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 ...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.展开更多
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
An analytical method is proposed to find geometric structures of stable,unstable and center manifolds of the collinear Lagrange points.In a transformed space,where the linearized equations are in Jordan canonical form...An analytical method is proposed to find geometric structures of stable,unstable and center manifolds of the collinear Lagrange points.In a transformed space,where the linearized equations are in Jordan canonical form,these invariant manifolds can be approximated arbitrarily closely as Taylor series around Lagrange points.These invariant manifolds are represented by algebraic equations containing the state variables only without the help of time.Thus the so-called geometric structure of these invariant manifolds is obtained.The stable,unstable and center manifolds are tangent to the stable,unstable and center eigenspaces,respectively.As an example of applicability,the invariant manifolds of L 1 point of the Sun-Earth system are considered.The stable and unstable manifolds are symmetric about the line from the Sun to the Earth,and they both reach near the Earth,so that the low energy transfer trajectory can be found based on the stable and unstable manifolds.The periodic or quasi-periodic orbits,which are chosen as nominal arrival orbits,can be obtained based on the center manifold.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
The stickiness effect suffered by chaotic orbits diffusing in the phase space of a dynamical system is studied in this paper.Previous works have shown that the hyperbolic structures in the phase space play an essentia...The stickiness effect suffered by chaotic orbits diffusing in the phase space of a dynamical system is studied in this paper.Previous works have shown that the hyperbolic structures in the phase space play an essential role in causing the stickiness effect.We present in this paper the relationship between the stickiness effect and the geometric property of hyperbolic structures.Using a two-dimensional area-preserving twist mapping as the model,we develop the numerical algorithms for computing the positions of the hyperbolic periodic orbits and for calculating the angle between the stable and unstable manifolds of the hyperbolic periodic orbit.We show how the stickiness effect and the orbital diffusion speed are related to the angle.展开更多
文摘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.
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supported by the National Natural Science Foundation of China (Grant Nos. 10832004 and 11102006)the FanZhou Foundation (Grant No. 20110502)
文摘An analytical method is proposed to find geometric structures of stable,unstable and center manifolds of the collinear Lagrange points.In a transformed space,where the linearized equations are in Jordan canonical form,these invariant manifolds can be approximated arbitrarily closely as Taylor series around Lagrange points.These invariant manifolds are represented by algebraic equations containing the state variables only without the help of time.Thus the so-called geometric structure of these invariant manifolds is obtained.The stable,unstable and center manifolds are tangent to the stable,unstable and center eigenspaces,respectively.As an example of applicability,the invariant manifolds of L 1 point of the Sun-Earth system are considered.The stable and unstable manifolds are symmetric about the line from the Sun to the Earth,and they both reach near the Earth,so that the low energy transfer trajectory can be found based on the stable and unstable manifolds.The periodic or quasi-periodic orbits,which are chosen as nominal arrival orbits,can be obtained based on the center manifold.
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.
基金supported by the National Natural Science Foundation of China(Grant Nos.11073012,11078001 and 11003008)the Qing Lan Project(Jiangsu Province)the National Basic Research Program of China(Grant Nos.2013CB834103 and 2013CB834904)
文摘The stickiness effect suffered by chaotic orbits diffusing in the phase space of a dynamical system is studied in this paper.Previous works have shown that the hyperbolic structures in the phase space play an essential role in causing the stickiness effect.We present in this paper the relationship between the stickiness effect and the geometric property of hyperbolic structures.Using a two-dimensional area-preserving twist mapping as the model,we develop the numerical algorithms for computing the positions of the hyperbolic periodic orbits and for calculating the angle between the stable and unstable manifolds of the hyperbolic periodic orbit.We show how the stickiness effect and the orbital diffusion speed are related to the angle.
