generalized Lorenz system introduced here corresponds to the case of, while τ = –1 is a single special case. This new class of chaotic systems is then analyzed, both analytically and numerically, showing its rich variety of dynamical behaviours, including bifurcation and chaos. Moreover, an algorithm for transforming the hyberbolic-type generalized Lorenz system into its canonical form, as well as its inverse scheme, are presented.
This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics.
Intuitionistic fuzzy (IF) theory has become main approach to representing imprecision and vagueness. The IF divergence measure (IFDivM) based on Jensen–Shannon divergence is perhaps the most widely used measure to compare the similarity of multiple intuitionistic fuzzy sets (IFSs). In the present paper, this IFDivM is examined and applied to multiple examples. It is found that some extant IFDivMs hardly satisfy the axiomatic definition, and in a few cases even unable to show divergence of trivial IFSs. To address these inconsistencies, a new IFDivM based on Jensen–Shannon divergence is proposed, free from these problems. The effectiveness of the proposed IFDivM is tested on several critical cases, and precise analysis of its properties is performed. It is proved that the proposed IFDivM satisfies the axiomatic definition of IFDivMs. To illustrate the practical significance of the IFDivM, a novel intuitionistic fuzzy (IF) TODIM method, based on the proposed IFDivM, is developed, termed as GIF-TODIM method. Unlike the existing IF-TODIM methods, GIF-TODIM does not suffer from the revere ordering inconsistencies. The proposed GIF-TODIM method and the proposed IFDivM are applied to a real-world case study on supplier selection. A detailed comparative analysis is performed taking the TOPSIS method and other IFDivMs as baselines. The role of attitude on the final choice is analyzed in great detail. It is found that the proposed GIF-TODIM method is indeed useful, effective, and superior to the counterpart methods, when it comes to real-world situations. Concomitantly, in the present work, it is also revealed that the TOPSIS method based on the 2-D Hamming distance is a special form of the proposed GIF-TODIM method, when decision-makers have the same attitude towards losses and gains. Thus, an interesting relationship between TOPSIS and TODIM is identified under the intuitionistic fuzzy environment, which is bound to propel significant research in the area of decision making under uncertain conditions. As a whole, the article offers comprehensive analyses of IFDivMs and the TODIM method under the intuitionistic fuzzy environment.
No abstract is provided for this article.
In this paper, the methodologies of dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] are applied to find the solutions in the form of [Formula: see text] for the Hirota-type peakon equation posed by [Anco & Mobasheramini, 2017; Anco et al., 2021]. For the function [Formula: see text] therein, under the parameter conditions of [Formula: see text] [Formula: see text] or [Formula: see text], the existence of some possible bounded solutions (solitary wave solutions, periodic wave solutions and periodic peakons, as well as compactons) is proved, with exact explicit parametric representations obtained, except for the peakon solution.
No abstract is provided for this article.
When implemented in the digital domain with time, space and value discretized in the binary form, many good dynamical properties of chaotic systems in continuous domain may be degraded or even diminish. To measure the dynamic complexity of a digital chaotic system, the dynamics can be transformed to the form of a state-mapping network. Then, the parameters of the network are verified by some typical dynamical metrics of the original chaotic system in infinite precision, such as Lyapunov exponent and entropy. This article reviews some representative works on the network-based analysis of digital chaotic dynamics and presents a general framework for such analysis, unveiling some intrinsic relationships between digital chaos and complex networks. As an example for discussion, the dynamics of a state-mapping network of the Logistic map in a fixed-precision computer is analyzed and discussed.
This paper discusses adaptive control of a class of discrete-time chaotic systems from a fuzzy control approach. Using the T–S model of discrete-time chaotic systems, an adaptive control algorithm is developed based on some conventional adaptive control techniques. The resulting adaptively controlled chaotic system is shown to be globally stable, and its robustness is discussed. A simulation example of the chaotic Henon map control is finally presented, to illustrate an application and the performance of the proposed control algorithm.
When computing a trajectory of a dynamical system, influence of noise can lead to large perturbations which can appear, however, with small probability. Then when calculating approximate trajectories, it makes sense to consider errors small on average, since controlling them in each iteration may be impossible. Demand to relate approximate trajectories with genuine orbits leads to various notions of shadowing (on average) which we consider in the paper. As the main tools in our studies we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center which in turn implies the average shadowing property. Finally, we study connections among sensitivity, transitivity, equicontinuity and (average) shadowing.
