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The purpose of this paper is to construct a multiwing chaotic system that has hidden attractors with multiple stable equilibrium points. Because the multiwing hidden attractors chaotic systems are safer and have more dynamic behaviors, it is necessary to construct such a system to meet the needs of developing engineering.

By introducing a multilevel pulse function into a three-dimensional chaotic system with two stable node–foci equilibrium points, a hidden multiwing attractor with multiple stable equilibrium points can be generated. The switching behavior of a hidden four-wing attractor is studied by phase portraits and time series. The dynamical properties of the multiwing attractor are analyzed via the Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. Furthermore, the hardware experiment of the proposed four-wing hidden attractors was carried out.

Not only unstable equilibrium points can produce multiwing attractors but stable node–foci equilibrium points can also produce multiwing attractors. And this system can obtain 2N + 2-wing attractors as the stage pulse of the multilevel pulse function is N. Moreover, the hardware experiment matches the simulation results well.

This paper constructs a new multiwing chaotic system by enlarging the number of stable node–foci equilibrium points. In addition, it is a nonautonomous system that is more suitable for practical projects. And the hardware experiment is also given in this article which has not been seen before. So, this paper promotes the development of hidden multiwing chaotic attractors in nonautonomous systems and makes sense for applications.

This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems.

A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C₀ algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components.

The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system.

The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

This paper provides an introduction to three key concepts derived from chaos theory, with practical examples of each, which the authors believe can offer marketers an enriched understanding of the process by which they set out to construct an alternative future. A central tenet of chaos theory is that the future is inherently unpredictable in detail, but can be predicted in broad terms: what appears to be random (e.g. the unexpected results of a new marketing strategy) may turn out to be determined by a phenomenon known as an attractor. Marketing managers are setting out to “construct a future”. The broad outcomes of their actions may be predictable in general terms, but the path by which their organisations reach those outcomes, and the detailed stages through which they pass, are not – however sound their conventional marketing planning may be. Practising marketing managers may understandably have reservations about transferring concepts which have not yet been fully tested in the natural sciences, where they have their origin, into marketing planning. Nevertheless, many are not in fact unique to chaos theory. Therefore, they are used here as a metaphor, to encourage managers to take a different view of the strategic issues facing their organisations, and to question the conventional wisdom that executive action can be taken on the basis of extensive rational analysis, in the expectation that it will achieve predetermined objectives. This paper thereby offers novel insights into the process of redefining markets and the opportunities that they provide. The attractor metaphor holds out a number of challenges for innovative marketers, as well as some cautionary lessons.

The purpose of this study is to solve the issues of time-consuming and complicated computation of traditional measures, as well as the underutilization of two-dimensional (2D) phase-trajectory projection matrix, so a new set of features were proposed based on the projection of attractors trajectory to characterize the friction-induced attractors and to reveal the tribological behavior during the running-in process.

The frictional running-in experiments were conducted by sliding a ball against a static disk, and the friction coefficient was collected to reconstruct the friction-induced attractors. The projection of the attractors in 2D subspace was then mapped and the distribution of phase points was adapted to conduct the feature extraction.

The evolution of the proposed moment measures could be described as “initial rapid decrease/increase- midterm gradual decrease/increase- finally stable,” which could effectively reveal the convergence degree of the friction-induced attractors. Moreover, the measures could also describe the relative position of the attractors in phase–space domain, which reveal the amplitude evolution of signals to some extent.

The proposed measures could reveal the evolution of tribological behaviors during the running-in process and meet the more precise real-time running-in status identification.

This paper is aimed at investigating a novel chemical oscillating chaotic system with different attractors at fixed parameters. The typical dynamical behavior of the new chemical oscillating system is discussed, and it is found that the state selection is dependent on initial values. Then, the stabilization problem of the chemical oscillating attractors is investigated analytically and numerically. Subsequently, the novel electronic circuit of the proposed chemical oscillating chaotic system are constructed, and the influences of the changes of circuit parameters on chemical oscillating chaotic attractors are investigated.

The different attractors of the novel chemical oscillating chaotic system are investigated by changing the initial values under fixed parameters. Moreover, the active control and adaptive control methods are presented to make the chemical oscillating chaotic systems asymptotically stable at the origin based on the Lyapunov stability theory. The influences on chemical oscillating chaotic attractors are also verified by changing the circuit parameters.

It is found that the active control method is easier to be realized by using physical components because of its less control signal and lower cost. It is also confirmed that the adaptive control method enjoys strong anti-interference ability because of its large number of selected controllers. What can be seen from the simulation results is that the chaotic circuits are extremely dependent on circuit parameters selection. Comparisons between MATLAB simulations and Multisim simulation results show that they are consistent with each other and demonstrate that changing attractors of the chemical oscillating chaotic system exist. It is conformed that circuit parameters selection can be effective to control and realize chaotic circuits.

