The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) mea...The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.展开更多
Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of thi...Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.展开更多
Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597- 606 （2013）] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the e...Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597- 606 （2013）] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy： A brief survey of recent results. Nonlinearity, 17, 493-502 （2004）]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.展开更多
In studying a dynamical system, one finds that there frequently exist some kinds of disturbance or false phenomenon. In order to remove them, we introduce a concept of the measure centre and in order to determine the ...In studying a dynamical system, one finds that there frequently exist some kinds of disturbance or false phenomenon. In order to remove them, we introduce a concept of the measure centre and in order to determine the structure of the measure centre, we also introduce another concept of the weakly almost periodic point. We totally determine the structure of the measure centre and exhibit an example to show that the measure centre may be contained properly in the motion centre and there is a system which is chaotic on the nonwandering set but has zero topological entropy.展开更多
Libration-point missions have been very useful and successful. Due to the unstable natures of most of these orbits, the long-time stationkeeping demands frequent maneuvers and precise orbit determinations. Earth-based...Libration-point missions have been very useful and successful. Due to the unstable natures of most of these orbits, the long-time stationkeeping demands frequent maneuvers and precise orbit determinations. Earth-based tracking will have to undertake much more responsibilities with the increasing number of libration missions. An autonomous navigation system could offer a better way to decrease the need for Earth-based tracking. Nevertheless, when an autonomous navigation system is applied, there are three important factors affecting autonomous navigation accuracy, i.e., the accuracy of initial conditions, the accuracy of measurements, and the accuracy of onboard dynamics for propagation. This paper focuses on analyzing the influence from the third factor and finding an appropriate navigation dynamics, which can satisfy the requirement of estimation accuracy but not cause too much burden for onboard computation. When considering the restricted three-body model and the bicircular restricted four-body model as navigation dynamics, the astrin- gency is not shown during the simulations. Meanwhile, when considering the influences of the Sun＇s direct and indirect perturbations and the eccentricity of the Moon＇s orbit, a new navigation dynamic model with the standard ephemerides is proposed. The simulation shows the feasibility of the proposed model.展开更多
Spacecrafts in periodic or quasi-periodic orbits near the collinear libration points are proved to be excellent platforms for scientific investigations of various phenomena.Since such periodic or quasi-periodic orbits...Spacecrafts in periodic or quasi-periodic orbits near the collinear libration points are proved to be excellent platforms for scientific investigations of various phenomena.Since such periodic or quasi-periodic orbits are exponentially unstable,the station-keeping maneuver is needed. A station-keeping strategy which is found by an analytical method is presented to eradicate the dominant unstable component of the libration point trajectories.The inhibit force transforms the unstable component to a stable component.In this method,it is not necessary to determine a nominal orbit as a reference path.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.10971236 and 11261039)the Foundation from the Jiangxi Education Department (Grant No. GJJ11295)+1 种基金the Natural Science Foundation of Jiangxi Province of China (Grant No. 20114BAB201006)the Foundation of Sun Yat-sen University Advanced Center
文摘The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.
基金Supported by the National Natural Science Foundation of China(Grant No.11661054)
文摘Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.
基金Supported by National Natural Science Foundation of China(Grant No.11261039)National Natural Science Foundation of Jiangxi Province(Grant No.20132BAB201009)the Innovation Fund Designated for Graduate Students of Jiangxi Province
文摘Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597- 606 （2013）] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy： A brief survey of recent results. Nonlinearity, 17, 493-502 （2004）]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.
基金the National Education Foundation of Chinathe National Basic Research Project "Nonlinear Science".
文摘In studying a dynamical system, one finds that there frequently exist some kinds of disturbance or false phenomenon. In order to remove them, we introduce a concept of the measure centre and in order to determine the structure of the measure centre, we also introduce another concept of the weakly almost periodic point. We totally determine the structure of the measure centre and exhibit an example to show that the measure centre may be contained properly in the motion centre and there is a system which is chaotic on the nonwandering set but has zero topological entropy.
基金was supported by the National Natural Science Foundation of China(No.61021002).
文摘Libration-point missions have been very useful and successful. Due to the unstable natures of most of these orbits, the long-time stationkeeping demands frequent maneuvers and precise orbit determinations. Earth-based tracking will have to undertake much more responsibilities with the increasing number of libration missions. An autonomous navigation system could offer a better way to decrease the need for Earth-based tracking. Nevertheless, when an autonomous navigation system is applied, there are three important factors affecting autonomous navigation accuracy, i.e., the accuracy of initial conditions, the accuracy of measurements, and the accuracy of onboard dynamics for propagation. This paper focuses on analyzing the influence from the third factor and finding an appropriate navigation dynamics, which can satisfy the requirement of estimation accuracy but not cause too much burden for onboard computation. When considering the restricted three-body model and the bicircular restricted four-body model as navigation dynamics, the astrin- gency is not shown during the simulations. Meanwhile, when considering the influences of the Sun＇s direct and indirect perturbations and the eccentricity of the Moon＇s orbit, a new navigation dynamic model with the standard ephemerides is proposed. The simulation shows the feasibility of the proposed model.
基金supported by the National Natural Science Foundation of China(10832004)the Fundamental Research Funds for the Central Universities(YWF-10- 02-049)
文摘Spacecrafts in periodic or quasi-periodic orbits near the collinear libration points are proved to be excellent platforms for scientific investigations of various phenomena.Since such periodic or quasi-periodic orbits are exponentially unstable,the station-keeping maneuver is needed. A station-keeping strategy which is found by an analytical method is presented to eradicate the dominant unstable component of the libration point trajectories.The inhibit force transforms the unstable component to a stable component.In this method,it is not necessary to determine a nominal orbit as a reference path.
