Two-Dimensional Manifolds with Computation V-Function of ODE Systems

The computation of stable or unstable manifold of two-dimensional is developed, which is an efficient method in studying stable structure analysis of system character geometrically. The Lorentz stable manifold is computed by the fixed arclength method and the hyperbolic equilibrium is a saddle. The two-dimensional stable structure of Lorentz manifold is significant in people’s usual view. We also introduce the V-function to compute the V-manifold correspondingly. The defined V-function is smooth in the unstable direction of the manifold. Especially, the routh to period-doubling attractor on manifold surface is discussed too.

Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

Galerkin-based quasi-smooth manifold element(QSME)method for anisotropic heat conduction problems in composites with complex geometry

The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method(FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element(QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional(2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity.The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom(DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

Stability for manifolds of equilibrium states of generalized Birkhoff system

Stability for the manifolds of equilibrium states of a generalized Birkhoff system is studied. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to the generalized BirkhoiYian system and two propositions on the stability of the manifolds of equilibrium states of the system are obtained. An example is given to illustrate the application of the results.

Gating system optimization of low pressure casting A356 aluminum alloy intake manifold based on numerical simulation

To eliminate the shrinkage porosity in low pressure casting of an A356 aluminum alloy intake manifold casting, numerical simulation on fi lling and solidifi cation processes of the casting was carried out using the ProCAST software. The gating system of the casting is optimized according to the simulation results. Results show that when the gating system consists of only one sprue, the fi lling of the molten metal is not stable; and the casting does not follow the sequence solidifi cation, and many shrinkage porosities are observed through the casting. After the gating system is improved by adding one runner and two in-gates, the fi lling time is prolonged from 4.0 s to 4.5 s, the fi lling of molten metal becomes stable, but this casting does not follow the sequence solidifi cation either. Some shrinkage porosity is also observed in the hot spots of the casting. When the gating system was further improved by adding risers and chill to the hot spots of the casting, the shrinkage porosity defects were eliminated completely. Finally, by using the optimized gating system the A356 aluminum alloy intake manifold casting with integrated shape and smooth surface as well as dense microstructure was successfully produced.

Nonlinear consensus protocols for multi-agent systems based on centre manifold reduction

Nonlinear consensus protocols for dynamic directed networks of multi-agent systems with fixed and switching topologies are investigated separately in this paper. Based on the centre manifold reduction technique, nonlinear consensus protocols are presented. We prove that a group of agents can reach a β-consensus, the value of which is the group decision value varying from the minimum and the maximum values of the initial states of the agents. Moreover, we derive the conditions to guarantee that all the agents reach a β-consensus on a desired group decision value. Finally, a simulation study concerning the vertical alignment manoeuvere of a team of unmanned air vehicles is performed. Simulation results show that the nonlinear consensus protocols proposed are more effective than the linear protocols for the formation control of the agents and they are an improvement over existing protocols.

Inertial Manifolds for 2D Generalized MHD System

In this paper, we prove the existence of inertial manifolds for 2D generalized MHD system under the spectral gap condition.

Evaluation of the submarine debris-flow hazard risks to planned subsea pipeline systems: a case study in the Qiongdongnan Basin, South China Sea

The ever-increasing deepwater oil and gas development in the Qiongdongnan Basin,South China Sea has initiated the need to evaluate submarine debris-flow hazard risks to seafloor infrastructures.This paper presents a case study on evaluating the debris-flow hazard risks to the planned pipeline systems in this region.We used a numerical model to perform simulations to support this quantitative evaluation.First,one relict failure interpreted across the development site was simulated.The back-analysis modeling was used to validate the applicability of the rheological parameters.Then,this model was applied to forecast the runout behaviors of future debris flows originating from the unstable upslope regions considered to be the most critical to the pipeline systems surrounding the Manifolds A and B.The model results showed that the potential debris-flow hazard risks rely on the location of structures and the selection of rheological parameters.For the Manifold B and connected pipeline systems,because of their remote distances away from unstable canyon flanks,the potential debris flows impose few risks.However,the pipeline systems around the Manifold A are exposed to significant hazard risks from future debris flows with selected rheological parameters.These results are beneficial for the design of a more resilient pipeline route in consideration of future debris-flow hazard risks.

