Approximate general solution of 2-degrees of freedom dynamical systems using `les solutions precieuses'

This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions. The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented here conservation of the energy constant between 10^-9 and 10^-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of sections. This revised version was published online in July 2006 with corrections to the Cover Date.     

一类迁移方程逆问题的求解

迁移方程在原子核物理、气象预报和人口理论等方面具有广泛的应用，此类问题的逆问题更具有其实用性。作者在已有工作的基础上，应用奇异积分方程理论，给出了求解此类问题的具体表达式。     

Dynamical causes of asymmetry in the arrangement of gaps in the asteroid belt

The centers of the gaps observed in the asteroid belt are displaced toward Jupiter from their positions that correspond to the exact commensurability between the mean motions of an asteroid and Jupiter. Using the current theory of stability and nonlinear oscillations of Hamiltonian systems, we point out the dynamical causes of this asymmetry. Our analysis is performed in terms of the plane circular restricted three-body problem. The orbits that correspond to Poincaré periodic solutions of the first kind are taken as unperturbed asteroid orbits.     

Classes of Families of Generalized Periodic Solutions to the Beletsky Equation

For the equation describing plane oscillations and rotations of a satellite, we consider families of symmetric generalized periodic solutions with integral rotation number p. We give new confirmations of the hypothesis: there are only four classes of these families with topologically different structures, namely, the classes of families of periodic solutions with p≥ 1, p= 0, p=−1, and p≤−2. Besides, we demonstrate that the vertices of cusps of these families are placed on some analytical curves, and the same is true for the multiple intersections of these families with other families.     

Stationary Orbits in Resonant Extrasolar Planetary Systems

We present a catalog of stable and unstable apsidal corotation resonance (ACR) for the resonant planar planetary three-body problem, including both symmetric and asymmetric solutions. Calculations are performed with a new approach based on a numerical determination of the averaged Hamiltonian function. It has the advantage of being very simple to use and, with the exception of the immediate vicinity of the collision curve, yields precise results for any values of the eccentricities and semimajor axes. The present catalog includes results for the 3/2, 3/1, and 4/1 mean-motion resonances. The 5/1 and 5/2 commensurabilities are also discussed briefly. These results complement our previous results for the 2/1 (Beaugé et al. 2006, MNRAS, 365, 1160–1170), and give a broad picture of the structure of many important planetary resonances.     

A minimizing property of eulerian solutions

In this paper, we prove that the elliptical solutions and collision ejection solutions with the Eulerian collinear configuration for planar 3-body problems are the variational minimizers of the Lagrangian action restricted on a suitable loop space.     

Numerical Investigation of the Galactic Problem by Computing Families of Periodic Solutions

The galactic dynamical system expressed by a third-order axisymmetric polynomial potential is investigated numerically by computing periodic solutions. We define as Sthe compact set of initial conditions generating bounded motions, and as S ^p , with S ^p ? S, the countable set of all initial conditions generating periodic solutions. Then, we consider the subsets S _s ^p and S _a ^p of S ^p , where S _s ^p ∪ S _a ^p = S ^p , S _s ^p S _a ^p = Ø, the first of which corresponds to symmetric periodic solutions, and the second to asymmetric solutions. Then, we approximate the set S _s ^p , leaving treatment of the set S _a ^p of asymmetric solutions for a future publication. The set S _s ^p is known to be dense in S (‘Last Geometric Theorem of Poincar;’, Birkhoff, 1913). Using a computer programme capable to locate all elements of the set S _s ^p that generate symmetric periodic solutions that re-enter after intersecting the axis of symmetry from 1 to ntimes. The results of the approximation of S _s ^p in the total domain and in the sample sub-domains of zooming, we present in graphical form as family curves in the (x, C) plane. The solutions located with the largest periods re-enter after 440 galaxy revolutions while the families calculated fully (initial conditions, period, energy, stability co-efficient) include solutions that re-enter after 340 galaxy revolutions. To advance further the approximation of the set S _s ^p thus obtained, we applied the same procedure inside eight sub-domains of the domain Sinto which we ‘zoomed’ through selection of finer search steps and double maximum periods. The family curves thus calculated presented in the (x, C) plane do not intersect anywhere in some sub-domains and their pattern resembles that of laminar flow. In other sub-domains, however, we found family curves from which branching families emanate. The concepts of completeand non-completeapproximation of S _s ^p in sub-domains of laminar and sub-domains with branching family curves, respectively, is introduced. Also, the concept of basic family of order1, 2, ..., n, are defined. The morphology of individual periodic solutions of all families is investigated, and the types of envelopes found are described. The approximate set S _s ^p was also checked by computing Poincar; sections for energy values corresponding to the mean energy range of the eight sub-domains of zooming mentioned above. These sections show that most parts of the compact domain in Sgenerating non-periodic but bounded solutions correspond to with well-shaped tori that intersect the x-axis, a fact that implies that dominant to exclusive type of periodic solutions are the symmetric ones with two normal crossings of this axis. The presence of non-symmetric periodic solutions as well as of chaotic regions is encountered. All calculations reported here were performed using the variable step R-K 8th-order direct integration and setting the allowable energy variation Δ C= |C _start? C _end| < 10^?13. The output, consisting of many thousands of families and their properties (initial conditions, morphology, stability, etc.), is stored in a directory entitled ‘Atlas of the Symmetric Periodic Solution of the Galactic Motion Problem’.     

