Numerical Continuation of Resonances and Bound States in Coupled Channel Schr¨odinger Equations

In this contribution,we introduce numerical continuation methods and bifurcation theory,techniques which find their roots in the study of dynamical systems,to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schrodinger equation.We extend previous work on the subject[1]to systems of coupled equations.Bound and resonant states of the Schrodinger equation can be determined through the poles of the S-matrix,a quantity that can be derived from the asymptotic form of the wave function.We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties,thus making it amenable to numerical continuation.This allows us to automate the process of tracking bound and resonant states when parameters in the Schrodinger equation are varied.We have applied this approach to a number of model problems with satisfying results.

Existence and Uniqueness of 2π-Periodic Solution About Duffing Equation and Its Numerical Method

A new provement of the existence and uniqueness about periodic boundary value Duffing equation is established by using global inverse function theorem. An algorithm for solving differential equation that has a large convergence domain is given. Finally, a numerical example is given.

Nonlinear aeroelastic analysis of a twin-tail boom UAV with rudder freeplay nonlinearities

The issue of nonlinear structural freeplays in aircraft has always been a significant con-cern.This study investigates the aeroelastic characteristics of a twin-tail boom Unmanned Aerial Vehicle(UAV)with simultaneous freeplay nonlinearity in its left and right rudders.A comprehen-sive Limit Cycle Oscillation(LCO)solution route is proposed for complex aircraft with multiple freeplays,which can consider both accuracy and effciency.For the first time,this study reveals the unique LCO characteristics exhibited by twin-tail boom UAVs with rudder freeplays and pro-vides simulations and explanations of interesting phenomena observed during actual flight.The governing equations are established using the free-interface component mode synthesis method,and the LCOs of the system are mainly solved through the improved time-domain numerical con-tinuation method and frequency-domain numerical continuation method.Furthermore,the study investigates the influence of the left and right rudder freeplay size ratio on the LCO characteristics.The results demonstrate that the twin-tail boom UAV exhibits two stable LCO types:close and dif-fering left and right rudder amplitudes.The proposed method successfully describes the complete LCO behaviors of the system.Overall,this study makes significant contributions to our understand-ing of the aeroelastic behavior of twin-tail boom UAVs with rudder freeplays.

Bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals

This study focuses on the stability and local bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals analytically,and numerically.The analytical results are obtained using thenormal form technique and numerical results are obtained using the numerical continuation method.For this model,a number of bifurcations are studied,including the transcritical(pitchfork)and fip bifurcations,the Neimark-Sacker(NS)bifurcations,and the strong resonance bifurcations.We especially determine the dynamical behaviors of the model for higher iterations up to fourth-order.Numerical simulation is employed to present a closed invariant curve emerging about an NS point,and its breaking down to several closed invariant curves and eventuality giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

Parameter effects on high-speed UAV ground directional stability using bifurcation analysis

A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle(UAV)to veer off the runway.In order to investigate the combined effects of the key structural and operational parameters on the UAV ground directional stability from a global perspective,a fully parameterized mathematical high-speed UAV ground nonlinear dynamic model is developed considering several nonlinear factors.The bifurcation analysis procedure of a UAV ground steering system is introduced,following which the simulation efficiency is greatly improved comparing with the time-domain simulation method.Then the numerical continuation method is employed to investigate the influence of the nose wheel steering angle and the global stability region is obtained.The bifurcation parameter plane is divided into several parts with different stability properties by the saddle nodes and the Hopf bifurcation points.We find that the UAV motion states will never cross the bifurcation curve in the nonlinear system.Also,the dual-parameter bifurcation analyses are presented to give a complete description of the possible steering performance.It is also found that BT bifurcation appears when the UAV initial rectilinear velocity and the tire frictional coefficient vary.In addition,results indicate that the influence of tire frictional coefficient has an opposite trend to the influence of initial rectilinear velocity.Overall,using bifurcation analysis method to identify the parameter regions of a UAV nonlinear ground dynamic system helps to improve the development efficiency and quality during UAV designing phase.

