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• A normal form is derived for codimension-two focus-fold bifurcation. • Observations of focus-fold bifurcation are extended from piecewise linear systems to general piecewise smooth systems. • Two types of sliding limit cycles are bifurcated from the focus-fold singularity for general piecewise smooth systems. • Two nested crossing limit cycles are bifurcated from the focus-fold singularity for general piecewise smooth systems. • Convergence in finite time is proved for sliding limit cycles. We consider codimension-two focus-fold bifurcation for general piecewise smooth (PWS) planar systems. Some changes of variables and parameters are employed to derive a normal form, which is topologically equivalent to the original system. Making use of the normal form, we verify that observations of focus-fold bifurcation in some special piecewise linear systems also hold for general PWS systems, including the existence of two types of sliding limit cycles and coexistence of crossing and sliding limit cycles. It is shown that two nested crossing limit cycles can be bifurcated from the focus-fold singularity. This result is the improvement of the work [Han and Zhang (2010) [29] ], in which only one bifurcated limit cycle is obtained. In addition, the sliding limit cycles are proved to be convergent in finite time. [ABSTRACT FROM AUTHOR]

In this work, a new perturbation approach is developed based on Bogdanov-Takens bifurcation theory, which enables the Poincaré-Lyapunov method for switching systems with linear type centers to be applied for studying the center conditions of planar switching polynomial systems associated with a nilpotent equilibrium point. The new method is then applied to consider a class of quadratic switching nilpotent systems, and a complete classification is given on the conditions of the nilpotent equilibrium point to be a center. Moreover, based on one of the center conditions, an example is constructed to show the existence of seven small-amplitude limit cycles around the nilpotent equilibrium point, which is a new lower bound on the number of limit cycles in such systems. [ABSTRACT FROM AUTHOR]

Autophagy, an evolutional conserved lysosomal degradation process, has been implicated to play an important role in cellular defense against a variety of microbial infection. Interestingly, numerous studies found that some pathogens, especially positive-single-strand RNA viruses, actually hijacked autophagy machinery to promote virus infection within host cells, facilitating different stages of viral life cycle, from replication, assembly to egress. Enterovirus, a genus of positive-strand RNA virus, can cause various human diseases and is one of main public health threat globally, yet no effective clinical intervention is available for enterovirus infection. Here we summarized recent literature on how enteroviruses regulate and utilize autophagy process to facilitate their propagation in the host cells. The studies on the interplay between enterovirus and autophagy not only shed light on the molecular mechanisms underlying how enterovirus hijacks cellular components and pathway for its own benefits, but also provide therapeutic option against enterovirus infection. [ABSTRACT FROM AUTHOR]

In this paper, a Filippov ratio-dependent predator–prey model is proposed to describe the effect on behavioral refuges caused by prey instinct anti-predator behavior. The proposed model extends the classical ratio-dependent predator–prey model by combining a prey–predator ratio that describes the behavioral refuges make sense once it is less than a certain threshold. One of the prominent mathematical features of our model, distinguishing from the classical one, is that there exist singular points on discontinuous surface whose characteristics determine the main dynamical behaviors of this model. The complete analysis of topological structures of orbits near all the singular points is presented. We show 14 types of system behaviors realized for various parameter values. In particular, globally stable pseudo-equilibrium and globally finite time stable canard cycle are shown to exist in some ranges of parameter values, which cannot be achieved in classical model. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems. [Copyright &y& Elsevier]

Abstract: In this work, we consider a variational restoration model for multiplicative noise removal problem. By using a maximum a posteriori estimator, we propose a strictly convex objective functional whose minimizer corresponds to the denoised image we want to recover. We incorporate the anisotropic total variation regularization in the objective functional in order to preserve the edges well. A fast alternating minimization algorithm is established to find the minimizer of the objective functional efficiently. We also give the convergence of this minimization algorithm. A broad range of numerical results are given to prove the effectiveness of our proposed model. [Copyright &y& Elsevier]

