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Hierarchical networks are frequently encountered in animal groups, gene networks, and artificial engineering systems such as multiple robots, unmanned vehicle systems, smart grids, wind farm networks, and so forth. The structure of a large directed hierarchical network is often strongly influenced by reverse edges from lower- to higher-level nodes, such as lagging birds’ howl in a flock or the opinions of lower-level individuals feeding back to higher-level ones in a social group. This study reveals that, for most large-scale real hierarchical networks, the majority of the reverse edges do not affect the synchronization process of the entire network; the synchronization process is influenced only by a small part of these reverse edges along specific paths. More surprisingly, a single effective reverse edge can slow down the synchronization of a huge hierarchical network by over 60%. The effect of such edges depends not on the network size but only on the average in-degree of the involved subnetwork. The overwhelming majority of active reverse edges turn out to have some kind of “bunching” effect on the information flows of hierarchical networks, which slows down synchronization processes. This finding refines the current understanding of the role of reverse edges in many natural, social, and engineering hierarchical networks, which might be beneficial for precisely tuning the synchronization rhythms of these networks. Our study also proposes an effective way to attack a hierarchical network by adding a malicious reverse edge to it and provides some guidance for protecting a network by screening out the specific small proportion of vulnerable nodes.

Abstract Recent studies suggest that circular RNA (circRNA)‐mediated post‐translational modification of RNA‐binding proteins (RBP) plays a pivotal role in metastasis of hepatocellular carcinoma (HCC). However, the specific mechanism and potential clinical therapeutic significance remain vague. This study attempts to profile the regulatory networks of circRNA and RBP using a multi‐omics approach. Has_circ_0006646 (circ0006646) is an unreported circRNA in HCC and is associated with a poor prognosis. Silencing of circ0006646 significantly hinders metastasis in vivo. Mechanistically, circ0006646 prevents the interaction between nucleolin (NCL) and the E3 ligase tripartite motif‐containing 21 to reduce the proteasome‐mediated degradation of NCL via K48‐linked polyubiquitylation. Furthermore, the change of NCL expression is proven to affect the phosphorylation levels of multiple proteins and inhibit p53 translation. Moreover, patient‐derived tumor xenograft and lentivirus injection, which is conducted to simulate clinical treatment confirmed the potential therapeutic value. Overall, this study describes the integrated multi‐omics landscape of circRNA‐mediated NCL ubiquitination degradation in HCC metastasis and provides a novel therapeutic target.

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems.

Abstract In this paper, we propose a new type of nonlinear strict distance and similarity measures for intuitionistic fuzzy sets (IFSs). Our proposed methods not only have good properties, but also improve the drawbacks proposed by Mahanta and Panda (Int J Intell Syst 36(2):615–627, 2021) in which, for example, their distance value of $$d_{_{\textrm{MP}}}(\langle \mu , \nu \rangle , \langle 0, 0\rangle )$$ d MP ( ⟨ μ , ν ⟩ , ⟨ 0 , 0 ⟩ ) is always equal to the maximum value 1 for any intuitionistic fuzzy number $$\langle \mu , \nu \rangle \ne \langle 0, 0\rangle $$ ⟨ μ , ν ⟩ ≠ ⟨ 0 , 0 ⟩ . To resolve these problems in Mahanta and Panda (Int J Intell Syst 36(2):615–627, 2021), we establish a nonlinear parametric distance measure for IFSs and prove that it satisfies the axiomatic definition of strict intuitionistic fuzzy distances and preserves all advantages of distance measures. In particular, our proposed distance measure can effectively distinguish different IFSs with high hesitancy. Meanwhile, we obtain that the dual similarity measure and the induced entropy of our proposed distance measure satisfy the axiomatic definitions of strict intuitionistic fuzzy similarity measure and intuitionistic fuzzy entropy. Finally, we apply our proposed distance and similarity measures to pattern classification, decision making on the choice of a proper antivirus face mask for COVID-19, and medical diagnosis problems, to illustrate the effectiveness of the new methods.

