Bifurcation of limit cycles and isochronous center at infinity for a class of differential systems

In this paper, we study a seventh degree polynomial differential system with full linear terms and cubic terms. The conditions of infinity to be a center and to be a fine focus of the highestorder are given and it is proved that this system has eight limit cycles in the neighborhood of infinity. Moreover, the conditions of infinity to be an isochronous center for a rational system associated the seventh degree polynomial differential system are obtained.     

On uniqueness and analyticity in thermoviscoelastic solids with voids

In this paper we consider the most general system proposed to describe the thermoviscoelasticity with voids. We study two qualitative properties of the solutions of this theory. First, we obtain a uniqueness result when we do not assume any sign to the internal energy. Second we extend some previous results and prove the analyticity of the solutions. The impossibility of localization in time of the solutions is a consequence. Last result we present corresponds to the analyticity of solutions in case that the dissipation is not very strong, but with suitable coupling terms.     

Numerical solution of fuzzy Camassa-Holm equation by using Homotopy analysis methods

In this paper, a fuzzy Camassa-Holm equation is solved by using the  homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed method are proved.     

On the integrability of the differential systems in dimension two and of the polynomial differential systems in arbitrary dimension

This is a survey on recent results providing sufficient conditions for the existence of a first integral, first for vector fields defined on real surfaces, and second for polynomial vector fields in \(R^n\) or \(C^n\) with \(n\geq 2\). We also provide an open question and some applications based on the existence of such first integrals.               

Resonances of the SD oscillator due to the discontinuous phase

Resonant phenomenon of a harmonically excited system with multiple well dynamics plays a very important role in nonlinear dynamics research. In this paper, we investigate resonant behaviors of a discontinuous forced SD system with snap-through buckling and double-well dynamics. Firstly, a discontinuous time dependent Hamiltonian is derived from the discontinuous stage of SD oscillator providing a Du±ng type nonlinearity with snap-through buckling and double-well dynamics. This system comprises two subsystems connected at x = 0, where the system is discontinuous. We construct a series of generating functions and canonical transformations to get the canonical form of the system to reveal the complicated resonant behavior of the system. Furthermore, we introduce a composed winding number to explorer the complicated resonant phenomena of the system.This formulation for resonant phenomena given in this paper generalizes the formulation of n!0 = m! in regular perturbation theory, where n and m are relative prime integers,!0 and ! are the natural frequency and external frequencies respectively. Understanding the resonant behaviors of the SD oscillator at discontinuous stage enables us to further reveal the vibrational energy transition mechanism of the smooth and discontinuous nonlinear dynamic system.     

Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model

In this article, we derive Ricker’s [22, 23] type nonlinear boundary condition for an age structured population dynamic model by using a singular perturbation. The question addressed in this paper is the convergence of the singularly perturbed system. We first obtain a finite time convergence for a fixed initial distribution. Then we focus on the convergence uniformly of the singularly perturbed system with respect to the initial distribution in bounded sets.     

Double Hopf bifurcation and chaos in Liu system with delayed feedback

In this paper, we consider the stability of equilibria, Hopf and double Hopf bifurcation in Liu system with delay feedback. Firstly, we identify the critical values for stability switches and Hopf bifurcationusing the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liusystem can be controlled to be stable at the same time, and the stable bifurcating periodic solutions occur in the neighborhood of the two equilibria at the same time. Secondly, by applying the normal form method and center manifold theory,the normal form near the double Hopf bifurcation, as well as classifications of local dynamics are analyzed. Furthermore, we give the bifurcation diagram to illustrate numerically that a family of stable periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the Liu system with delay can appear the phenomenon of ``chaos switchover''.     

Exact tail asymptotics for fluid models driven by an M/M/c queue

Queueing Systems - In this paper, we investigate exact tail asymptotics for the stationary distribution of a fluid model driven by the M / M / c queue, which is...     

Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations

This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones.  It first introduces the concept of generalized traveling wave solutions of time recurrent and space periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It then proves that in the direction of any unit vector \(\xi\), there is \(c^*(\xi)\) such that for any \(c>c^*(\xi)\), a generalized traveling wave solution in the direction of \(\xi\) with averaged propagation speed \(c\) exists. It also proves that if the time recurrent and space periodic monostable equation is indeed time periodic, then \(c^*(\xi)\) is the minimal wave speed in the direction of \(\xi\) and the generalized traveling wave solution in the direction of \(\xi\) with averaged speed \(c>c^*(\xi)\) is a periodic traveling wave solution with speed \(c\), which recovers the existing results on the existence of periodic traveling wave solutions in the direction of \(\xi\) with speed greater than the minimal speed in that direction.     