THE INVARIANT MANIFOLD METHOD AND THE CONTROLLABILITY OF NONLINEAR CONTROL SYSTEM

The problem of controllability of nonlinear control system is a significant field which has an extensive prospect of application. A. M. Kovalev of Ukraine Academy of Science applied the oriented manifold method developed in dynamics:of rigid body to nonlinear control system for the first time,and obtained a series of efficient results. Based on Kovalev's oriented manifold method, firstly, by invariant manifold method the problem of controllability of nonlinear control system was studied and the necessary condition of the controllability of a kind of affine nonlinear system was given out. Then the realization of the necessary condition wars discussed. At last, the motion of a rigid body with two rotors,vas, investigated and the necessary condition which is satisfied by this system,vas proved.

INVARIANT SUB-MANIFOLDS AND MODES OF NONLINEAR AUTONOMOUS SYSTEMS

A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.

MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD

For a given T>0,we prove,under the global ARS-condition and using the Nehari manifold method,the existence of a T-periodic solution having the W-symmetry introduced in[21],for the hamiltonian system z+V'(z)=0,z∈R^N,N∈N^*.Moreover,such a solution is shown to have T as a minimal period without relaying to any index theory.A multiplicity result is also proved under the same condition.

SLOW MANIFOLD AND PARAMETER ESTIMATION FOR A NONLOCAL FAST-SLOW DYNAMICAL SYSTEM WITH BROWNIAN MOTION

We establish a slow manifold for a fast-slow dynamical system with anomalous diffusion,where both fast and slow components are influenced by white noise.Furthermore,we verify the exponential tracking property for the established random slow manifold,which leads to a lower dimensional reduced system.Alongside this we consider a parameter estimation method for a nonlocal fast-slow stochastic dynamical system,where only the slow component is observable.In terms of quantifying parameters in stochastic evolutionary systems,the provided method offers the advantage of dimension reduction.

ITERATIVE ALGORITHMS FOR SYSTEM OF VARIATIONAL INCLUSIONS IN HADAMARD MANIFOLDS

In this paper,we consider system of variational inclusions and its several spacial cases,namely,alternating point problems,system of variational inequalities,etc.,in the setting of Hadamard manifolds.We propose an iterative algorithm for solving system of variational inclusions and study its convergence analysis.Several special cases of the proposed algorithm and convergence result are also presented.We present application to constraints minimization problems for bifunctions in the setting of Hadamard manifolds.At the end,we illustrate proposed algorithms and convergence analysis by a numerical example.The algorithms and convergence results of this paper either improve or extend known algorithms and convergence results from linear structure to Hadamard manifolds.

INERTIAL MANIFOLDS FOR NONAUTONOMOUS INFINITEDIMENSIONAL DYNAMICAL SYSTEMS

In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is discussed Under the spectral gap condition, It is proved that there exist inertial manifolds for a class of nonautonomous evolution equations.

EAPPROXIMATE INERTIAL MANIFOLDS FOR THE SYSTEM OF THE J-J EQUATIONS

In this paper foe Liapunov functionals has been constructed.the decay property of the high dimensional modes of the J-J equations in the Josephson junctions is obtained,and thus the approxtmate inertial manifolds are given.

Discovering exact,gauge-invariant,local energy–momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas.However,exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite continuous effort.Without such local conservation laws,energy and momentum can be instantaneously transported across spacetime,which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory.The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds.To overcome this difficulty,we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact,local conservation laws,in particular the energy–momentum conservation laws,for the electromagnetic gyrokinetic system.A weak Euler–Lagrange(EL)equation is established to replace the standard EL equation for the particles.It is discovered that an induced weak EL current enters the local conservation laws,and it is the new physics captured by the high-order field theory on heterogeneous manifolds.A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy–momentum conservation laws electromagnetic gauge-invariant.

INVARIANT MANIFOLDS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.

INVARIANT MANIFOLDS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.

Finslerian Mechanical Systems on Manifolds

The goal of this paper is to present Finslerian mechanical systems on manifolds.In conclusion,some differential geometrical and physical results on the related mechanic systems have been given.