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The Cucker-Smale (C-S) flocking model can be applied to design several mobile autonomous agent systems such as unmanned aerial vehicles and robot systems. In these applications, the problem of collision is usually a nonnegligible issue that should be addressed. In the literature, extra bonding/repulsive forces are added to solve the problem of collision. However, all the known results are limited to the symmetric coupling topology. In this paper, we address this problem under a group-hierarchical multileadership consisting of symmetric coupling topologies and the hierarchical leadership. We obtain the following three main results. First, we show that the augmented C-S model achieves asymptotic flocking provided, roughly, that the ratio between the rate of bounding force and communication rate is sufficiently large. Second, we prove that, for a given augmented C-S model, there exists a set of initial states, which can be explicitly described, such that their corresponding solutions maintain some minimum distance between them. Finally, we show that, for a given set of initial states, we can control the system so that its corresponding solution is collision free by suitably choosing bounding forces, which are independent of given initial states. This in turn suggests that the corresponding system can be collision free by having tolerance for error in their initial states. [ABSTRACT FROM AUTHOR]

The role of vaccine success is introduced into an epidemic spreading model consisting of three states: susceptible, infectious, and vaccinated. Moreover, the effect of three types, namely, contact, local, and global, of infection awareness and immunization awareness is also taken into consideration. The model generalizes those considered in Pastor-Satorras and Vespignani [Phys. Rev. E 63, 066117 (2001)], Pastor-Satorras and Vespignani [Phys. Rev. E 65, 036104 (2002)], Moreno et al. [Eur. Phys. J. B 26, 521-529 (2002)], Wu et al. [Chaos 22, 013101 (2012)], and Wu et al. [Chaos 24, 023108 (2014)]. Our main results contain the following. First, the epidemic threshold is explicitly obtained. In particular, we show that, for any initial conditions, the epidemic eventually dies out regardless of what other factors are whenever some type of immunization awareness is considered, and vaccination has a perfect success. Moreover, the threshold is independent of the global type of awareness. Second, we compare the effect of contact and local types of awareness on the epidemic thresholds between heterogeneous networks and homogeneous networks. Specifically, we find that the epidemic threshold for the homogeneous network can be lower than that of the heterogeneous network in an intermediate regime for intensity of contact infection awareness while it is higher otherwise. In summary, our results highlight the important and crucial roles of both vaccine success and contact infection awareness on epidemic dynamics. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to address synchronous chaos on the Julia set of complex-valued coupled map lattices (CCMLs). Our main results contain the following. First, we solve an inf min max problem for which its solution gives the fastest synchronized rate in a certain class of coupling matrices. Specifically, we show that for the class of real circulant matrices of size 4, the coupling weights, possible complex numbers, yielding the fastest synchronized rate can be exactly obtained. Second, for individual map of the form fc(z) = z² + c with |c| < ¼, we show that the corresponding CCMLs acquires global synchrony on its Julia set with the number of the oscillators being 3 or 4 for the diffusive coupling. For c = 0 and -2, the corresponding CCMLs obtain local synchronization if and only if the number of oscillators is less than or equal to 5. Global synchronization for the individual map of the form gc(z) = z³ + cz is also reported. [ABSTRACT FROM AUTHOR]

An SIS model based on the microscopic Markov-chain approximation is considered in this paper. It is assumed that the individual vaccination behavior depends on the contact awareness, local and global information of an epidemic. To better simulate the real situation, the vaccine failure rate is also taken into consideration. Our main conclusions are given in the following. First, we show that if the vaccine failure rate α is zero, then the epidemic eventually dies out regardless of what the network structure is or how large the effective spreading rate and the immunization response rates of an epidemic are. Second, we show that for any positive α, there exists a positive epidemic threshold depending on an adjusted network structure, which is only determined by the structure of the original network, the positive vaccine failure rate and the immunization response rate for contact awareness. Moreover, the epidemic threshold increases with respect to the strength of the immunization response rate for contact awareness. Finally, if the vaccine failure rate and the immunization response rate for contact awareness are positive, then there exists a critical vaccine failure rate αc >0 so that the disease free equilibrium (DFE) is stable (resp., unstable) if α<αc (resp., α>αc). Numerical simulations to see the effectiveness of our theoretical results are also provided. [ABSTRACT FROM AUTHOR]

In this work, we study the cluster synchronization of chemically coupled and generally formulated networks which are allowed to be nonidentical. The sufficient condition for the existence of stably synchronous clusters is derived. Specifically, we only need to check the stability of the origins of m decoupled linear systems. Here, m is the number of subpopulations. Examples of nonidentical networks such as Hindmarsh-Rose (HR) neurons with various choices of parameters in different subpopulations, or HR neurons in one subpopulation and FitzHugh-Nagumo neurons in the other subpopulation are provided. Explicit threshold for the coupling strength that guarantees the stably cluster synchronization can be obtained. [ABSTRACT FROM AUTHOR]

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes. [ABSTRACT FROM AUTHOR]