On the existence of bubble-type solutions of nonlinear singular problems

Considered in this paper is a class of singular boundary value problem, arising in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are sought: \((p(t)u0)0 = c(t)p(t)f(u); u0(0) = 0; u(+1) = L > 0\) in the half-line \([0;+1)\), where \(p(0) = 0\). We are interested in strictly increasing solutions of this problem in \([0;1)\) having just one zero in \((0;+1) \)and finite limit at zero, which has great importance in applications or pure and applied mathematics. Su±cient conditions of the existence of such solutions are obtained by applying the critical point theory and by using shooting argument [9,10] to better analysis the properties of certain solutions associated with the singular di®erential equation. To the authors' knowledge, for the first time, the above problem is dealt with when f satis¯es non-Lipschitz condition. Recent results in the literature are generalized and signi¯cantly improved.     

\( {\small \bf Communication Letter:}\) Four limit cycles in quadratic near-integrable system

In this note, we report of obtaining 4 limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least 4 limit cycles with (3.1) distribution. This result provides a positive answer to an open problem in this area.     

Explicit soliton solutions of the Kaup-Kupershmidt equation through the dynamical system approach

In this paper, we study the traveling wave solutions of the Kaup-Kupershmidt (KK) equation through using the dynamical system approach, which is an integrable fifth-order wave equation. Based on Cosgrove's work [3] and the phase analysis method of dynamical systems, infinitely many soliton solutions are presented in an explicit form. To guarantee the existence of soliton solutions, we discuss the parameters range as well as geometrical explanation of soliton solutions.     

Cauchy problem for the Zakharov System Arising from Ion-Acoustic Modes with low regularity data

We prove local well-posedness results for the Zakharov System Arising from Ion-Acoustic Modes in two spacial dimension with large initial data in low regularity Sobolev space \(   (\dot{H}^1 \bigcup H^{\frac{1}{2}})\times L^2 \times H^{-1}  \).  Using ”derivative sharing”, the local well-posedness results in \( (\dot{H}^1 \bigcup H^{\frac{1}{2}-\delta})\times H^{\delta} \times H^{-1+\delta}\)  are also obtained, for any  0 \(\leq \delta \leq 1/2 \).     

Cross-diffusion induced instability and stability in reaction-diffusion systems

In a reaction-diffusion system, diffusion can induce the instability of a uniform equilibrium  which is stable with respect to a constant perturbation, as shown by Turing in 1950s. We show that cross-diffusion can destabilize  a uniform equilibrium  which is stable for the kinetic and self-diffusion reaction systems; on the other hand, cross-diffusion can also stabilize  a uniform equilibrium which is stable for the kinetic system but unstable for the self-diffusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.     

On the Diophantine equation $$x^2+3^a41^b=y^n $$ x 2 + 3 a 41 b = y n

Periodica Mathematica Hungarica - In this paper we find all positive integer solutions (x, y, n, a, b) of the equation in the title for non negative integers a...     

Formulas for Calculating the $ 3j $ -Symbols of the Representations of the Lie Algebra $ {\mathfrak{gl}}_{3} $ for the Gelfand–Tsetlin Bases

Siberian Mathematical Journal - We give a simple explicit formula for an arbitrary $ 3j $ -symbol for the Lie algebra  $ {\mathfrak{gl}}_{3} $ . The symbol is...     

Heights of Minor Faces in 3-Polytopes

Siberian Mathematical Journal - Each 3-polytope has obviously a face  $ f $ of degree $ d(f) $ at most 5 which is called minor. The height $ h(f) $ of $ f $ is...     

Random attractor of stochastic Zakharov lattice system

In this paper, we consider a lattice system of stochastic Zakharov equation with white noise. We first show that the solutions of the system determine a continuous random dynamical system with random absorbing set. And then we prove the random  asymptotic compact on the random absorbing set. Finally, we obtain the existence of a random attractor for the system.     

Locally Finite Groups with Prescribed Structure of Finite Subgroups

Siberian Mathematical Journal - Let  $ {\mathfrak{M}} $ be a set of finite groups. Given a group  $ G $ , denote the set of all subgroups of  $ G $...     

Queue length asymptotics for the multiple-server queue with heavy-tailed Weibull service times

Queueing Systems - We study the occurrence of large queue lengths in the GI / GI / d queue with heavy-tailed Weibull-type service times. Our analysis hinges on a...