Abstract Liver transplantation (LT) stands as the gold standard for treating end‐stage liver disease and hepatocellular carcinoma, yet postoperative complications continue to impact survival rates. The liver's unique immune system, governed by a microenvironment of diverse immune cells, is disrupted during processes like ischemia–reperfusion injury posttransplantation, leading to immune imbalance, inflammation, and subsequent complications. In the posttransplantation period, immune cells within the liver collaboratively foster a tolerant environment, crucial for immune tolerance and liver regeneration. While clinical trials exploring cell therapy for LT complications exist, a comprehensive summary is lacking. This review provides an insight into the intricacies of the liver's immune microenvironment, with a specific focus on macrophages and T cells as primary immune players. Delving into the immunological dynamics at different stages of LT, we explore the disruptions after LT and subsequent immune responses. Focusing on immune cell targeting for treating liver transplant complications, we provide a comprehensive summary of ongoing clinical trials in this domain, especially cell therapies. Furthermore, we offer innovative treatment strategies that leverage the opportunities and prospects identified in the therapeutic landscape. This review seeks to advance our understanding of LT immunology and steer the development of precise therapies for postoperative complications.

It is important for a networked-system to have both good controllability and robustness, where the former measures the ability that the networked system can be properly steered via control to any target state form any initial state within a finite time duration, while the later reflects how well this system can regain its controllability after being destructively attacked. Empirical observations suggest that multi-loop structures are beneficial for enhancing the controllability robustness. The Henneberg-growth mechanism in social networks offers a natural manner to generate multi-clique structures that can be developed to multi-loops by assigning proper edge directions. In this paper, a series of random polygon networks are generated, forming random triangle, rectangle, pentagon and hexagon networks. Then, their controllability robustness is investigated. A realistic measure is designed for characterizing their controllability robustness, which can be used to filter out trivial performances after the network is severely destructed, so that the computation and analysis become much more efficient. Extensive simulation results suggest that, for random polygon networks, 1) non-loop polygon structures are inferior to the loop polygons, confirming that the multi-loop structures are indeed beneficial for controllability robustness; 2) polygons with more sides possess better controllability robustness; and 3) the correlation between controllability robustness and connectivity robustness is weak, implying that the two objectives cannot be enhanced in the same way.

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the dataset, that is, establishing the integrated taxonomic relation among points, lines, and simplices. Here, the simplicial network composed of all-order simplices in a simplicial complex is essential. Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values, a method based on the concept of critical simplices can be developed by searching all-order spanning trees. Employing this new method, not only the Morse function values with the theoretical minimum number of critical simplices can be obtained, but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly. Finally, this method is used to analyze some examples and compared with other methods, showing its effectiveness and feasibility.

In this article, dynamical robustness of a directed complex network with additive noise is inverstigated. The failure of a node in the network is modeled by injecting noise into the node. Under the framework of mean-square stochastic stability, a new robustness metric is formulated to characterize the robustness of the network in terms of synchronization to the additive noise. It is found that the node dynamics plays a pivotal role in dynamical robustness of the directed network. Numerical simulations are shown for illustration and verification.

The study of high-order networks as attracted significant attention recently. The authors introduce the concept of computability for searching maximum clique in large networks and an optimized algorithm for finding cavities with different orders.

The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.

A high-quality data transmission scheme in wireless body area networks is implemented by a joint source-channel coding (JSCC) system with $M$ -ary differential chaos shift keying modulation over an on-body channel. Current DP-LDPC code pairs, which have shown good bit-error ratio performance over additive white Gaussian noise channel and Rayleigh fading channel, cannot perform well over on-body channel. In this paper, the DP-LDPC codes are redesigned for the JSCC system with new methods depending on the on-body channel characteristics. A protomatrix dimension-restrictive searching algorithm for DP-LDPC codes in the JSCC system is proposed according to the on-body channel bandwith and source statistic, which is effective to predesign the dimension of the code pair for a new JSCC system. In this way, the source-channel code rates are appropriately allocated to leave more source redundancy for improving the error floor. Moreover, the joint protomatrix is reconstructed into two kinds of compact structures by simplifying the edge linking relationships and modifying joint protograph extrinsic information transfer algorithm, which decreases the induction of searching entries to further accelerate the code searching speed. The searched code pairs show better bit-error ratio performances compared with existing DP-LDPC codes. The power consumption in the new joint coding system is lower than the separate coding system in WBAN on the physical layer.

