A Mechanical Approach to the Dynamical Behavior of Non-Autonomous Van der Pol Equation

A mechanical reasoning method,based on Ritt-Wu’s method,for analytic equa-lities is developed to obtain a result on the Lyapunov characteristic exponent(LCE)ofthe following non-autonomous Van der Pol system     

Exact Vacuum Solutions to the Einstein Equation

In this paper,the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation.This procedure of resolution is based on a canonical form of the metric.According to this procedure,the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution.     

Chaos,feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit

In this work, stability analysis of the fractional-order modified Autonomous Van der Pol–Duffing (MAVPD) circuit is studied using the fractional Routh–Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, the fractional Routh–Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh–Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.     

Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère Equation

By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the a...     

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.     

Lie Symmetry Analysis and Exact Solutions for?the?Extended mKdV Equation

By using Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation, the all exact solutions based on the Lie group method are given. Especially, the bifurcations and traveling wave solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlinear science.     

Exact Periodic Solutions to Generalized BBM Equation and Relevant Conclusions

In this paper, we consider generalized BBM equation with nonlinear terms of high order.In the case of p=1/2, p=1 and p=2, the exact periodic solutions to G-BBM equation are obtained by means of proper transformation, which degrades the order of nonlinear terms. And we prove that if p ≠ 1/2,p ≠ 1 or p ≠ 2, G-BBM equation does not exist this kind of periodic solution.     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

联合Duffing方程和Van der Pol方程的非线性分数阶微分方程

本文研究了Adomian分解方法在非线性分数阶微分方程求解中的应用. 利用Riemann-Liouville分数阶导数和Adomian分解方法, 将Duffing方程和Van der Pol方程联合在一个分数阶方程中,并获得了此方程的解析近似解.     

Chaos generalized synchronization of new Mathieu–Van der Pol systems with new Duffing–Van der Pol systems as functional system by GYC partial region stability theory

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve generalized chaos synchronization. via using the GYC partial region stability theory, the new Lyapunov function used is a simple linear homogeneous function of states and the lower order controllers are much more simple and introduce less simulation error. Numerical simulations are given for new Mathieu–Van der Pol system and new Duffing–Van der Pol system to show the effectiveness of this strategy.     

Hydrodynamic Coherence and Vortex Solutions of the Euler–Helmholtz Equation

The form of the general solution of the steady-state Euler–Helmholtz equation (reducible to the Joyce–Montgomery one) in arbitrary domains on the plane is considered. This equation describes the dynamics of vortex hydrodynamic structures.     

Analytic Solutions of Klein-gordon Equation and Generalized Schrdinger Equation

Inmanyapproximationcases,wecansumlotsofphysicalphenomenonsuptoKlein_gor donequationutt- (uxx+uyy) +α2 u +g(uu )u =0 ,(1 )whereg(z)isafunctionofzandu iscojugatecomplexnumberofu .ManyscholarshavebeeninterestedinanalyticsolutionofEq .(1 ) .Sinceitisestablihed .Papers [1 ,2 ]and [3]viewedrespecrtivelyaccuratesolutionandanalyticsolutionofEq .(1 )wheng(z) =βz .Inpa per [4] ,weobtainedaclassofanalyticsolutionofEq .(1 )wheng(z) =βz1 /k,k∈R+ andaclassofanalyticsolutionofgeneralizedSchrodingerequ…     

Bifurcation of Periodic Orbits and Chaos in Duffing Equation

Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.     

On the Existence of Harmonic Solutions of the Forced Lienard Equation

In this paper, we present a method to study the existence of the harmonicsolutions of the forcde Lienard equation′s equivalent system     

Folding of Set-Theoretical Solutions of the Yang–Baxter Equation

We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang–Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution.     

Regularity of Solutions of the Cahn-Hilliard Equation with Non-constant Mobility

In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

On the Existence of Solutions to the Operator Riccati Equation and the tan Θ Theorem

Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d > 0 be the distance between the spectra of A and C. Under these assumptions we prove that the best possible value of the constant c in the condition guaranteeing the existence of a (bounded) solution to the operator Riccati equation XA–CX+XBX = B* is equal to We also prove an extension of the Davis-Kahan tan theorem and provide a sharp estimate for the norm of the solution to the Riccati equation. If C is bounded, we prove, in addition, that the solution X is a strict contraction if B satisfies the condition and that this condition is optimal.