类星体3C 446射电光变非线性特性分析

类星体有剧烈、大幅度的光变现象, 光变研究有助于建立与观测相符的理论模型. 这篇文章从密歇根大学射电天文台数据库收集了类星体3C 446射电4.8、8.0和14.5GHz波段的长期观测数据. 传统的线性方法难以分析复杂的光变现象, 文章采用了集合经验模态分解(Ensemble Empirical Mode Decomposition, EEMD)方法和非线性分析方法相结合, 从混沌动力学特性、分形特性和周期性多角度对类星体光变随时间演化的规律进行了较全面的分析, 并重点对比分析了除去周期成分或混沌成分前后, 光变的周期性和非线性特性是否存在明显区别. 分析结果表明, 类星体3C 446射电波段光变资料由周期成分、趋势成分和混沌成分组成, 光变具有周期性、混沌性和分形特性. 除去混沌成分和趋势成分后的光变周期与原始光变资料的周期完全相同, 而两者的混沌和分形特性有明显不同. 从饱和关联维数来看, 重构动力学系统时, 除去周期成分和趋势成分后的光变资料比原始光变资料需要更多的独立参量, Kolmogorov熵值表明前者信息的损失率比后者大, 系统的混沌程度更高, 系统也更复杂, Hurst值表明后者自相似性和长程相关性比前者略强.     

Chaotic behaviour of solar radio flux

The presence of a chaotic attractor is investigated in time series of 10.7 cm solar flux. The correlation dimension and the Kolomogorov entropy have been calculated for the time period 1964–1984. The values found for the Kolmogorov entropy show that chaos is indeed present. The correlation dimension found for high solar activity is 3.3 and for low solar activity is 4.5, indicating that a low-dimensionsion chaotic attractor is present in the time series analysed.     

Separation of periodic, chaotic, and random components in solar activity

Two new techniques were applied to search for chaotic behavior in solar activity. A mixture of periodic and chaotic components in a time series makes it difficult to find chaotic behavior. The singular spectrum analysis (SSA) method (Broomhead and King, 1986) was used to separate periodic and irregular components in solar activity (e.g., sunspot number and 10.7 cm flux). The nonlinear prediction method (Sugihara and May, 1990) was applied for each component to examine whether it has a chaotic characteristic. The result suggests that are are dominant periodic components and highly irregular (random) components in solar activity.     

Low-dimensional chaos and fractal properties of long-term sunspot activity

Two primary solar-activity indicators sunspot numbers(SNs)and sunspot areas(SAs)in the time interval from November 1874 to December 2012 are used to determine the chaotic and fractal properties of solar activity.The results show that(1)the long-term solar activity is governed by a low-dimensional chaotic strange attractor,and its fractal motion shows a long-term persistence on large scales;(2)both the fractal dimension and maximal Lyapunov exponent of SAs are larger than those of SNs,implying that the dynamical system of SAs is more chaotic and complex than SNs;(3)the predictions of solar activity should only be done for short-to mid-term behaviors due to its intrinsic complexity;moreover,the predictability time of SAs is obviously smaller than that of SNs and previous results.     

Three-body dynamics: intermittent chaos with strange attractor

We have studied the structure of chaos in three-body dynamics using the concept of intermittency, implying that violent states of a system alternate in time with quasi-regular states producing together a non-stationary and evolving pattern of unpredictable behaviour. Computer simulations are produced to demonstrate explicitly sporadic short violent bursts in quasi-regular hierarchical states of the systems. This is seen both in orbits and in the long time series generated by the system. The time series prove to be similar in shape to what is observed in various physical experiments with laboratory chaotic systems when they reveal the so-called type-III intermittency. The new effective methods of time series analysis enable us to discover a strange attractor with a fractal dimension slightly above 2. This shows that three-body dynamics has the same intrinsic qualitative structure and quantitative measure of chaos as the widely known chaotic system, the Lorenz attractor.     