A model of integrate-and-fire oscillators is studied. In the special case of identical oscillators, the model was first proposed and analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645-1662]. We assume, as in Mirollo and Strogatz's model, that each oscillator xi evolves according to a map fi. Our main results are to demonstrate that the concavity structure of fi plays an important role in determining whether Peskin's second conjecture holds true. Specifically, the following statements are proved. First, the system of convex oscillators (i.e., f″i < 0 for all i), in general, synchronizes when the oscillators are not quite identical. Second, the system of a certain class of concave oscillators (i.e., f″i > 0 for all i) will not achieve synchrony for initial conditions in a set of positive measure when the oscillators are nearly identical. Third, the system of concave oscillators may achieve synchrony under certain sufficient conditions, provided that the oscillators are not quite nonidentical and that its concavity is small. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to address the synchronous chaos in coupled map lattices with general connectivity topology. Our main results contain the following. First, the master stability functions also hold for general connectivity topology with coupling through a nonlinear function that needs to be exactly the individual chaotic map. Second, the synchronization curve, composed of pieces of transverse Lyapunov exponent curves, is constructed. Third, necessary and sufficient conditions on coupling strength for yielding the synchronous chaos of the system are given. Moreover, the coupling strength dc giving the fastest convergence rate of the initial values toward the synchronous state is explicitly obtained. It is also proved that such dc is independent of the choice of the individual map. Finally, our results here can be applied to address questions of wavelength bifurcations and size instability. [ABSTRACT FROM AUTHOR]

We consider difference equations Phl(yn, yn+1, ..., yn+m) = 0, n\in \mathbb{Z} , of order m with parameter l close to that exceptional value l0 for which the function Ph depends on two variables: \Phi _{\lambda _{0}}(x_{0},\ldots, x_{m})=\xi (x_{N},x_{N+L}) with 0 [?] N, N + L [?] m. It is also assumed that for the equation x(x, y) = 0, there is a branch y = ph(x) with positive topological entropy htop(ph). Under these assumptions we prove that in the set of bi-infinite solutions of the difference equation with l in some neighbourhood of l0, there is a closed (in the product topology) invariant set to which the restriction of the shift map has topological entropy arbitrarily close to htop(ph)/|L|, and moreover, orbits of this invariant set depend continuously on l not only in the product topology but also in the uniform topology. We then apply this result to establish chaotic behaviour for Arneodo-Coullet-Tresser maps near degenerate ones, for quadratic volume preserving automorphisms of \mathbb{R}^{3} and for several lattice models including the generalized cellular neural networks (CNNs), the time discrete version of the CNNs and coupled Chua's circuit. [ABSTRACT FROM AUTHOR]

Based on the concept of matrix measures, we study global stability of synchronization in networks. Our results apply to quite general connectivity topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized. [ABSTRACT FROM AUTHOR]

In 1998, Chen et al. [1998] found an error in Marotto's paper [1978]. It was pointed out by them that the existence of an expanding fixed point z of a map F in Br(z), the ball of radius r with center at z does not necessarily imply that F is expanding in Br(z). Subsequent efforts (see e.g. [Chen et al., 1998; Lin et al., 2002; Li & Chen, 2003]) in fixing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give sufficient conditions so that F is "globally" expanding. This, in turn, gives more satisfying definitions of a snap-back repeller. We then use those results to show the existence of chaotic backward traveling waves in a discrete time analogy of one-dimensional Cellular Neural Networks (CNNs). Some computer evidence of chaotic traveling waves is also given. [ABSTRACT FROM AUTHOR]

Controlling chaos via wavelet transform was recently proposed by Wei, Zhan, and Lai [Phys. Rev. Lett. 89, 284103 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue λ1(α,β) of the (wavelet) transformed coupling matrix C(α,β) for each α and β. Here β is a mixed boundary constant and α is a scalar factor. In particular, β=1 (respectively, 0) gives the nearest neighbor coupling with periodic (respectively, Neumann) boundary conditions. The first, rigorous work to understand the eigenvalues of C(α,1) was provided by Shieh et al. [J. Math. Phys. (to be published)]. The purpose of this paper is twofold. First, we apply a different approach to obtain the explicit formulas for the eigenvalues of C(α,1) and C(α,0). This, in turn, yields some new information concerning λ1(α,1). Second, we shed some light on the question whether the wavelet method works for general coupling schemes. In particular, we show that the wavelet method is also good for the nearest neighbor coupling with Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive a sharper lower bound for the spatial entropy of two-dimensional golden mean. [ABSTRACT FROM AUTHOR]

Subcarrier multiplexing by chaotic multitone modulation is investigated. Optical chaotic light can be achieved by injecting multiple subcarriers into a self-pulsating laser diode. Synchronization between two identical chaotic systems (drive and response) can be implemented provided that the conditional Lyapunov exponents are all negative. By adding amplitude modulation (AM) signals to each subcarrier, the two systems become asymptotically synchronized. The AM signals in each subcarrier can be recovered by the introduction of a filtering process where the Lyapunov exponent of the synchronized error function matches the cutoff frequency of a first order low pass filter. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the one-dimensional discrete Schrödinger equation with general, symmetric boundary conditions. Our results primarily concern the number of energy states lying in the wells. [ABSTRACT FROM AUTHOR]

We consider a cellular neural network (CNN) with a bias term z in the integer lattice Z² on the plane R². We impose a symmetric coupling between nearest neighbors, and also between next-nearest neighbors. Two parameters, a and ε, are used to describe the weights between such interacting cells. We study patterns that can exist as stable equilibria. In particular, the relationship between mosaic patterns and the parameter space (z, a; ε) can be completely characterized. This, in turn, addresses the so-called learning problem in CNNs. The complexities of mosaic patterns are also addressed. [ABSTRACT FROM AUTHOR]

In this paper we rigorously show the existence of solutions of a matrix equation which arises in the design of micro electronical circuits. This equation was studied by Szidarovszky and Palusinsk [Appl. Math. Comput. \textbf{64}, 115-119(1994)], who also presented an iterative algorithm for its solution. We show, via an example, that this algorithm could converge extremely slow in certian cases. The solution can then be used to minimize the reflection coefficients of the active signals. [ABSTRACT FROM AUTHOR]