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In this paper, we study bifurcation of limit cycles from the equator of piecewise polynomial systems with no singular points at infinity. We develop a method for computing the Lyapunov constants at infinity of piecewise polynomial systems. In particular, we consider cubic piecewise polynomial systems and study limit cycle bifurcations in the neighborhood of the origin and infinity. Moreover, an example is presented to show 11 limit cycles bifurcating from infinity. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of coupled time-delayed neural networks with discontinuous activations. Based on the new analysis techniques, framework of nonsmooth analysis, differential inclusion theory, both discontinuous controller and novel controller with continuous part are considered to realize the finite-time synchronization for discontinuous coupled neural networks. Some criteria are investigated in detail on constructing suitable Lyapunov functions cover 1-norm and quadratic-form, which were considered separately in the past. Finally, numerical examples are provided to show the correctness of our analysis. Our results are essentially new and they extend previously known researches. [ABSTRACT FROM AUTHOR]

This paper deals with the finite-time synchronization issue of time-varying delayed neural networks (DNNs) with discontinuous activations. Based on master-slave concept, several sufficient conditions are given to guarantee the finite-time synchronization of discontinuous DNNs. In order to control the synchronization error to converge zero in a finite time, we design three classes of novel switching state-feedback controllers which involve time-delays and discontinuous factors. The analysis in this paper employs the extended differential inclusion theory, the famous finite-time stability theorem, inequality techniques and generalized Lyapunov approach. Moreover, the upper bounds of the settling time of synchronization are estimated. Finally, the validity of proposed design method and theoretical results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this study, without assuming the symmetry or irreducibility of the coupling configuration matrix, we investigate the synchronization of linearly coupled dynamical systems with discontinuous identical nodes and time delay. First, we design a feedback controller to realize the complete synchronization and exponential synchronization of discontinuous complex networks using a generalized Lyapunov approach. Second, we propose a set of adaptive controllers and adaptive laws to achieve complete synchronization. Third, we study the adaptive pinning synchronization of the discontinuous complex networks. The results of numerical simulations are presented to demonstrate the effectiveness of the controllers and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the global stability has been investigated for a novel class of competitive neural networks with mixed time delays and discontinuous activations. Without presuming the boundedness of activation functions, we draw two set of sufficient conditions ensuring the existence, uniqueness of the equilibrium, global exponential asymptotic stability of the solution and the associated output of the solution converging to the output equilibrium point in measure, by linear matrix inequality, M -matrix, Leray–Schauder alternative theorem in multivalued analysis, general Lyapunov method, topological degree theory of set-valued map. Meanwhile, we present a result on the global convergence in finite time of given neural networks. The results in the literature are generalized and significantly improved. Finally, one example and simulations are presented to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, under the assumption that the activation functions only satisfy global Lipschitz conditions, a novel LMI-based sufficient condition for global asymptotic stability of equilibrium point of a class of BAM neural networks with reaction–diffusion terms and distributed delays is obtained by using degree theory, LMI method, inequalities technique and constructing Lyapunov functionals. In our results, the assumptions for boundedness and monotonicity in existing papers on the activation functions are removed. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the global dynamics of equilibrium point for delayed competitive neural networks with different time scales and discontinuous activations. Employing the Leray–Schauder alternative theorem in multivalued analysis, linear matrix inequality technique and generalized Lyapunov-like method, we obtain some new sufficient conditions ensuring the existence, uniqueness and global stability and the global convergence in finite time of the equilibrium point, the results of this paper improve and extend previously known results. Finally, two examples and numerical simulations are conducted to validate the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a class of neural networks with mixed time-varying delays and discontinuous activations, new criteria for the existence and exponential stability of the periodic solution are established by using the Mawhin-like coincidence theorem, M-matrix theory, differential inequality techniques and Lyapunov functional. The obtained results are new and complement previously known results. Finally, we present a numerical example with simulations to demonstrate the effectiveness of the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the dynamical behaviors of a novel class of Cohen–Grossberg neural networks with not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activations, which may be unbounded or nonmonotonic. Some sufficient conditions are derived to ensure the existence, uniqueness of the equilibrium, global exponential asymptotic stability of the solution and the associated output of the solution converging to the output equilibrium point in measure by using the Leray–Schauder alternative theorem in multivalued analysis, matrix theory and generalized Lyapunov-like approach. Some recent results in the literature are generalized and significantly improved. One example is provided to demonstrate the effectiveness of the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, a delayed multispecies ecological competition-predator system with Holling-III functional response is studied. By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functionals, some sufficient conditions are derived for the existence, uniqueness and global asymptotic stability of positive periodic solutions to the system. [ABSTRACT FROM AUTHOR]

Abstract: Nonlinear dynamical systems described by differential equations with discontinuous right-hand side and solutions intended in Filippov sense are considered. Based on Filippov differential inclusion and a new chain rule for differentiating regular functions along Filippov solution trajectories, different kinds of stability and convergence results are presented. Moreover, we investigate the stability and convergence for the corresponding perturbation system of the discontinuous system and some new criteria are addressed. [Copyright &y& Elsevier]