Analysis of Sunspot Activity Cycles

A nonlinear analysis of the daily sunspot number for each of cycles 10 to 23 is used to indicate whether the convective turbulence is stochastic or chaotic. There is a short review of recent papers considering sunspot statistics and solar activity cycles. The differences in the three possible regimes – deterministic laminar flow, chaotic flow, and stochastic flow – are discussed. The length of data sets necessary to analyze the regimes is investigated. Chaos is described and a chronology of recent results that utilize chaos and fractals to analyze sunspot numbers follows. The parameters necessary to describe chaos – time lag, phase space, embedding dimension, local dimension, correlation dimension, and the Lyapunov exponents – are determined for the attractor for each cycle. Assuming the laminar regime is unlikely if chaos is not indicated in a cycle by the calculations, the regime must be stochastic. The sunspot numbers in each of cycles 10 to 19 indicate stochastic behavior. There is a transition from stochastic to chaotic behavior of the sunspot numbers in cycles 20, 21, 22, and 23. These changes in cycles 20 – 23 may indicate a change in the scale of turbulence in the convection zone that could result in a change in the convective heat transfer and a change in the size of the convection region for these four cycles.     

The Transitional Time Scale from Stochastic to Chaotic Behavior for Solar Activity

Following the progression of nonlinear dynamical system theory, many authors have used varied methods to calculate the fractal dimension and the largest Lyapunov exponent ₁ of the sunspot numbers and to evaluate the character of the chaotic attractor governing solar activity. These include the Grassberger–Procaccia algorithm, the technique provided by Wolf et al., and the nonlinear forecasting approach based on the method of distinguishing between chaos and measurement errors in time series described by Sugihara and May. In this paper, we use the Grassberger–Procaccia algorithm to estimate the other character of the chaotic attractor. This character is time scale of a transition from high-dimensional or stochastic at shorter times to a low-dimensional chaotic behavior at longer times. We find that the transitional time scale in the monthly mean sunspot numbers is about 8 yr; the low-dimensional chaotic behavior operates at time scales longer than about 8 yr and a high-dimensional or stochastic process operates at time scales shorter than about 8 yr.     

The route to chaos during a pulsation event

A time series analysis of a pulsation event in solar radio emission provides an evolution from a regular doubly periodic phase to an irregular behaviour. Applying some techniques developed in the theory of nonlinear dynamic systems to this irregular stage suggests that there exists a low-dimensional attractor. Estimates of the maximum Lyapunov exponent give some evidence to deterministic chaos. The sudden transition from a regular to a chaotic structure is identified as a part of the Ruelle-Takens-Newhouse route to chaos which is typical in nonlinear systems. It is checked whether this pulsation event may be interpreted in terms of known pulsation models. Consequences for models, which are suitable to describe such an evolution, are discussed.     

控制太阳活动的混沌吸引子(英)

随着非线性科学研究的进展,可利用表征太阳活动的太阳活动指数组成的时间序列来寻找或许存在的太阳混沌吸引子,并计算其关联维数、最大Lyapunov指数以及其它特征量．文中综述了用非线性科学的某些概念来研究太阳活动的进展及其在太阳活动预报方面的一些应用．     

Can a solar pulsation event be characterized by a low-dimensional chaotic attractor?

Using the theory of nonlinear dynamical systems a time-series analysis of a pulsation event in solar radio emission suggests that there exists a low-dimensional attractor. The power spectrum cannot be interpreted as a superposition of periodic components. Estimates of the maximum Lyapunov exponent and the Kolmogorov entropy give some hints to deterministic chaos. Consequences for the physical modelling of the event are discussed.     

Chaotic Feature in the Light Curve of 3C 273

Some nonlinear dynamical techniques, including state-space reconstruction and correlation integral, are used to analyze the light curve of 3C 273. The result is compared with a chaotic model. The similarities between them suggest there is a low-dimension chaotic attractor in the light curve of 3C 273.     

Discrete Virial Theorem

We reexamine the classical virial theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Hénon–Heiles system, weak chaos in the standard map, and a 4D Froeschlé map.     