The joint shuffled scheduling decoding (JSSD) algorithm can reduce the decoding complexity of the joint source-channel coding system (JSCC) based on double protograph low-density parity-check (P-LDPC) codes. However, the JSSD algorithm will not work when the linking matrix between check nodes (CNs) of the source P-LDPC and variable nodes (VNs) of the channel P-LDPC is adopted in such a system, and this linking matrix has a significant influence on the system performance. In this paper, a generalized joint shuffled scheduling decoding (GJSSD) algorithm is designed to work for the system, and the JSSD algorithm can be regarded as a special case of this algorithm. Simulations show that the proposed GJSSD algorithm can reduce the decoding complexity with performance improvement when compared with the joint belief-propagation (JBP) algorithm.

The 2019 novel coronavirus disease (COVID-19) outbreak has become a worldwide problem. Due to globalization and the proliferation of international travel, many countries are now facing local epidemics. The existence of asymptomatic and pre-symptomatic transmissions makes it more difficult to control disease transmission by isolating infectious individuals. To accurately describe and represent the spread of COVID-19, we suggest a susceptible-exposed-infected-hospitalized-removed (SEIHR) model with human migrations, where the “exposed” (asymptomatic) individuals are contagious. From this model, we derive the basic reproduction number of the disease and its relationship with the model parameters. We find that, for highly contagious diseases like COVID-19, when the adjacent region's epidemic is not severe, a large migration rate can reduce the speed of local epidemic spreading at the price of infecting the neighboring regions. In addition, since “infected” (symptomatic) patients are isolated almost immediately, the transmission rate of the epidemic is more sensitive to that of the “exposed” (asymptomatic) individuals. Furthermore, we investigate the impact of various interventions, e.g. isolation and border control, on the speed of disease propagation and the resultant demand on medical facilities, and find that a strict intervention measure can be more effective than closing the borders. Finally, we use some real historical data of COVID-19 caseloads from different regions, including Hong Kong, to validate the modified SEIHR model, and make an accurate prediction for the third wave of the outbreak in Hong Kong.

The crowdsourcing system is an internet-based distributed problem-solving and production organization model, which has been applied in human–computer interaction, databases, natural language processing, machine learning and other fields. It guides the public to complete some tasks through specific strategies and methods. However, rational and selfish workers in crowdsourcing systems will submit solutions of different qualities in order to maximize their own benefits. Therefore, how to choose optimal strategies for selfish workers to maximize their benefits is important and crucial in such a scenario. In this paper, we propose a decision optimization method with incomplete information in a crowdsourcing system based on zero-determinant (ZD) strategies to help workers make optimal decisions. We first formulate the crowdsourcing problem, where workers have “winner-takes-all” rules as an iterated game with incomplete information. Subsequently, we analyze the optimal decision of workers in crowdsourcing systems in terms of ZD strategies, for which we find conditions to reach the maximum payoff of a focused worker. In addition, the analysis helps understand what solutions selfish workers will submit under the condition of having incomplete information. Finally, numerical simulations illustrate the performances of different strategies and the effects of the parameters on the payoffs of the focused worker.

The loss of complexity in ageing hypothesis (LOCH) has found support from EEG studies, most of which adopted signal-domain complexity measures. The present study adopted the largest Lyapunov exponent (LLE) to measure complexity from a nonlinear dynamical systems perspective. A total of 144 participants were included and divided into young, young-old and old-old groups. Both sensor-space and source-space results showed significantly lower LLE for older than younger adults. The age-related differences were region-dependent, being most prominent in the frontal region, followed by bilateral temporal regions. The occipital region showed non-significant differences. Significant reduction of LLE in the posterior cingulate was also observed by virtue of the source-space analysis. We also evaluated the relationships between LLE and other complexity measures. The most intriguing result was the negative correlation between LLE and Lempel-Ziv complexity (LZC). The age-related decrease in LLE indicated a higher regularity in dynamics, while the higher LZC indicated a higher randomness in the signal domain. The new findings support the LOCH by demonstrating the simultaneous increase in regularity and randomness.

