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ISSN 2562-2854 (print)

ISSN 2562-2862 (online)

 

We develop a general symbolic dynamics framework to examine the dynamics of an analogue of the integrate-and-fire neuron model of recurrent inhibitory loops with delayed feedback, which incorporates the firing procedure and absolute refractoriness. We first show that the interaction of the delay, the inhibitory feedback and the absolute refractoriness can generate three basic types of oscillations, and these oscillations can be pinned together to form interesting coexisting periodic patterns in the case of short feedback duration. We then develop a natural symbolic dynamics formulation for the segmentation of a typical trajectory in terms of the basic oscillatory patterns, and use this to derive general principles that determine whether a periodic pattern can and should occur.

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In this survey we discuss some recent progress on the ideal free distribution theory in patch models, with the emphasis on two patches. We show that dispersal strategies leading to the ideal free distributions of organisms are generally evolutionarily stable. We will also study the existence of evolutionarily stable dispersal strategies when dispersal strategies do not lead to the ideal free distributions. Applications to some river models are given.

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In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form x′′ + cx′ + a(t)x + b(t)x3 = h(t), where c is a real parameter, a(t), b(t) and h(t) are continuous T–periodic functions. Our results are proved using three different results on the averaging theory of first order.

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By virtue of the KP hierarchy reduction method, the N-dark soliton solutions of the multi-component Maccari system are constructed. Taking the coupled Maccari system for instance, the N-dark soliton solutions are further obtained in terms of determinants. In addition, in contrast with bright- bright soliton collisions, the dynamical analysis shows that the collisions of dark-dark solitons are elastic and there is no energy exchange among solitons in different components. What’s more, we also investigate the dark-dark soliton bound states including stationary and moving ones.

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Han et al. [Han et al., Polynomial Hamiltonian systems with a nilpotent critical point, J. Adv. Space Res. 2010, 46, 521–525] successfully studied local behavior of an isolated nilpotent critical point for polynomial Hamiltonian systems. In this paper, we extend the previous result by analyzing the global phase portraits of polynomial Hamiltonian systems. We provide 12 non-topological equivalent classes of global phase portraits in the Poincaré disk of cubic polynomial Hamiltonian systems with a nilpotent center or saddle at origin under some conditions of symmetry.

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In this paper, first of all we give the necessary and sufficient conditions of the center of a class of planar quintic differential systems by using reflecting function method, and provide a simple proof of this results. Secondly, We use the reflecting integral to research the equivalence of the Abel equation and some complicated equations and derive their center conditions and discuss their integrability.

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This paper presents the existence of solutions for a class of Cauchy problems with integral condition for impulsive fractional integro-differential equations. Based on definition of solution for impulsive fractional integro- differential equations, the existence theorems of solutions of fractional differential equation are obtained by applying fixed point methods. Finally, three examples are given to demonstrate the feasibility of the obtained results.

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Chaotic behavior for the Duffing-van der Pol (DVP) oscillator is investigated both analytically and numerically. The critical curves separating the chaotic and non-chaotic regions are obtained. The chaotic feature on the system parameters are discussed in detail. The conditions for subharmonic bifurcations are also obtained. Numerical results are given, which verify the analytical ones.

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In this paper, we study the solitary waves for the generalized nonautonomous dual-power nonlinear Schrödinger equations (DPNLS) with variable coefficients, which could be used to describe the saturation of the nonlinear refractive index and the solitons in photovoltaic-photorefractive materials such as LiNbO3, as well as many nonlinear optics problems. We generalize an explicit similarity transformation, which maps generalized nonautonomous DPNLS equations into ordinary autonomous DPNLS. Moreover, solitary waves of two concrete equations with space-quadratic potential and optical super-lattice potential are investigated.

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A class of boundary value problems (BVPs) of even order neutral partial functional differential equations with continuous distribution delay and nonlinear diffusion term are studied. By applying the integral average and Riccati’s method, the high-dimensional oscillatory problems are changed into the one-dimensional ones, and some new sufficient conditions are obtained for oscillation of all solutions of such boundary value problems under first boundary condition. The results generalize and improve some results of the latest literature.

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