Abstract: Many existing algorithms taking the seminorm in for regularization have achieved great success in image processing. However, this paper considers the total bounded variation regularization based approach to perform image deblurring. Based on this novel model, we introduce an extended split Bregman iteration to obtain the optimum solution quickly. We also provide the rigorous convergence analysis of the iterative algorithm here. Compared with the results of the ROF method, numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a ring of identical neurons with self-feedback and delays. Based on the normal form approach and the center manifold theory, we derive some formula to determine the direction of Hopf bifurcation and stability of the Hopf bifurcated synchronous periodic orbits, phase-locked oscillatory waves, standing waves, mirror-reflecting waves, and so on. In addition, under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation. Despite the fact that the slowly oscillatory synchronous periodic solution of the scalar equation is stable, we show that the corresponding synchronized periodic solution is unstable if the number of the neurons is large or arbitrary even. [Copyright &y& Elsevier]

Abstract: In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form [Copyright &y& Elsevier]

Abstract: In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Rayleigh equation with a deviating argument of the form [Copyright &y& Elsevier]

Abstract: Some sufficient conditions are obtained for the existence and global exponential stability of a periodic solution to the general bidirectional associative memory (BAM) neural networks with distributed delays by using the continuation theorem of Mawhin''s coincidence degree theory and the Lyapunov functional method and the Young''s inequality technique. These results are helpful for designing a globally exponentially stable and periodic oscillatory BAM neural network, and the conditions can be easily verified and be applied in practice. An example is also given to illustrate our results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction–diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption. [Copyright &y& Elsevier]

Abstract: By applying the continuation theorem of coincidence degree theory, we establish the existence of 2π-periodic solutions for a class of nonlinear nth order differential equations with delays. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a class of pseudo monotone semiflows, which only enjoy some weak monotonicity properties and are defined on product-ordered topological spaces. Under certain conditions, several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium, which improve and extend some corresponding results already known. Some applications to delay differential equations are presented. [Copyright &y& Elsevier]

Consider the retarded Lie´nard equation x¨+f1(x)x˙+f2(x)x˙2+gx(t−h)=e(t), where h is a nonnegative constant, f1, f2, and g are continuous functions on R=(−∞,+∞), and e(t) is a continuous function on R+=[0,+∞). We obtain some new sufficient conditions, as well as some new necessary and sufficient conditions for all solutions and their derivatives to be bounded, which substantially extend and improve some important results in the literature. [Copyright &y& Elsevier]

We consider an artificial neural network where the signal transmission is of a digital (McCulloch–Pitts) nature and is delayed due to the finite switching speed of neurons (amplifiers). For a particular connection topology, we show that all solutions starting from nonoscillatory initial states will be eventually synchronized and stabilized at a unique limit cycle, and hence such a network can be used as a synchronized oscillator. [Copyright &y& Elsevier]

Abstract The objective of this paper is to study the number and stability of limit cycles for planar piecewise linear (PWL) systems of node–saddle type with two linear regions. Firstly, we give a thorough analysis of limit cycles for Liénard PWL systems of this type, proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle. These conditions can be easily verified directly according to the parameters in the systems, and play an important role in giving birth to two limit cycles for general PWL systems. In this step, the tool of a Bendixon-like theorem is successfully employed to derive the existence of a limit cycle. Secondly, making use of the results gained in the first step, we obtain parameter regions where the general PWL systems have at least one, at least two and no limit cycles respectively. In addition for the general PWL systems, some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively. Finally, some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles. Highlights • The piecewise linear (PWL) system is of node–saddle type. • A necessary and sufficient condition is presented for the uniqueness of a limit cycle for Liénard PWL systems. • The number of limit cycles are given in different parameter regions for general PWL systems. • A Poincaré–Bendixon-like theorem for Filippov systems is successfully employed to obtain the existence of a limit cycle. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept named the weak flocking behavior. That is, if the complex system has the weak flocking property, the agents do not need have the same velocities to keep together when t → ∞. This shows the biggest difference between weak flocking behavior and the flocking behavior. The advantage of this concept is that its practical significance—the agents don't need always keep in step in lots of the complex systems but they still keep together. Then, we propose a weak flocking model with two agents and study two different mechanisms for this model. Moreover, we also consider a system with N agents. Under the weak link mechanism, the system with N agents can keep weak flocking. [ABSTRACT FROM AUTHOR]