The different attractors of the novel chemical oscillating chaotic system are investigated by changing the initial values under fixed parameters. The characteristic of the chemical oscillating attractor is that the basin of attraction of the three-dimensional attractor is located in the first quadrant of the eight quadrants of the three-dimensional space, and the ranges of the three variables are positive. This is because the concentrations of the three chemical substances are all positive.

The purpose of this paper is to compare and contrast two formal models of escalation and de-escalation: the attractor landscape model and the S-shaped reaction function model. Also, the paper aims to enumerate conditions that affect the shape and location of reaction functions and, hence, the stability of less and more escalated states.

Both models are presented together with geometric proofs of the main assertions of the second model. Overlap and comparative strengths of the models are reviewed. Parts of the social science literature are synthesized in a discussion of the antecedents of stability.

Though derived from totally different traditions, these models are similar in their basic assumptions and predictions. Each model has value. The attractor landscape model is easier to grasp and contains a concept of resistance to escalation that is not found in the S-shaped reaction function model. The latter model looks at individual parties rather than the dyad as a whole and, thus, offers an explanation for most of the phenomena described by the former model. It also allows identification of many variables that affect the shape and location of reaction functions and, hence, can be viewed as antecedents of escalation and de-escalation.

Seven testable hypotheses are presented in the Conclusions section. Laboratory tasks for testing such hypotheses have yet to be developed and there is only one study employing real-life measures. However, it is clear that once research on these phenomena really begins, new variables will be found that moderate the strength of the effects hypothesized.

The models provide concepts for thinking about how to avoid runaway escalation and promote runaway de-escalation. The variables mentioned in the hypotheses suggest ways to diminish the likelihood of runaway escalation and can also be used for constructing measures of the likelihood of that phenomenon. The theories also imply that when the likelihood of runaway escalation increases, disputants should be doubly careful to avoid initiating escalative behavior.

The article is original in that the S-shaped reaction function model is refined and further developed and the proofs are new. The comparison between the models is also new, as is most of the enumeration of conditions affecting the stability of low and high escalation. The value of the article is to provide concepts and theory for thinking about escalation and de-escalation, and testable hypotheses for studying these phenomena.

The purpose of this paper is to present a simulation and realization of the different types of chaotic attractors using the generalized Chua's circuit equations.

This paper presents n+n‐scroll and 2D n‐scroll chaotic attractors by introducing multiple breakpoints to the nonlinearity. Two piece‐wise linear elements controlled with x‐ and y‐state space variables are used. Chaotic scrolls are originated through both x‐ and y‐axes. The current feedback operational amplifiers are preferred in the experimental circuits because of their wide bandwidth.

It is possible to increase the number of the scrolls in two directions by varying the number of breakpoints in the piece‐wise linear characteristics or changing the location of equilibrium points of the system on 2D‐plane. Theoretically developed behaviors are also experimentally tested.

The excellent adaptation is observed between theoretical and experimental results. This paper also provides useful information about how to build the multiple breakpoints nonlinear elements using the simulation results.

The purpose of this work is to comprehensively reveal the spatial distribution and evolution features of a running-in attractor.

The friction coefficient signals extracted from wear experiments are reconstructed. A projected matrix is obtained based on the reconstructed matrix. Then the approach of three-dimensional (3D) histogram of phase points is proposed, which is used to intuitively characterize the complex properties of the running-in attractor.

The space occupied by the running-in attractor gradually contracts, then stabilizes and finally expands; the maximum of phase points number in a certain bin initially decreases, then keeps stable and finally increases rapidly; yet the percentage of bins number storing phase points shows an inverse variation tendency. Consequently, 3D histogram evolves from a nonuniform state to a uniform state then returns back to the nonuniform state, which indicates the evolution rule of “formation, stabilization and disappearance” of the running-in attractor.

Characterization on the features of the running-in attractor can provide valuable information about friction systems and their dynamic behaviors.

The paper shows that it is possible to reconstruct the dynamical attractors of demand at different levels of the supply chain by using time series duplication and techniques for normalisation. The objective of reconstructing dynamical attractors is to learn more about the long‐term dynamical behaviour of supply chains. Typical patterns that can be encountered through phase space reconstruction are discussed. Based on the analysis of real life supply chains first results are presented on how attractors can be used to better understand the dynamical behaviour of supply chains. The cases show that clear attractors can be identified for consumer and retailer demand. When this demand is compared with supply the phase space analysis becomes an effective tool for identifying distortion in the supply chain. The paper concludes by presenting two examples on how a better understanding of demand attractors have been used to improve operational and tactical planning.