A Fast Local Level Set Method for Inverse Gravimetry

We propose a fast local level set method for the inverse problem of gravimetry.The theoretical foundation for our approach is based on the following uniqueness result:if an open set D is star-shaped or x3-convex with respect to its center of gravity,then its exterior potential uniquely determines the open set D.To achieve this purpose constructively,the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes.To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set.The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domainΩ.To overcome this difficulty,we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain containing the unknown set D.The third challenge is how to speed up the level set inversion process.Based on some features of the underlying inverse gravimetry problem such as the potential density being constant inside the unknown domain,we propose a novel numerical approach which is able to take advantage of these features so that the computational speed is accelerated by an order of magnitude.We carry out numerical experiments for both two-and three-dimensional cases to demonstrate the effectiveness of the new algorithm.

Bifurcation analysis of dual-sidestay landing gear locking performance considering joint clearance

Sidestay lock mechanism is an important part of landing gear system,and the locking performance can be analyzed based on changes in its stability.However,during numerical continuation analysis of fully-rigid dual-sidestay landing gear without clearance,it has been found that the appearance of bifurcation points does not necessarily imply that both sidestay links can be locked synchronously.This problem reveals the limitations of fully-rigid model with ideally-articulated in solving dual-sidestay mechanisms with extremely high motion sensitivity.Therefore,this study proposes a bifurcation analysis method for synchronous locking of dual-sidestay landing gears,which takes into consideration the joint clearance.For in-depth analysis of this problem,we initially build kinematic and mechanical models of a landing gear mechanism that consider joint clearance.Then,the models are solved based on continuation.The fundamental causes of synchronous locking are discussed in detail,and the number of bifurcation points is found to be closely related to whether the landing gear is completely locked.Finally,the effects of structural parameters on the synchronous locking are analyzed,and the feasible region of parameters satisfying synchronous locking condition is given,which agrees well with the test results.

Exact Bivariate Polynomial Factorization over Q by Approximation of Roots

Factorization of polynomials is one of the foundations of symbolic computation.Its applications arise in numerous branches of mathematics and other sciences.However,the present advanced programming languages such as C++ and J++,do not support symbolic computation directly.Hence,it leads to difficulties in applying factorization in engineering fields.In this paper,the authors present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients.The proposed method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library.In addition,the numerical computation part often only requires double precision and is easily parallelizable.

Multiphase Flow and Thermo-Mechanical Behaviors of Solidifying Shell in Continuous Casting Mold

The metallurgical phenomena occurring in the continuous casting mold have a significant influence on the performance and the quality of steel product.The multiphase flow phenomena of molten steel,steel/slag interface and gas bubbles in the slab continuous casting mold were described by numerical simulation,and the effect of electromagnetic brake（EMBR） and argon gas blowing on the process were investigated.The relationship between wavy fluctuation height near meniscus and the level fluctuation index F,which reflects the situation of mold flux entrapment,was clarified.Moreover,based on a microsegregation model of solute elements in mushy zone with δ/γ transformation and a thermo-mechanical coupling finite element model of shell solidification,the thermal and mechanical behaviors of solidifying shell including the dynamic distribution laws of air gap and mold flux,temperature and stress of shell in slab continuous casting mold were described.

Giant Flexoelectric Effect in Snapping Surfaces Enhanced by Graded Stiffness

Flexoelectricity is present in nonuniformly deformed dielectric materials and has size-dependent properties, making it useful for microelectromechanical systems. Flexoelectricity is small compared to piezoelectricity;therefore, producing a large-scale flexoelectric effect is of great interest. In this paper, we explore a way to enhance the flexoelectric effect by utilizing the snap-through instability and a stiffness gradient present along the length of a curved dielectric plate. To analyze the effect of stiffness profiles on the plate, we employ numerical parameter continuation. Our analysis reveals a nonlinear relationship between the effective electromechanical coupling coefficient and the gradient of Young’s modulus. Moreover, we demonstrate that the quadratic profile is more advantageous than the linear profile. For a dielectric plate with a quadratic profile and a modulus gradient of − 0.9, the effective coefficient can reach as high as 15.74 pC/N, which is over three times the conventional coupling coefficient of piezoelectric material. This paper contributes to our understanding of the amplification of flexoelectric effects by harnessing snapping surfaces and stiffness gradient design.