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 plate 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.     

Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearisation of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to Hénon-like strange attractors, as a consequence of the Torus-Breakdown Theory. The mechanism for the creation of horseshoes and strange attractors is also discussed. Theoretical results are applied to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.

     

非自旋航天器混沌姿态运动及其参数开闭环控制

研究万有引力场中受大气阻力且存在结构内阻尼的非自旋航天器在椭圆轨道上平面天平动的混沌及其参数开闭环控制问题．在建立数学模型的基础上确定出现混沌的必要条件并数值验证混沌的存在性，提出非线性振动系统混沌运动的参数开闭环控制并应用于控制航天器的混沌姿态运动．     

Inertial Manifolds for the 3D Modified-Leray-$$\alpha $$ Model with Periodic Boundary Conditions

The existence of an inertial manifold for the modified Leray-\(\alpha \) model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. Moreover, an adaptation of the Perron method for constructing inertial manifolds in the particular case of zero spatial averaging is suggested.     

A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections.     

Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds

In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier–Taylor parameterizations of local stable/unstable manifolds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in Castelli et al. (SIAM J Appl Dyn Syst 14(1):132–167, 2015) in order to obtain a high order Fourier–Taylor series expansion of the parameterization. The main result of the present work is an a-posteriori theorem which provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with computer assistance. The argument relies on a sequence of preliminary computer assisted proofs where we validate the numerical approximation of the periodic orbit, its stable/unstable normal bundles, and the jets of the manifold to some desired order M. We illustrate our method by implementing validated computations for two dimensional manifolds in the Lorenz equations in \(\mathbb {R}^3\) and a three dimensional manifold of a suspension bridge equation in \(\mathbb {R}^4\).     

Transversality for Cyclic Negative Feedback Systems

Transversality of stable and unstable manifolds of hyperbolic periodic trajectories is proved for monotone cyclic systems with negative feedback. Such systems in general are not in the category of monotone dynamical systems in the sense of Hirsch. Our main tool utilized in the proofs is the so-called cone of high rank. We further show that stable and unstable manifolds between a hyperbolic equilibrium and a hyperbolic periodic trajectory, or between two hyperbolic equilibria with different dimensional unstable manifolds also intersect transversely.     

Global solution for coupled nonlinear Klein-Gordon system

The globed solution for a coupled nonlinear Klein-Gordon system in two-dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.     

Homoclinic Shadowing

A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in Rⁿ is presented. It is a computer-assisted technique with two main components. First, a global Newton’s method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.     

DYNAMICAL CHARACTER FOR A PERTURBED COUPLED NONLINEAR SCHRODINGER SYSTEM

The dynamical character for a perturbed coupled nonlinear Schrodinger system with periodic boundary condition was studied. First, the dynamical character of perturbed and unperturbed systems on the invariant plane was analyzed by the spectrum of the linear operator. Then the existence of the locally invariant manifolds was proved by the singular perturbation theory and the fixed-point argument.     

DYNAMICAL CHARACTER FOR A PERTURBED COUPLED NONLINEAR SCHRDINGER SYSTEM

Introduction WeconsidertheperturbedcouplednonlinearSchr dingerequations(CNLS)iq1t=q1xx 2[|q1|2 |q2|2-ω2]q1 iε(^D1q1-r1),iq2t=q2xx 2[|q1|2 |q2|2-ω2]q2 iε(^D2q2-r2),(1)whereqjis2πperiodicandeveninx.rjisconstant,^Djisboundeddissipativeoperatorand assumedtotaketheform^Djqj=-αjqj βj^Bqj,(2)αjandβjarepositiveconstants,j=1,2.^BisaFouriertruncationofLaplaceoperator xx,i.e.,^Bcos(kx)=-k2cos(kx),k0isasmallperturbation parameter.Th…     

Periodic and Homoclinic Motions in Forced,Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.     

Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension

We study the qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise herein, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria are studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.     

Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems

We consider a special class of monotone dynamical systems and show that in this special class the stable and unstable manifolds of two hyperbolic periodic orbits always intersect transversally. The proof is based on the existence of a family of positively invariant nested cones.This paper is dedicated to Jack Hale on the occasion of his 60th birthday.     

The Linearized Crocco Equation

In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions, coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator associated with the adjoint linearized Crocco equation – with homogeneous boundary conditions – we first study degenerate parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂_x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which cannot be obtained in a classical way.     

Handling complex boundaries on a Cartesian grid using surface singularities

This paper considers flow around arbitrarily shaped objects. The boundary conditions on the solid boundaries have been applied by replacing the boundary with a surface force distribution on the surface, such that the required boundary conditions are satisfied. The velocity on the boundary is determined by interpolation or by local (Gaussian space) average. The source terms are determined iteratively as part of the solution. They are then averaged and are smoothed out to nearby computational grid points. The method has been applied both to test problems as well as to more complex engineering problems, where there are not many real competitive alternatives to the proposed method. Simulations of creeping flow around a sphere were studied in order to evaluate the performance of different, competitive approaches of imposing boundary conditions. Using local averaging first‐order accuracy is obtained; this can be improved by using a Lagrangian polynomial instead, although the convergence is then considerably slower. Simulations of flows around spheres in the Reynolds number range 1–1000 have been carried out. Finally, the approach was used to describe the impellers in a turbine agitated mixer. For these cases, the results show overall good agreement with other computational and experimental results. Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamics of a hyperchaotic Lorenz-type system

This paper discusses the complex dynamics of a new four-dimensional continuous-time autonomous hyperchaotic Lorenz-type system. The local dynamics, such as the stability, pitchfork bifurcation, and Hopf bifurcation at equilibria of this hyperchaotic system are analyzed by using the parameter-dependent center manifold theory and the normal form theory. The existence of homoclinic and heteroclinic orbits of this hyperchaotic system is further rigorously studied. More exactly, under some special parameter conditions, the fact that this hyperchaotic system has no homoclinic orbit but has two and only two heteroclinic orbits are proved.     

Dynamics of a multi-DOF beam system with discontinuous support

This paper deals with the long term behaviour of periodically excited mechanical systems consisting of linear components and local nonlinearities. The particular system investigated is a 2D pinned-pinned beam, which halfway its length is supported by a one-sided spring and excited by a periodic transversal force. The linear part of this system is modelled by means of the finite element method and subse1uently reduced using a Component Mode Synthesis method. Periodic solutions are computed by solving a two-point boundary value problem using finite differences or, alternatively, by using the shooting method. Branches of periodic solutions are followed at a changing design variable by applying a path following technique. Floquet multipliers are calculated to determine the local stability of these solutions and to identify local bifurcation points. Also stable and unstable manifolds are calculated. The long term behaviour is also investigated by means of standard numerical time integration, in particular for determining chaotic motions. In addition, the Cell Mapping technique is applied to identify periodic and chaotic solutions and their basins of attraction. An extension of the existing cell mapping methods enables to investigate systems with many degress of freedom. By means of the above methods very rich complex dynamic behaviour is demonstrated for the beam system with one-sided spring support. This behaviour is confirmed by experimental results.     

应用突变理论分析Euler压杆的稳定性

建立了Euler压杆问题的能量表达式。对此模型，应用突变理论进行稳定分析，得出了系统全部的分叉集与突变流形。突变流形为一族层层嵌套、互不交叉的抛物线。分析的结论要比有限差分法、有限单元法、大范围非线形分析等数值计算的结果简洁、明了，有助于压杆后屈曲问题的研究。     

Melnikov function and Poincaré map

In this paper we give the relationship between Melnikov function and Poincare map, and a new proof for Melnikov’s method. The advantage of our paper is to give a more explicit solution and to make Melnikov function for the subharmonics bifurcation and Melnikoy function which the stable manifolds and unstable manifolds intersect transversely into a formula.