Our objective is to find a necessary and sufficient condition for consensus over a random network generated by i.i.d. stochastic matrices. We show that the consensus problem in all different types of convergence (almost surely, in probability, and in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L^{p}$</tex> </formula> for every <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p\geq 1$</tex></formula> ) are actually equivalent, thereby obtain the same necessary and sufficient condition for all of them. The main technique we used is based on the stability in a projected subspace of the concerned infinite sequences.
When multi-agent systems are equipped with embedded microprocessors of limited resources, it is important to design an appropriate real-time scheduling of control tasks. In this paper, we examine the synthesis of event-triggered information transmission and distributed estimation for a tracking control problem of leader-follower multi-agent systems. For a second-order leader-follower multi-agent system, we use a decentralized event-triggered strategy in the dynamic tracking control design. Then, we analyze the input-to-state stability of the closed-loop multi-agent system with the proposed event-triggered tracking control. Finally, we present a numerical simulation to validate the proposed design.
It is well known that many economic data series show chaotic behaviors. In this Letter, we further investigate the complex dynamical behaviors of the daily data series, including opening quotation, closing quotation, maximum price, minimum price, and total exchange quantum, in Shenzhen stock exchange and Shanghai stock exchange, which are two representative stock exchanges in mainland China. The maximum Lyapunov exponents, correlation dimensions, and frequency spectra are calculated for these time series. Our results indicate that some daily data series of stock exchanges display low-dimensional chaotic behaviors, and some other daily data series do not show any chaotic behavior. Moreover, we introduce a weighted one-rank local-region approach for predicting short-term daily data series of stock exchange.
No abstract is provided for this article.
Bifurcations of a general class of traveling wave solutions are analyzed. In particular, the existence of solitary wave, kink and anti-kink wave solutions, and uncountably infinite periodic wave solutions and breather solutions of a general class of traveling wave equations is proved. Also, the existence of breaking wave solution is discussed in detail. Under different parametric conditions, several sufficient conditions for the existence of these solutions are derived. Sufficient simulation results are provided to visualize the theoretical results.
Bifurcation control has attracted increasing attention in recent years. A simple and unified state-feedback methodology is developed in this paper for Hopf bifurcation controls for both continuous-time and discrete-time systems. The control task can be either shifting an existing bifurcation or creating a new one. Some computer simulations are included to illustrate the methodology and verify the theoretical results.
We study the evolution of spatiotemporal dynamics and synchronization transition on small-world Hodgkin-Huxley (HH) neuronal networks that are characterized with channel noises, ion channel blocking and information transmission delays. In particular, we examine the effects of delay on spatiotemporal dynamics over neuronal networks when channel blocking of potassium or sodium is involved. We show that small delays can detriment synchronization in the network due to a dynamic clustering anti-phase synchronization transition. We also show that regions of irregular and regular wave propagations related to synchronization transitions appear intermittently as the delay increases, and the delay-induced synchronization transitions manifest as well-expressed minima in the measure for spatial synchrony. In addition, we show that the fraction of sodium or potassium channels can play a key role in dynamics of neuronal networks. Furthermore, We found that the fraction of sodium and potassium channels has different impacts on spatiotemporal dynamics of neuronal networks, respectively. Our results thus provide insights that could facilitate the understanding of the joint impact of ion channel blocking and information transmission delays on the dynamical behaviors of realistic neuronal networks.
In this work, a graphical approach is developed from an engineering frequency-domain approach enabling prediction of period-doubling bifurcations (PDB's) starting from a small neighborhood of Hopf bifurcation points useful for analysis of multiple oscillations of periodic solutions for time-delayed feedback systems. The proposed algorithm employs higher order harmonic-balance approximations (HBA's) for the predicted periodic solutions of the time-delayed systems. As compared to the same study of feedback systems without time delays, the HBA's used in the new algorithm include only some simple modifications. Two examples are used to verify the graphical algorithm for prediction: one is the well-known time-delayed Chua's circuit (TDCC) and the other is a time-delayed neural-network model.
In this brief, the state PI regulator method is introduced for stabilizing unstable equilibrium points and tracking set-point targets for a class of chaotic systems. Based on the theory of nonlinear ordinary dif- ferential equations, a new criterion is derived for designing the controller gains for stabilization and tracking, in which control parameters can be se- lected via the pole placement technique of linear control theory. More im- portantly, according to our numerical studies, this method is robust against system parametric variations, can strongly reject external constant dis- turbances, and possesses an attractive low-frequency-disturbance response property. The design method is applied to the chaotic Lorenz system and Chua's circuit with a cubic nonlinearity, showing satisfactory simulation results.
