Regular and chaotic orbits in the dynamics of exoplanets

Many of exoplanetary systems consist of more than one planet and the study of planetary orbits with respect to their long-term stability is very interesting. Furthermore, many exoplanets seem to be locked in a mean-motion resonance (MMR), which offers a phase protection mechanism, so that, even highly eccentric planets can avoid close encounters. However, the present estimation of their initial conditions, which may change significantly after obtaining additional observational data in the future, locate most of the systems in chaotic regions and consequently, they are destabilized. Hence, dynamical analysis is imperative for the derivation of proper planetary orbital elements. We utilize the model of spatial general three body problem, in order to simulate such resonant systems through the computation of families periodic orbits. In this way, we can figure out regions in phase space, where the planets in resonances should be ideally hosted in favour of long-term stability and therefore, survival. In this review, we summarize our methodology and showcase the fact that stable resonant planetary systems evolve being exactly centered at stable periodic orbits. We apply this process to co-orbital motion and systems HD 82943, HD 73526, HD 128311, HD 60532, HD 45364 and HD 108874.     

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L(1) and the other around L(2), with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L(1) and L(2) as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena. (c) 2000 American Institute of Physics.     

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem can be traced back to Newton in 1687,but it is still an open question today.Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem.Although triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce.However,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are stable.Compared with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass region.We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations.This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice.Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses.Note that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.     

Cycling chaotic attractors in two models for dynamics with invariant subspaces

Nonergodic attractors can robustly appear in symmetric systems as structurally stable cycles between saddle-type invariant sets. These saddles may be chaotic giving rise to "cycling chaos." The robustness of such attractors appears by virtue of the fact that the connections are robust within some invariant subspace. We consider two previously studied examples and examine these in detail for a number of effects: (i) presence of internal symmetries within the chaotic saddles, (ii) phase-resetting, where only a limited set of connecting trajectories between saddles are possible, and (iii) multistability of periodic orbits near bifurcation to cycling attractors. The first model consists of three cyclically coupled Lorenz equations and was investigated first by Dellnitz et al. [Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1243-1247 (1995)]. We show that one can find a "false phase-resetting" effect here due to the presence of a skew product structure for the dynamics in an invariant subspace; we verify this by considering a more general bi-directional coupling. The presence of internal symmetries of the chaotic saddles means that the set of connections can never be clean in this system, that is, there will always be transversely repelling orbits within the saddles that are transversely attracting on average. Nonetheless we argue that "anomalous connections" are rare. The second model we consider is an approximate return mapping near the stable manifold of a saddle in a cycling attractor from a magnetoconvection problem previously investigated by two of the authors. Near resonance, we show that the model genuinely is phase-resetting, and there are indeed stable periodic orbits of arbitrarily long period close to resonance, as previously conjectured. We examine the set of nearby periodic orbits in both parameter and phase space and show that their structure appears to be much more complicated than previously suspected. In particular, the basins of attraction of the periodic orbits appear to be pseudo-riddled in the terminology of Lai [Physica D 150, 1-13 (2001)].     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three Body Problem. Part II

We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski [12]. Electronic Supplementary Material: Supplementary material is available in the online version of this article at An erratum to this article is available at .     

Topologically inequivalent orbits induced by noise

In this Letter we extend the concept of stochastic resonance. We show that in forced excitable systems noise can be responsible for the appearance of recurrences presenting a robust topological organization inequivalent to the periodic orbits of the deterministic system. As in stochastic resonance, these new structures are most pronounced at an optimal noise intensity.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

An automated algorithm for stability analysis of hybrid dynamical systems

There are many hybrid dynamical systems encountered in nature and in engineering, that have a large number of subsystems and a large number of switching conditions for transitions between subsystems. Bifurcation analysis of such systems poses a problem, because the detection of periodic orbits and the computation of their Floquet multipliers become difficult in such systems. In this paper we propose an algorithm to solve this problem. It is based on the computation of the fundamental solution matrix over a complete period–where the orbit may contain transitions through a large number of subsystems. The fundamental solution matrix is composed of the exponential matrices for evolution through the subsystems (considered linear time invariant in this paper) and the saltation matrices for the transitions through switching conditions. This matrix is then used to compose a Newton-Raphson search algorithm to converge on the periodic orbit. The algorithm–which has no restriction of the complexity of the system–locates the periodic orbit (stable or unstable), and at the same time computes its Floquet multipliers. The program is written in a sufficiently general way, so that it can be applied to any hybrid dynamical system.     

