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 comp...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.展开更多
Driven by curiosity about possible flight options for the Chang＇e-2 spacecraft after it remains at the Sun-Earth L2 point, effective approaches were developed for designing preliminary fuel-optimal near-Earth asteroi...Driven by curiosity about possible flight options for the Chang＇e-2 spacecraft after it remains at the Sun-Earth L2 point, effective approaches were developed for designing preliminary fuel-optimal near-Earth asteroid flyby trajectories. The approaches include the use of modified unstable manifolds, grid search of the manifolds＇ parameters, and a two-impulse maneuver for orbital phase matching and z-axis bias change, and are demonstrated to be effective in asteroid target screening and trajectory optimization. Asteroid flybys are expected to be within a distance of 2 × 10^7 km from the Earth owing to the constrained Earth-spacecraft communication range. In this case, the spacecraft＇s orbital motion is significantly affected by the gravities of both the Sun and the Earth, and therefore, the concept of the ＂he- liocentric oscillating-Kepler orbit＂ is proposed, because the classical orbital elements of the flyby trajectories referenced in the heliocentric inertial frame oscillate significantly with respect to time. The analysis and results presented in this study show that, among the asteroids whose orbits are the most accurately predicted, ＂Toutatis＂, ＂2005 NZ6＂, or ＂2010 CL19＂ might be encountered by Chang＇e-2 in late 2012 or 2013 with total impulses less than 100 rn/s.展开更多
A class of large scale geophysical fluid flows are modelled by the quasi-geostrophic equation. An averaging principle for quasi-geostrophic motion under rapidly oscil-lating ( non-autonomous) forcing was obtained, bot...A class of large scale geophysical fluid flows are modelled by the quasi-geostrophic equation. An averaging principle for quasi-geostrophic motion under rapidly oscil-lating ( non-autonomous) forcing was obtained, both on finite but large time intervals and on the entire time axis. This includes comparison estimate, stability estimate, and convergence result between quasi-geostrophic motions and its averaged motions. Furthermore, the existence of almost periodic quasi-geostrophic motions and attractor convergence were also investigated.展开更多
The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and emp...The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.展开更多
OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and g...OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and gives a mathematics proof of it.展开更多
A diffeomorphism f:M→M is pointwise partially hyperbolic on an open invariant subset N⊂M if there is an invariant decomposition TNM=Eu⊕Ec⊕Es such that Dxf is strictly expanding on Eu x and contracting on Esx at eac...A diffeomorphism f:M→M is pointwise partially hyperbolic on an open invariant subset N⊂M if there is an invariant decomposition TNM=Eu⊕Ec⊕Es such that Dxf is strictly expanding on Eu x and contracting on Esx at each x∈N.We show that under certain conditions f has unstable and stable manifolds,and admits a nite or an in nite u-Gibbs measure μ.If f is pointwise hyperbolic on N,then μ is a Sinai-Ruelle-Bowen(SRB)measure or an in nite SRB measure.As applications,we show that some almost Anosov di eomorphisms and gentle perturbations of Katok's map have the properties.展开更多
文摘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.
基金supported by the State Key Laboratory of Astronautic Dynamics(2011ADL-DW0202)
文摘Driven by curiosity about possible flight options for the Chang＇e-2 spacecraft after it remains at the Sun-Earth L2 point, effective approaches were developed for designing preliminary fuel-optimal near-Earth asteroid flyby trajectories. The approaches include the use of modified unstable manifolds, grid search of the manifolds＇ parameters, and a two-impulse maneuver for orbital phase matching and z-axis bias change, and are demonstrated to be effective in asteroid target screening and trajectory optimization. Asteroid flybys are expected to be within a distance of 2 × 10^7 km from the Earth owing to the constrained Earth-spacecraft communication range. In this case, the spacecraft＇s orbital motion is significantly affected by the gravities of both the Sun and the Earth, and therefore, the concept of the ＂he- liocentric oscillating-Kepler orbit＂ is proposed, because the classical orbital elements of the flyby trajectories referenced in the heliocentric inertial frame oscillate significantly with respect to time. The analysis and results presented in this study show that, among the asteroids whose orbits are the most accurately predicted, ＂Toutatis＂, ＂2005 NZ6＂, or ＂2010 CL19＂ might be encountered by Chang＇e-2 in late 2012 or 2013 with total impulses less than 100 rn/s.
文摘A class of large scale geophysical fluid flows are modelled by the quasi-geostrophic equation. An averaging principle for quasi-geostrophic motion under rapidly oscil-lating ( non-autonomous) forcing was obtained, both on finite but large time intervals and on the entire time axis. This includes comparison estimate, stability estimate, and convergence result between quasi-geostrophic motions and its averaged motions. Furthermore, the existence of almost periodic quasi-geostrophic motions and attractor convergence were also investigated.
文摘The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.
文摘OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and gives a mathematics proof of it.
基金The third author was supported by National Natural Science Foundation of China(Grant Nos.11871120 and 11671093).
文摘A diffeomorphism f:M→M is pointwise partially hyperbolic on an open invariant subset N⊂M if there is an invariant decomposition TNM=Eu⊕Ec⊕Es such that Dxf is strictly expanding on Eu x and contracting on Esx at each x∈N.We show that under certain conditions f has unstable and stable manifolds,and admits a nite or an in nite u-Gibbs measure μ.If f is pointwise hyperbolic on N,then μ is a Sinai-Ruelle-Bowen(SRB)measure or an in nite SRB measure.As applications,we show that some almost Anosov di eomorphisms and gentle perturbations of Katok's map have the properties.
