SOME EXTENDED RESULTS OF“SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY

In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed.We assume that the free surface is almost periodic in the horizontal direction.Using conformal mappings,one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition.By virtue of the Hilbert transform,the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable.The bifurcation theory ensures that we can obtain an existence result.Our existence result generalizes and covers the recent result in[15].Moreover,our result implies a non-uniqueness result at the same bifurcation point.

Bifurcation analysis and exact traveling wave solutions for (2+1)-dimensional generalized modified dispersive water wave equation

We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane system corresponding to the GMDWW equation. By using the special orbits in the phase portraits, we analyze the existence of the traveling wave solutions. When some parameter takes special values, we obtain abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions for the GMDWW equation.

Topological aspect of vortex lines in two-dimensional Gross-Pitaevskii theory

Using the C-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross Pitaevskii theory in （3＋l）-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the function indicates that the vortices can only be generated from the zero points of Ф and are quantized in terms of the Hopf indices and Brouwer degrees. The C-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.

External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip

In this paper, external bifurcations of heterodimensional cycles connecting three saddle points with one orbit flip, in the shape of “∞”, are studied in three-dimensional vector field. We construct a poincaré return map between returning points in a transverse section by establishing a locally active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the bifurcation equations are obtained under different conditions. Near the double heterodimensional cycles, the authors prove the preservation of “∞”-shape double heterodimensional cycles and the existence of the second and third shape heterodimensional cycle and a large 1-heteroclinic cycle connecting with P1 and P3. The coexistence of a 1-fold large 1-heteroclinic cycle and the “∞”-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.

Bifurcation on synchronous full annular rub of rigid-rotor elastic-support system

An aero-engine rotor system is simplified as an unsymmetrical-rigid-rotor with nonlinear-elastic-support based on its characteristics. Governing equations of the rubbing system, obtained from the Lagrange equation, are solved by the averaging method to find the bifurcation equations. Then, according to the two-dimensional constraint bi- furcation theory, transition sets and bifurcation diagrams of the system with and without rubbing are given to study the influence of system eccentricity and damping on the bi- furcation behaviors, respectively. Finally, according to the Lyapunov stability theory, the stability region of the steady-state rubbing solution, the boundary of static bifurcation, and the Hopf bifurcation are determined to discuss the influence of system parameters on the evolution of system motion. The results may provide some references for the designer in aero rotor systems.

Scale-up and thermal stability analysis of fluidized bed reactors for ethylene polymerization

A set of hydrodynamic similarity laws is applied to the scale-up of ethylene polymerization fluidized bed reactors(FBRs)under the condensed mode operation.The thermal stability of open-loop controlled FBRs is investigated by the homotopy continuation method.And the Hopf bifurcation point is selected as an index of the thermal stability similarity.The simulation results show the similarity in state variables,operation parameters,the space-time yield(STY),and the thermal stability of FBRs with different scales.Furthermore,the thermal stability behaviors and similarity of the closed-loop controlled FBRs with different scales are analyzed.The observed similar trend of Hopf bifurcation curves reveals the similarity in the thermal stability of closed-loop controlled FBRs with different scaling ratios.In general,the results of the thermal stability similarity confirm that the hydrodynamics scaling laws proposed in the work are applicable to the scale-up of FBRs under the condensed mode operation.

Adaptivity and A Posteriori Error Control for Bifurcation Problems Ⅰ:The Bratu Problem

This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied.In particular,we study in detail the numerical approximation of the Bratu problem,based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method.A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual(DWR)approach.Numerical experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.

Modeling the transmission dynamics of a time-delayed epidemic model with saturated treatment rate

In this paper,an attempt has been made to explore a new delayed epidemiological model assuming that the disease is transmitted among the susceptible population and possessing nonlinear incidence function along with a saturated treatment rate.Due attention is paid to the positivity and boundedness of the solutions and the bifurcation of the dynamical system as well.Basic reproduction number is being calculated,and considering the latent period as a bifurcation parameter,it has been examined that a Hopf-bifurcation occurs near the endemic equilibrium point while the parameter attains critical values.We have also discussed the stability and direction of Hopf-bifurcation near the endemic equilibrium point,the global stability analysis and the optimal control theory.We conclude that the system reveals chaotic dynamics through a specific time-delay value.Numerical simulations are being performed in order to explain the accuracy and effectiveness of the acquired theoretical results.

LIMIT CYCLES FOR A CLASS OF NONPOLYNOMIAL PLANAR VECTOR FIELDS (II)

In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.