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.

THE INVARIANT MANIFOLDS FOR A PERTURBED QUINTIC-CUBIC SCHRDINGER EQUATION

In this paper, the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions.

Research on the transfers to Halo orbits from the view of invariant manifolds

This paper discusses the evolutions of invariant manifolds of Halo orbits by low-thrust and lunar gravity. The possibility of applying all these manifolds in designing low-thrust transfer, and the presence of single-impulse trajectories under lunar gravity are also explained. The relationship between invafiant manifolds and the altitude of the perigee is investigated using a Poincare map. Six types of single-impulse transfer trajectories are then attained from the geometry of the invariant manifolds. The evolutions of controlled manifolds are surveyed by the gradient law of Jacobi energy, and the following conclusions are drawn. First, the low thrust （acceleration or deceleration） near the libration point is very inefficient that the spacecraft free-flies along the invariant manifolds. The purpose is to increase its velocity and avoid stagnation near the libration point. Second, all con- trolled manifolds are captured because they lie inside the boundary of Eatlh＇s gravity trap in the configuration space. The evo- lutions of invariant manifolds under lunar gravity are indicated from the relationship between the lunar phasic angle and the altitude of the perigee. Third and last, most of the manifolds have preserved their topologies in the circular restricted three-body problem. However, the altitudes of the perigee of few manifolds are quite non-continuous, which can be used to generate single-impulse flyby trajectories.

Out-of-plane equilibrium points and invariant manifolds about an asteroid with gravitational orbit–attitude coupling perturbation

By considering the spacecraft as an extended,rigid body with a prior known attitude instead of a point mass,the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynamics in close proximity to an asteroid,because it includes the perturbation caused by the gravitational orbit–attitude coupling of the spacecraft(GOACP).The GOACP is defined as the difference between the gravity acting on a non-spherical,extended body(the real case of a spacecraft)and the gravity acting on a point mass(the approximation of a spacecraft in classical orbital dynamics).Inplane equilibrium points that are within the principal planes of the asteroid have been investigated for the attitude-restricted orbital dynamics in previous studies,including equatorial and in-plane non-equatorial equilibrium points.In this study,out-of-plane equilibrium points outside the principal planes of the asteroid were examined.Out-ofplane equilibrium points cannot exist in the classical point-mass orbital dynamics but do exist in the attitude-restricted orbital dynamics owing to the effects of the GOACP.The previously investigated in-plane equilibrium points and the out-of-plane ones examined in this study provide a complete map of the equilibrium points in close proximity to an asteroid with the GOACP.Equatorial and in-plane non-equatorial equilibrium points have extended the longitude and latitude ranges of the classical equilibrium points without the GOACP,respectively,while the out-of-plane ones examined in the present study extend both the longitude and latitude ranges.Additionally,the invariant manifolds of out-of-plane equilibrium points were calculated,and the results indicated that the attitude of spacecraft significantly affects the invariant manifolds.In practice,these equilibrium points can provide natural hovering positions for operations in proximity to asteroids,and their invariant manifolds can be used for transfers to or from the equilibrium points.

Orbits Heteroclinic to Invariant Manifolds

Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].

Smoothness of invariant manifolds and foliations for in nite dimensional random dynamical systems--Dedicated to Professor Shantao Liao

In this paper,we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces.We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations,which preserve the CN,τ;(ω)Holder smoothness of the random system in the space variable and the measurability of the random system in the sample point.Moreover,we also prove that the stable foliation is CN-1;(ω)in the base point.

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 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.

Comparison of Invariant Manifolds for Model Reduction in Chemical Kinetics

A modern approach to model reduction in chemical kinetics is often based on the notion of slow invariant manifold.The goal of this paper is to give a comparison of various methods of construction of slow invariant manifolds using a simple Michaelis-Menten catalytic reaction.We explore a recently introduced Method of Invariant Grids(MIG)for iteratively solving the invariance equation.Various initial approximations for the grid are considered such as Quasi Equilibrium Manifold,Spectral Quasi Equilibrium Manifold,Intrinsic Low Dimensional Manifold and Symmetric Entropic Intrinsic Low Dimensional Manifold.Slow invariant manifold was also computed using the Computational Singular Perturbation(CSP)method.A comparison between MIG and CSP is also reported.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields,the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates.The initial growth directions depend on the Jacobians of Poincarémap on that cycle,for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincarésections.The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems.Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration.A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

PERSISTENCE OF RESONANT INVARIANT TORI WITH NON-HAMILTONIAN PERTURBATION

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone’s for the three dimensional non-Hamiltonian systems.

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.

Lunar landing trajectory design based on invariant manifold

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied. Considering angle between the ecliptic plane and lunar orbit plane, the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems, the sun-earth-spacecraft in the ecliptic plane and the earth-moon-spacecraft in the lunar orbit plane. Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect, a general method to design low energy lunar landing trajectory is given. It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory. The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given. In addition, some rules of selecting parameters with respect to orbit design are provided. The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.