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

A two-layer αω-dynamo model with dynamic feedback on the ω-effect

In an attempt to produce a simple representation of an interface dynamo, I examine a dynamo model composed of two one-dimensional (radially averaged) pseudo-spherical layers, one in the convection zone and possessing an α-effect, and the other in the tachocline and possessing an ω-effect. The two layers communicate by means of an analogue of Newton's law of cooling, and a dynamical back-reaction of the magnetic field on ω is provided. Extensive bifurcation diagrams are calculated for three separate values of η, the ratio of magnetic diffusivities of the two layers. I find recognizable similarities to, but also dramatic differences from, the comparable one-layer model examined by Roald &38; Thomas. In particular, the solar-like dynamo mode found previously is no longer stable in the two-layer version; in its place there is a sequence of periodic, quasi-periodic and chaotic modes probably created in a homoclinic bifurcation. These differences are important enough to provide support for the view that the solar dynamo cannot be meaningfully modelled in one dimension.     

Chaotic mixing in noisy Hamiltonian systems

This paper summarizes an investigation of the effects of low-amplitude noise and periodic driving on phase-space transport in three-dimensional Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and/or a surrounding environment. A new diagnostic tool is exploited to quantify the extent to which, over long times, different segments of the same chaotic orbit evolved in the absence of such perturbations can exhibit very different amounts of chaos. First-passage-time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase-space transport, allowing 'sticky' chaotic orbits trapped near regular islands to become unstuck on surprisingly short time‐scales. The effects of small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly logarithmically in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of friction related to the noise by a fluctuation–dissipation theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that there is little power at frequencies comparable with the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed.     

Radiocarbon evidence of the global stochasticity of solar activity

An estimate of the dimension of the attractor of the dynamic system responsible for solar activity is obtained from the time series of carbon 14 experimental data (4300 BC to 1950 AD). According to this estimate the attractor is a fractal, in shape close to a 3-torus. The attractor's trajectories characterizing the evolution of the magnetic field exhibit irregular long-term behaviour.     

Ordered and chaotic Bohmian trajectories

We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.     

A three dimensional investigation of two dimensional orbits

Orbits in the principal planes of triaxial potentials are known to be prone to unstable motion normal to those planes, so that three dimensional investigations of those orbits are needed even though they are two dimensional. We present here an investigation of such orbits in the well known logarithmic potential which shows that the third dimension must be taken into account when studying them and that the instability worsens for lower values of the forces normal to the plane. Partially chaotic orbits are present around resonances, but also in other regions. The action normal to the plane seems to be related to the isolating integral that distinguishes regular from partially chaotic orbits, but not to the integral that distinguishes partially from fully chaotic orbits.     

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.     

Stable Chaos versus Kirkwood Gaps in the Asteroid Belt: A Comparative Study of Mean Motion Resonances

We have shown, in previous publications, that stable chaos is associated with medium/high-order mean motion resonances with Jupiter, for which there exist no resonant periodic orbits in the framework of the elliptic restricted three-body problem. This topological “defect” results in the absence of the most efficient mechanism of eccentricity transport (i.e., large-amplitude modulation on a short time scale) in three-body models. Thus, chaotic diffusion of the orbital elements can be quite slow, while there can also exist a nonnegligible set of chaotic orbits which are semiconfined (stable chaos) by “quasi-barriers” in the phase space. In the present paper we extend our study to all mean motion resonances of order q≤9 in the inner main belt (1.9-3.3 AU) and q≤7 in the outer belt (3.3-3.9 AU). We find that, out of the 34 resonances studied, only 8 possess resonant periodic orbits that are continued from the circular to the elliptic three-body problem (regular families), namely, the 2/1, 3/1, 4/1, and 5/2 in the inner belt and the 7/4, 5/3, 11/7, and 3/2 in the outer belt. Numerical results indicate that the 7/3 resonance also carries periodic orbits but, unlike the aforementioned resonances, 7/3-periodic orbits belong to an irregular family. Note that the five inner-belt resonances that carry periodic orbits correspond to the location of the main Kirkwood gaps, while the three outer-belt resonances correspond to gaps in the distribution of outer-belt asteroids noted by Holman and Murray (1996, Astron. J.112, 1278-1293), except for the 3/2 case where the Hildas reside. Fast, intermittent eccentricity increase is found in resonances possessing periodic orbits. In the remaining resonances the time-averaged elements of chaotic orbits are, in general, quite stable, at least for times t∼250 Myr. This slow diffusion picture does not change qualitatively, even if more perturbing planets are included in the model.