Singular Points, Closed Orbits, Stable Manifolds and Unstable Manifolds of Second Order Autonomous Birkhoff Systems

Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and Conclusion The criteria for singular points, closed orbits and hyperbolic equilibrium points of a second order autonomous Birkhoff system are given. Moreover the stability of equilibria, stable manifolds and unstable manifolds are obtained.

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.

All-Electric Subsea Control Systems and the Effects on Subsea Manifold Valves

Subsea development is moving constantly toward simplification,digitalization,and cost-out strategies because the exploration and production of hydrocarbons are moving toward deeper and remote sea water areas.Usage of all-electric subsea technology instead of hydraulic technology is growing and will be the future of subsea systems due to the former’s environmental and functional advantages and reduced costs.The benefits of all-electric subsea systems are health,safety,and environment(HSE)and improved reliability,flexibility,and functionality compared with traditional hydraulic-electrical systems.Existing electrohydraulic technology for a typical subsea system,hydraulic and electric actuators,and subsea manifold valves including valve types and selection philosophy have been reviewed in this paper.Some major worldwide oil companies such as Equinor and Schlumberger have successful experiences with subsea electric actuators.Considering the benefits of all-electric technology especially in terms of cost and HSE,as well as successful experiences of two major oil companies,further research in this area is warranted.One of the gaps in existing reviewed literature is the effect of using all-electric actuators for manifold valves.Thus,three main questions related to electric actuator selection,requirement of safety integrity level(SIL),and effect of using electric actuators on manifold valve selection have been addressed and answered.Forty hydraulic actuated manifold valves from nine past subsea projects in different parts of the world,mainly Africa and Australia,have been selected for the analysis of all-electric actuators.Results show that 93%of the valves require spring-return electric actuators,whereas 7%can be operated with conventional electric actuators without any spring.The manifold valves do not require SIL certification because they are not connected to an emergency shut down system.Introducing the electric actuators to the manifold valve will not change the valve selection philosophy.

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.

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.

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.

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.

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.

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.

APPROXIMATE 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.

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.

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.

Delineating homogeneous domains of fractured rocks using topological manifolds and deep learning

Determining homogeneous domains statistically is helpful for engineering geological modeling and rock mass stability evaluation.In this text,a technique that can integrate lithology,geotechnical and structural information is proposed to delineate homogeneous domains.This technique is then applied to a high and steep slope along a road.First,geological and geotechnical domains were described based on lithology,faults,and shear zones.Next,topological manifolds were used to eliminate the incompatibility between orientations and other parameters(i.e.trace length and roughness)so that the data concerning various properties of each discontinuity can be matched and characterized in the same Euclidean space.Thus,the influence of implicit combined effect in between parameter sequences on the homogeneous domains could be considered.Deep learning technique was employed to quantify abstract features of the characterization images of discontinuity properties,and to assess the similarity of rock mass structures.The results show that the technique can effectively distinguish structural variations and outperform conventional methods.It can handle multisource engineering geological information and multiple discontinuity parameters.This technique can also minimize the interference of human factors and delineate homogeneous domains based on orientations or multi-parameter with arbitrary distributions to satisfy different engineering requirements.