Abstract: Without assuming that the neuron activations are bounded, some delay-independent criteria for interval delayed neural networks with discontinuous neuron activations are derived to guarantee global robust stability by using the generalized Lyapunov method and linear matrix inequality (LMI) technique. The obtained results improve and extend those given in earlier literature, and two numerical examples are also given to show the effectiveness of our results. [Copyright &y& Elsevier]

Abstract: In this paper, a general class of neural networks with arbitrary constant delays is studied, whose neuron activations are discontinuous and may be unbounded or nonmonotonic. Based on the Leray–Schauder alternative principle and generalized Lyapunov approach, conditions are given under which there is a unique equilibrium of the neural network, which is globally asymptotically stable. Moreover, the existence and global asymptotic stability of periodic solutions are derived, where the neuron inputs are periodic. The obtained results extend previous works not only on delayed neural networks with Lipschitz continuous neuron activations, but also on delayed neural networks with discontinuous neuron activations. [Copyright &y& Elsevier]

By using the Banach fixed point theorem and constructing suitable Lyapunov function, some sufficient conditions are obtained ensuring existence, uniqueness and global stability of almost periodic solution of the BAM neural networks with variable coefficients and delays. These results are helpful to design global exponential stable BAM networks and almost periodic oscillatory BAM networks. [Copyright &y& Elsevier]

Abstract In this paper, we study the finite-time synchronization issue for delayed fuzzy cellular neural networks with discontinuous activations. Under the framework of differential inclusions, by utilizing the discontinuous state feedback control method and constructing Lyapunov functionals, new and useful finite-time synchronization criteria for the considered networks are established, which significantly generalize and improve recent works in literature. Finally, two examples with simulations are presented to show the effectiveness of the synchronization schemes. [ABSTRACT FROM AUTHOR]

This paper investigates the finite-time robust synchronization problem of mixed time-delayed neural networks (DNNs) with discontinuous activations and uncertain external perturbations. Two classes of simple discontinuous controllers (i.e., switching state-feedback controller and switching adaptive controller) are designed such that the response system can be finite-time robustly synchronized with a drive system. Without using the well-known finite-time stability theorem given by Forti in 2006, the finite-time robust synchronization control of discontinuous DNNs can be realized via generalized Lyapunov–Krasovskii functionals approach and novel analysis techniques. Moreover, the upper bounds of the settling time for synchronization are estimated. The main tools of this paper involve functional differential inclusion theory in the sense of Filippov solutions and non-smooth analysis. Finally, two numerical examples are provided to illustrate the effectiveness of the designed control method and the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the global robust exponential stability of time-varying delayed neural networks with discontinuous activation functions and norm-bounded uncertainties. Based on the Lyapunov–Krasovskii stability theory, we originally analyze the global robust exponential stability of discontinuous neural networks with time-varying delays in view of the linear matrix inequalities (LMIs). Therefore, our results are brand new compared to previous literatures. A numerical example is given to validate the effectiveness of our results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the convergence dynamics of Cohen–Grossberg neural networks (CGNNs) with continuously distributed delays. Without assuming the differentiability and monotonicity of activation functions, the differentiability of amplification functions and the symmetry of synaptic interconnection weights, we construct suitable Lyapunov functionals and employ inequality technique to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, global exponential convergence, and even global exponential stability of equilibria. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks including Hopfield neural networks, bidirectional association memory neural networks and cellular neural networks. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and global exponential stability of periodic solutions for a nonlinear periodic system, arising from the description of the states of neurons in delayed Cohen–Grossberg type. By the continuation theorem of coincidence degree theory and Lyapunov functionals technique, we deduce some sufficient conditions ensuring existence as well as global exponential stability of periodic solution. Some existing results are improved and extended. Even corresponding to an autonomous system, our results that these conditions are milder and less restrictive than previous known criteria since the hypothesis of boundedness and differentiability on the activation function are dropped. The theoretical analyses are verified by numerical simulations. [Copyright &y& Elsevier]

This paper discusses the issue of periodicity and finite-time periodic synchronization of discontinuous complex-valued neural networks (CVNNs). Based on a modified version of Kakutani's fixed point theorem, general conditions are obtained to guarantee the periodicity of discontinuous CVNNs. Next, several criteria for finite-time periodic synchronization (FTPS) are given by using a new proposed finite-time convergence theorem. Different from the traditional convergence lemma, the estimated upper bound of the derivative of the Lyapunov function (LF) is allowed to be indefinite or even positive. In order to achieve FTPS, novel discontinuous control algorithms, including state-feedback control algorithm and generalized pinning control algorithm, are designed. In the generalized pinning control algorithm, a guideline is proposed to select neurons to pin the designed controller. Finally, two simulations are given to substantiate the main results. [ABSTRACT FROM AUTHOR]