This is an Editorial. It is amazing and also exciting to see a new journal Chaos Theory and Applications established recently. After chaos was coined with a precise model, the Lorenz system, more than half a century ago, there have already been many well-known journals on chaos such as, to name just a few specialized ones, Chaos, Chaos Solitons and Fractals, International Journal of Bifurcation and Chaos, Nonlinear Dynamics, and several Physical Review journals. Therefore, on the one hand, organizing a new journal on chaos needs a lot of courage and planning, and on the other hand, one can see that the chaos is still an ever-young subject for scientific research today. It is our expectation, therefore, that the new journal Chaos Theory and Applications could contribute more to this new direction of chaos research, along with other traditional topics.

p53 regulates the cellular response to genotoxic damage and prevents carcinogenic events. Theoretical and experimental studies state that the p53-Mdm2 network constitutes the core module of regulatory interactions activated by cellular stress induced by a variety of signaling pathways. In this paper, a strategy to control the p53-Mdm2 network regulated by p14ARF is developed, based on the pinning control technique, which consists into applying local feedback controllers to a small number of nodes (pinned ones) in the network. Pinned nodes are selected on the basis of their importance level in a topological hierarchy, their degree of connectivity within the network, and the biological role they perform. In this paper, two cases are considered. For the first case, the oscillatory pattern under gamma-radiation is recovered; afterward, as the second case, increased expression of p53 level is taken into account. For both cases, the control law is applied to p14ARF (pinned node based on a virtual leader methodology), and overexpressed Mdm2-mediated p53 degradation condition is considered as carcinogenic initial behavior. The approach in this paper uses a computational algorithm, which opens an alternative path to understand the cellular responses to stress, doing it possible to model and control the gene regulatory network dynamics in two different biological contexts. As the main result of the proposed control technique, the two mentioned desired behaviors are obtained.

Our aim is to propose an efficient algorithm for enhancing the contrast of dark images based on the principle of stochastic resonance in a global feedback spiking network of integrate-and-fire neurons. By linear approximation and direct simulation, we disclose the dependence of the peak signal-to-noise ratio on the spiking threshold and the feedback coupling strength. Based on this theoretical analysis, we then develop a dynamical system algorithm for enhancing dark images. In the new algorithm, an explicit formula is given on how to choose a suitable spiking threshold for the images to be enhanced, and a more effective quantifying index, the variance of image, is used to replace the commonly used measure. Numerical tests verify the efficiency of the new algorithm. The investigation provides a good example for the application of stochastic resonance, and it might be useful for explaining the biophysical mechanism behind visual perception.

This paper studies a system of coupled discrete fractional-order logistic maps, modeled by Caputo’s delta fractional difference, regarding its numerical integration and chaotic dynamics. Some interesting new dynamical properties and unusual phenomena from this coupled chaotic-map system are revealed. Moreover, the coexistence of attractors, a necessary ingredient of the existence of hidden attractors, is proved and analyzed.

Abstract This paper establishes a Markov chain model as a unified framework for describing the evolution processes in complex networks. The unique feature of the proposed model is its capability in addressing the formation mechanism that can reflect the “trichotomy” observed in degree distributions, based on which closed-form solutions can be derived. Important special cases of the proposed unified framework are those classical models, including Poisson, Exponential, Power-law distributed networks. Both simulation and experimental results demonstrate a good match of the proposed model with real datasets, showing its superiority over the classical models. Implications of the model to various applications including citation analysis, online social networks, and vehicular networks design, are also discussed in the paper.

Non-coherent capacity bounds for the M-ary differential chaos shift keying (DCSK) modulation system are derived and analyzed over multipath Rayleigh fading channels, deriving conditions on the channel state information/non-channel state information (CSI/NonCSI) and soft-decision/hard-decision (SD/HD), respectively. Meanwhile, the inter-symbol interference is modeled and analyzed mathematically. Through numerical simulations and analyses, it is found that: (1) the influences of the spreading factor β, multipath number L, modulation dimension M, CSI/NonCSI, and SD/HD on the non-coherent capacity bounds are significant; (2) there is a relatively broad range of code rates corresponding to the optimal system power in U-shaped capacity curves of the non-coherent reception; and (3) the U-shaped non-coherent capacity bounds are proven to exist by investigating the mechanism of the low density parity check coded the M-ary DCSK system. These results are useful as benchmarks for designing power-efficient coded M-ary DCSK systems.

