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.               

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''.     

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.     

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.     

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.     

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.     

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.     

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 \(\leq \delta \leq 1/2 \).     

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.     

Unique Solvability of a Linear Parabolic Problem with Nonlocal Time Data

Siberian Mathematical Journal - We consider a problem for a first-order differential equation in a Hilbert space. We replace the initial data with some condition that...     

Correction to: The invariant subspaces of the shift plus integer multiple of the Volterra operator on Hardy spaces

Archiv der Mathematik - We note that the main results (Proposition 1, Proposition 2, and Theorem 1) in [1] are only true for.     

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...     

On a new model of two-patch predator-prey system with migration of both species

In this paper we explore the dynamics of predator-prey in a two patch system. The two patches of the system are coupled with both the migration of the predator and the prey. The purpose of this exploration is to find upper and lower bounds for the populations and get an insight on the different possibilities with the three types of Holling functional responses. Also we discuss the stability and instability of the equilibrium solutions found in earlier papers. Numerical simulations are provided to graphically demonstrate the population dynamics of the system.     

Cesàro matrix and (1, 1; r)-convex sequences

Positivity - This paper deals with (1, 1; r)-convexity of sequences. First, we prove several results on the sets of (1, 1; r)-convex sequences for various values...     

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.     

Robust synchronization of parametrized nonautonomous discrete systems with applications to communication systems

We study synchronization of a coupled discrete system consisting of a Master System and a Slave System. The Master System usually exhibits chaotic or complicated behavior and transmits a signal with a chaotic component to the Slave System. The Slave System then recovers the original signal and removes the chaotic component. To ensure secured communication, the Master and the Slave systems must synchronize independent of the variation of the systems parameters and initial conditions. Here we develop a general approach and obtain some general results for synchronization of such coupled systems naturally arising from discretization of well-know continuous systems, and we illustrate general results with two specific examples: the discretized Lorenz system and a discretized nonlinear oscillator. We also present some simulations using MatLab to illustrate our discussions.     

A Class of Invertible Subelliptic Operators in S(m,g)-Classes

Results in Mathematics - Given a self-adjoint second order differential operator L with positive characteristic and subellipticity of order 1 ≤ τ < 2....     

On the number of \(n\)-dimensional invariant spheres in polynomial vector fields of \(\mathbb{C}^{n+1}\)

We study the polynomial vector fields \(\mathcal{X}= \displaystyle \sum_{i=1}^{n+1} P_i(x_1,\ldots,x_{n+1}) \frac{\partial}{\partial x_i}\) in \(\mathbb{C}^{n+1}\) with \(n\geq 1\) . Let \(m_i\) be the degree of the polynomial \(P_i\). We call \((m_1,\ldots,m_{n+1})\) the degree of \(\mathcal{X}\). For these polynomial vector fields \(\mathcal{X}\) and in function of their degree we provide upper bounds, first for the maximal number of invariant \(n\)-dimensional spheres, and second for the maximal number of \(n\)-dimensional concentric invariant spheres.     

Identities and Quasi-Identities of Pointed Algebras

Siberian Mathematical Journal - Each pointed enrichment of an algebra can be regarded as the same algebra with an additional finite set of constant operations. An algebra...     

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 $...