To investigate how consensus is reached on a large self-organized peer-to-peer network, we extended the naming game model commonly used in language and communication to Naming Game in Groups (NGG). Differing from other existing naming game models, in NGG everyone in the population (network) can be both speaker and hearer simultaneously, which resembles in a closer manner to real-life scenarios. Moreover, NGG allows the transmission (communication) of multiple words (opinions) for multiple intra-group consensuses. The communications among indirectly-connected nodes are also enabled in NGG. We simulated and analyzed the consensus process in some typical network topologies, including random-graph networks, small-world networks and scale-free networks, to better understand how global convergence (consensus) could be reached on one common word. The results are interpreted on group negotiation of a peer-to-peer network, which shows that global consensus in the population can be reached more rapidly when more opinions are permitted within each group or when the negotiating groups in the population are larger in size. The novel features and properties introduced by our model have demonstrated its applicability in better investigating general consensus problems on peer-to-peer networks.
The nodes in a community within a network are much more connected to each other than to the others outside the community in the same network. This phenomenon has been commonly observed from many real-world networks, ranging from social to biological even to technical networks. Meanwhile, the number of communities in some real-world networks, such as the Internet and most social networks, are evolving with time. To model this kind of networks, the present Letter proposes a multi-local-world (MLW) model to capture and describe their essential topological properties. Based on the mean-field theory, the degree distribution of this model is obtained analytically, showing that the generated network has a novel topological feature as being not completely random nor completely scale-free but behaving somewhere between them. As a typical application, the MLW model is applied to characterize the Internet against some other models such as the BA, GBA, Fitness and HOT models, demonstrating the superiority of the new model.
This paper proposes a novel stream encryption scheme with avalanche effect (SESAE). Using this scheme and an ideal pseudorandom number generator (PRNG) to generate d-bit segment binary key streams, one can encrypt a plaintext such that by using any key stream generated from a different seed to decrypt the ciphertext, the decrypted plaintext will become an avalanche-like text which has 2 d − 1 consecutive one’s with a high probability. As a cost, the required bits of the ciphertext are d times those of the plaintext. A corresponding avalanche-type encryption theorem is established. Two chaotic 12-bit segment PRNGs are designed. A generalized FIPS140 test and SESAE test for the two chaotic PRNGs, RC4 12-bit segment PRNG and 12-bit segment Matlab PRNG are implemented. The SESAE tests for 16-bit segment PRNGs are also compared. The results suggest that those PRNGs are able to generate the SESAEs which are similar to those generated via ideal PRNGs.
No abstract is provided for this article.
This brief presents the first observations of multivaluedness in four systems: a random process, a nonlinear nondynamical system, a nonlinear dynamical system with nonlinearly sensed input and output and an adaptive linear estimator. The preliminary findings reported here suggest the impact of multivaluedness in different types of networks to range from adverse to benign or even essential.
This chapter proposes a new approach to fuzzy adaptive controller design using only system input–output data. The design procedure consists of three steps: First, a fuzzy ARMAX model is identified using the available data; then, a fuzzy controller is derived based on a combination of sliding mode control (SMC) theory and fuzzy control methodology; finally, the controller is equipped with an adaptive mechanism to update all control parameters online. The resulting control systems generally possess strong model-adaptation capability, good tracking control performance, and robustness against perturbations and uncertainties. This approach is applicable to general multiple-input, multiple-output (MIMO) nonlinear and uncertain systems. In this chapter, the basic problem of modeling and control of an unknown multivariable dynamic system is studied with analysis, design, and simulations. The new controller only used input-output data, but has a strong prediction ability and satisfactory tracking performance. Several simulation examples have been shown to demonstrate the merits of the proposed FARMAX (fuzzy ARMAX).
The differential geometry approach for exact input-output linearizable nonlinear systems usually results in a complicated nonlinear controller that is difficult to implement. To overcome this difficulty, we propose to approximate the nonlinear controller by a canonical piecewise linear expression within an allowable error bound. This design procedure can reduce a tremendous amount of computation in the design and the synthesis. The resulting controller turns out to be fairly simple in general and can achieve many performance specifications. Some sufficient conditions for guaranteeing the closed-loop system stability using this controller design method is derived in the present paper. An application to a chemical reactor system is also briefly discussed.
Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated.