Multiresolution M-ary differential chaos shift keying (MR-M-DCSK) modulation using nonuniformly spaced phase constellation is a promising technique that can satisfy different bit-error-rate (BER) requirements within one symbol over multipath fading channels. However, the BER performance always deteriorates as the modulation order M becomes larger. To overcome this shortcoming, a new hierarchical square-constellation-based M-DCSK communication system using non-uniformly spaced distance constellation is proposed, which can be easily extended to other-constellation scenarios, e.g., rectangular, star, and asymmetrical square constellations. Furthermore, an adaptive transmission scheme is designed by modifying the distance between the two constellation points in the last hierarchical level. In addition, theoretical BER expressions of the proposed systems are derived over multipath Rayleigh fading channels, which are consistent with the simulated results. Analytical and simulated results show that the proposed system not only can provide lower energy consumption as compared with the conventional MR-M-DCSK system but also can efficiently adjust BER performance based on the signal-to-noise ratio information. Therefore, the proposed system can serve as a desirable alternative for energy-efficient short-range wireless-communication applications.

An improved multi-carrier M-ary differential chaos shift keying (MM-DCSK) system is presented, where differential modulation and demodulation are carried out across multiple carriers in the frequency domain, so channel estimation is not needed. For one data frame transmission, only one noninformation-bearing reference sub-carrier signal is required, and the reference of each information-bearing sub-carrier is its previous sub-carrier signal, thus high energy efficiency is attained. If channel response changes during a frame period, the time diversity is achieved. In addition, the peak-to-average power ratio (PAPR) is considered, and it is found that adjacent symbols with a large Euclidean distance achieve a low PAPR. Accordingly, a low-complexity PAPR reduction algorithm is proposed based on symbol-interleaving with only one inverse fast Fourier transform processor. The simulation results demonstrate that the system with the proposed algorithm dramatically reduces the PAPR. Analytical bit-error-rate expressions are derived and verified by simulations over additive white Gaussian noise and multi-path fading channels.

The well-known small-world network model was established by randomly rewiring edges, aiming to enhance the synchronizability of an undirected nearest-neighbor regular network. This paper demonstrates via extensive numerical simulations that randomly redirecting edges could enhance the robustness of the network controllability for directed snapback networks against both random and intentional node-removal and edge-removal attacks.

Differential-chaos-shift-keying cooperative communication (DCSK-CC) has attracted a great deal of attention in the past five years, because it can overcome the negative effect of wireless multipath fading. However, the conventional DCSK-CC system suffers from a major disadvantage of low data rate due to the resource consumption in cooperation. To address the issue, this paper proposes an efficient cooperative scheme, called partial-sequence CC (PS-CC) scheme, for the DCSK-CC system. In the PS-CC framework, the source and relays only transmit a partial chaotic sequence corresponding to one bit to the destination, while consuming the same transmission energy as DCSK-CC. To validate the feasibility of the DCSK PS-CC system, this paper carries out the error performance, diversity order, and data rate analyses under different cooperative conditions over multipath fading channels. Both numerical and simulated results show that the proposed PS-CC system outperforms the DCSK-based non-cooperative system and DCSK-CC systems over the range of signal-to-noise ratio under study. Moreover, the proposed system can enhance data rate and reduce the computational overhead of the relays as compared with the DCSK-CC system. In addition, it will be demonstrated that the DCSK PS-CC system preserves its advantages in typical ultra-wideband (UWB) channels. Overall, the PS-CC appears to be a desirable cooperative scheme for those applications featured by high-data-rate, low-power, and low-complexity, such as wireless sensor networks using the transmitted reference UWB technique.

This paper reports the finding of unstable limit cycles and singular attractors in a two-dimensional dynamical system consisting of an inductor and a bistable bi-local active memristor. Inspired by the idea of nested intervals theorem, a new programmable scheme for finding unstable limit cycles is proposed, and its feasibility is verified by numerical simulations. The unstable limit cycles and their evolution laws in the memristor-based dynamic system are found from two subcritical Hopf bifurcation domains, which are subdomains of twin local activity domains of the memristor. Coexisting singular attractors are discovered in the twin local activity domains, apart from the two corresponding subcritical Hopf bifurcation domains. Of particular interest is the coexistence of a singular attractor and a period-2 or period-3 attractor, observed in numerical simulations.

In this paper, a new memcapacitor model and its corresponding circuit emulator are proposed, based on which, a chaotic oscillator is designed and the system dynamic characteristics are investigated, both analytically and experimentally. Extreme multistability and coexisting attractors are observed in this complex system. The basins of attraction, multistability, bifurcations, Lyapunov exponents, and initial-condition-triggered similar bifurcation are analyzed. Finally, the memcapacitor-based chaotic oscillator is realized via circuit implementation with experimental results presented.