Evolution of the General Solution of the Restricted Problem Covering Symmetric and Escape Solutions

The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10⁻³. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions.     

湿度对盐溶液在壁画地仗中的毛细迁移影响研究

盐溶液在地仗中毛细迁移引起的盐分迁移、富集已成为壁画病害发生的重要原因。研究不同湿度条件下盐溶液在模拟地仗中的毛细迁移过程对可溶盐再分布的影响。分析不同湿度条件下毛细上升高度随时间的变化关系,含水率和电导率随高度变化关系,毛细饱和后试样表面变化情况及可溶盐含量随试样高度的分布。试验结果表明,空气相对湿度越低,毛细上升过程中水分向空气中迁移越多,水分难以向上迁移。试样的含水率随高度呈递减趋势,而电导率随试样高度增加而增大。可溶盐在毛细水上升过程中发生结晶分异,毛细前锋以NaCl结晶富集为主,亚前锋以Na2SO4结晶富集为主。其研究结果可以为壁画盐害的防治提供基础科学依据。     

气候变化与海洋生态系统：影响、适应和脆弱性——IPCC AR6 WGⅡ报告之解读

IPCC第六次评估报告(AR6)第二工作组报告第三章开展了气候变化对海洋的影响和风险,以及生态系统及其服务功能、脆弱性和适应评估。AR6明确指出,人为气候变化已经并将继续显著地改变全球和区域海洋的气候影响驱动因子,包括海温升高、海平面上升、海洋酸化和缺氧,以及营养盐浓度变化等海洋物理和化学因子。例如,20世纪80年代以来全球海洋热浪发生的频率已增加了1倍,到21世纪末期可能增加4～8倍。气候影响驱动因子的变化已经对海洋和海岸带生态系统造成了广泛而深远的影响:1)海洋变暖使得海洋物种自1950年代以来以(59.2±15.5) km/(10 a)的速率向极地方向迁移,导致热带海域生物量减少,中纬度海区热带化,极地和亚极地海区浮游植物生长期提前;2)频繁发生的海洋热浪事件已经接近甚至超过了某些海洋生物的耐受极限或其气候临界点,如暖水珊瑚的大规模白化、死亡,海草和大型海藻的大面积消失;3)海洋变暖、缺氧和酸化使得河口区生物群落结构改变,赤潮等有害藻华事件频发,近海和大洋浮游植物生物量和初级生产力下降;4)海平面上升导致海岸带红树林、盐沼和海草床等生态系统的退化;5)未来全球海洋生态系统面临的风险将不断加剧,尤其是在热带和北冰洋海区。其中,当全球升温1.5℃时(最快到21世纪40年代,SSP5-8.5情景),暖水珊瑚礁预计将减少70％～90％;当升温2℃时,几乎所有的(>99％)暖水珊瑚礁将会消失。目前人类社会采取的一些措施(如建立海洋保护区和红树林生态修复)已越来越不能应对日益增长的气候风险,迫切需要发展变革性的行动措施,推动海洋生态系统恢复力的发展,并需尽快采取强有力的减排措施以减缓全球变暖的影响。