Heteroclinic and Homoclinic Connections between the Sun-Earth Triangular Points and Quasi-Satellite Orbits for Solar Observations

Investigation of new orbit geometries exhibits a very attractive behavior for a spacecraft to monitor space weather coming from the Sun. Several orbit transfer mechanisms are analyzed as potential alternatives to monitor solar activity such as a sub-solar orbit or quasi-satellite orbit and short and long heteroclinic and homoclinic connections between the triangular points L4 and L5 and the collinear point L3 of the CRTBP （circular restricted three-body problem） in the Sun-Earth system. These trajectories could serve as channels through where material can be transported from L5 to L3 by performing small maneuvers at the departure of the Trojan orbit. The size of these maneuvers at L5 is between 299 m/s and 730 m/s depending on the transfer time of the trajectory and does not need any deterministic maneuvers at L3. Our results suggest that material may also be transported from the Trojan orbits to quasi-satellite orbits or even displaced quasi-satellite orbits.

First-round design of the flight scenario for Chang’e-2’s extended mission:take off from lunar orbit

Chang＇e-2, Chinese second lunar probe, was inserted into a 100 km altitude low lunar orbit on October 9th, 2010, its purpose is to continuously photograph the lunar surface and possibly chosen landing sites for future lunar missions. The probe will still carry considerable amount of propellant after completing all prescribed tasks in about six months. After the successful launch of Chang＇e-2, we began designing the probe＇s subsequent flight scenario, considering a total impulse of 1 100 m/s for takeoff from low lunar orbit and a maximum 3× 10^6 km distance for Earth-probe telecom- munication. Our first-round effort proposed a preliminary flight scenario that involves consecutive arrivals at the halo orbits around the Earth-Moon L1/L2 and Sun-Earth L1/L2 points, near-Earth asteroid flyby, Earth return, and lunar impact. The designed solution of Chang＇e-2＇s subsequent flight scenario is a multi-segment flight trajectory that serves as a reference for making the final decision on Chang＇e-2＇s extended mission, which is a flight to the Sun-Earth L2 point, and a possible scheme of lunar impact via Earth flyby after remaining at the Sun-Earth L2 point was also presented. The proposed flight trajectory, which possesses acceptable solution accuracy for mission analysis, is a novel design that effectively exploits the invariant manifolds in the circular restricted three-body problem and the patched-manifold-conic method.

Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear SchrSdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.

INVARIANT 3-TORUS BIFURCATIONS FROM PERIODIC MANIFOLDS

Bifurcation problems of high dimensional system with several parameters are considered. Assume that the system has an invariant manifold, which consists entirely of periodic orbits and has a center subspace with two dimensions in nor mal direction. The existence and the normal hyperbolicity of the 2-dimensional invariant torus and the 3-dimensional invariant torus are given. The phenome non of bifurcation for the 3-dimensional invariant torus from one single periodic orbit is discovered for the first time.

Rapid accessibility evaluation for ballistic lunar capture via manifolds: A Gaussian process regression application

In this study,a supervised machine learning approach called Gaussian process regression(GPR)was applied to approximate optimal bi-impulse rendezvous maneuvers in the cis-lunar space.We demonstrate the use of the GPR approximation of the optimal bi-impulse transfer to patch points associated with various invariant manifolds in the cis-lunar space.The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solutions of the optimal bi-impulsive Lambert transfer because the learned map is computationally efficient.This approach promises to be useful for aiding in preliminary mission design.The use of invariant manifolds as part of the transfer trajectory design offers unique features for reducing propellant consumption while facilitating the solution of trajectory optimization problems.Long ballistic capture coasts are also very attractive for mission guidance,navigation,and control robustness.A multi-input single-output GPR model is presented to represent the fuel costs(in terms of theΔV magnitude)associated with the class of orbital transfers of interest efficiently.The developed model is also proven to provide efficient approximations.The multi-resolution use of local GPRs over smaller sub-domains and their use for constructing a global GPR model are also demonstrated.One of the unique features of GPRs is that they provide an estimate of the quality of approximations in the form of covariance,which is proven to provide statistical consistency with the optimal trajectories generated through the approximation process.The numerical results demonstrate our basis for optimism for the utility of the proposed method.

Unknown system dynamics estimator-based impedance control for lower limb exoskeleton with enhanced performance

In this article, an unknown system dynamics estimator-based impedance control method is proposed for the lower limb exoskeleton to stimulate the tracking flexibility with the terminal target position when suffering parametric inaccuracies and unexpected disturbances. To reinforce the robust performance, via constructing the filtering operation-based dynamic relation, i.e., invariant manifold, the unknown system dynamics estimators are employed to maintain the accurate perturbation identification in both the hip and knee subsystem. Besides, a funnel control technique is designed to govern the convergence process within a minor overshoot and a higher steady-state precision. Meanwhile, an interactive complaint result can be obtained with the aid of the impedance control, where the prescribed terminal trajectory can be adjusted into the interaction variable-based target position by the force–position mapping, revealing the dynamic influence between the impedance coefficient (stiffness and damping) and the adjusted position magnitude. A sufficient stability analysis verifies the ultimately uniformly bounded results of all the error signals, and even the angle errors can be regulated within the predefined funnel boundary in the whole convergence. Finally, some simulations are provided to demonstrate the validity and superiority including the enhanced interaction flexibility and robustness.

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x＇= A（t）x ＋ f（t, x, λ）, t≠τi and x（τ＋i） = Bix（τi） ＋ gi（x（τi）, λ）, i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x＇= A（t）x, t≠τi and x（τ＋i） = Bix（τi）, i ∈ N admits a nonuniform （μ, ν）-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.