Epidemic spreading processes on multiplex networks have richer dynamical properties than those on single layered networks. To describe the intertwined processes on such networks, heterogeneous mean field (HMF) approach for continuous-time processes and microscopic Markov chain approach for discrete-time processes have been proposed. However, it has been shown that the time evolution of infected individuals and the final epidemic size obtained from these approaches have noticeable discrepancy comparing to those from Monte Carlo simulations. In this paper, we extend the approach of effective degree theory (EDT) on multiplex networks. We will show that predictions obtained from the EDT have excellent agreement with Monte Carlo simulations. Moreover, since the dynamics on multiplex networks involve more dynamical variables, which may invoke more computations, to reduce the computational burden, we further develop an approach based on partial effective degree theory (PEDT) for analyzing the dynamics on multiplex networks, where one layer adopts EDT and the other layer adopts the HMF. Our results show that PEDT has a good performance in predicting the target dynamical process.

In recent years, as natural and social sciences are rapidly evolving, classical chaos theoryand modern complex networks studies are gradually interacting each other with a great joineddevelopment [...]

This paper is concerned with invariance ( F 1 , F 2 ) -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of ( F 1 , F 2 ) -scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties P ( k ) and Q ( k ) of Furstenberg families are introduced. It is shown that, for any positive integer k, for any s ∈ [ 0 , 1 ] , Furstenberg family M ¯ ( s ) has properties P ( k ) and Q ( k ) , where M ¯ ( s ) denotes the family of all infinite subsets of Z + whose upper density is not less than s. Then, the following conclusion is obtained. D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ ) if and only if D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ [ m ] ) .

This paper is concerned with delayed generalized 2D discrete logistic systems of the form xm+1,n=f(m,n,xm,n,xm,n+1,xm−Ã,n−Ä) , where à and Ä are positive integers, f:ℕ02×â„Â3→℠is a real function, which contains the logistic map as a special case, and m and n are nonnegative integers, where ℕ0={0,1,…} and â„Â=(−∞,∞). Some sufficient conditions for this system to be stable and exponentially stable are derived.

Halo-chaos in high-current accelerator has become one of the key issues because it can cause excessive radioactivity from the accelerators and significantly limits the applications of the new accelerators in industrial and other fields. Some general engineering methods for chaos control have been developed, but they generally are unsuccessful for halo-chaos suppression due to many technical constraints. In this article, controllability condition for beam halo-chaos is analyzed qualitatively. Then Particles-in-Cell (PIC) simulations explore the nature of beam halo-chaos formation. A nonlinear control method and wavelet function feedback controller are proposed for controlling beam halo-chaos. After control of beam halo-chaos for initial proton beam with water bag distributions, the beam halo strength factor H is reduced to zero, and other statistical physical quantities of beam halo-chaos are doubly reduced. The results show that the developed methods in this paper are very effective for proton beam halo-chaos suppression. Potential application of the halo-chaos control method is finally pointed out.

Bifurcation control has attracted increasing attention in recent years. A simple and unified state-feedback methodology is developed in this paper for Hopf bifurcation control for discrete-time systems. The control task can be either shifting an existing Hopf bifurcation or creating a new Hopf bifurcation. Some computer simulations are included to illustrate the methodology and to verify the theoretical results.

In this paper, a switching manifold approach is developed for nonlinear feed-back control of chaotic systems. The design strategy is straightforward, and the nonlinear control law is the simple bang–bang control. Yet, this control method is very effective; for instance, several desired equilibria can be stabilized by using one control law with different initial conditions. Its effectiveness is verified by both theoretical analysis and numerical simulations. The Lorenz system simulation is shown for the purpose of illustration.

This paper investigates the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling. It is shown that for both types of coupling, the delay always plays a subtle role in either promoting or impairing synchronization. In particular, depending on the inherent oscillation period of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. For attractive coupling, the minima appear at every integer multiple of the average oscillation period, while for the repulsive coupling, they appear at every odd multiple of the half of the average oscillation period. The obtained results are robust to the variations of the dynamics of individual neurons, the system size, and the neuronal firing type. Hence, they can be used to characterize attractively or repulsively coupled scale-free neuronal networks with delays.