Reduction of the semisimple 1:1 resonance

The method of “averaging” is often used in Hamiltonian systems of two degrees of freedom to find periodic orbits. Such periodic orbits can be reconstructed from the critical points of an associated “reduced” Hamiltonian on a “reduced space”. This paper details the construction of the reduced space and the reduced Hamiltonian for the semisimple 1:1 resonance case. The reduced space will be a 2-sphere in R³, and the reduced differential equations will be Euler's equations restricted to this sphere. The orbit projection from the energy surface in phase space to this sphere will be the Hopf map. The results of the paper are related to problems in physics on “degeneracies” due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.     

Special features of galactic dynamics: Disc dynamics

This is a tutorial presentation of special features of galactic disc dynamics, which completes our introduction to galactic dynamics initially presented in [30]. The emphasis is on topics where galactic dynamics and celestial mechanics share common starting points and/or methods of approach. We start by giving some definitions and general notions on the link between observations and dynamical modeling of discs. Then we focus on the application of resonant Hamiltonian perturbation theory in disc resonances. By examining in detail the case of the Inner Lindblad resonance, we demonstrate how resonant perturbation theory leads to an orbital theory of spiral structure in normal galaxies. Passing to barred galaxies, the phase space structure and the role of chaos in the corotation region are analyzed. This is accomplished by a summary of the modern theory of invariant manifolds of unstable periodic orbits in the vicinity of L₁ or L₂, which can interpret the generation of spiral patterns by chaotic orbits beyond corotation. Some additional topics, potentially important for disc dynamics, are briefly commented.     

Symmetry and Resonance in Periodic FPU Chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This “Birkhoff–Gustavson normal form” retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems. So the periodic FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries. Received: 25 July 2000 / Accepted: 20 December 2000     

The quasi-periodic doubling cascade in the transition to weak turbulence

The quasi-periodic doubling cascade is shown to occur in the transition from regular to weakly turbulent behaviour in simulations of incompressible Navier–Stokes flow on a three-periodic domain. Special symmetries are imposed on the flow field in order to reduce the computational effort. Thus we can apply tools from dynamical systems theory such as continuation of periodic orbits and computation of Lyapunov exponents. We propose a model ODE for the quasi-period doubling cascade which, in a limit of a perturbation parameter to zero, avoids resonance related problems. The cascade we observe in the simulations is then compared to the perturbed case, in which resonances complicate the bifurcation scenario. In particular, we compare the frequency spectrum and the Lyapunov exponents. The perturbed model ODE is shown to be in good agreement with the simulations of weak turbulence. The scaling of the observed cascade is shown to resemble the unperturbed case, which is directly related to the well known doubling cascade of periodic orbits.     

ANALYTICAL CALCULATION OF BIFURCATION POINTS IN STANDARD MAPPING

For standard mapping, we calculate analytically the values of non-integrable parameter K where the period 1 to 4 orbits variate their stability via tangent bifurcation, period doubling bifurcation and period equal bifurcation.Hence, the analytical stable windows of periodic orbits are given and a complete picture can be seen for the behaviour of bifurcation of periodic motion in standard mapping.     

Quantum spectra and classical periodic orbit in the cubic billiard

Quantum billiards have attracted much interest in many fields. People have made a lot of researches on the two-dimensional (2D) billiard systems. Contrary to the 2D billiard, due to the complication of its classical periodic orbits, no one has studied the correspondence between the quantum spectra and the classical orbits of the three-dimensional (3D) billiards. Taking the cubic billiard as an example, using the periodic orbit theory, we find the periodic orbit of the cubic billiard and study the correspondence between the quantum spectra and the length of the classical orbits in 3D system. The Fourier transformed spectrum of this system has allowed direct comparison between peaks in such plot and the length of the periodic orbits, which verifies the correctness of the periodic orbit theory. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.     

非扩散洛伦兹系统的周期轨道

混沌系统的奇怪吸引子是由无数条周期轨道稠密覆盖构成的,周期轨道是非线性动力系统中除不动点之外最简单的不变集,它不仅能够体现出混沌运动的所有特征,而且和系统振荡的产生与变化密切相关,因此分析复杂系统的动力学行为时获取周期轨道具有重要意义.本文系统地研究了非扩散洛伦兹系统一定拓扑长度以内的周期轨道,提出一种基于轨道的拓扑结构来建立一维符号动力学的新方法,通过变分法数值计算轨道显得很稳定.寻找轨道初始化时,两条轨道片段能够被用作基本的组成单元,基于整条轨道的结构进行拓扑分类的方式显得很有效.此外,讨论了周期轨道随着参数变化时的形变情况,为研究轨道的周期演化规律提供了新途径.本研究可为在其他类似的混沌体系中找到并且系统分类周期轨道提供一种可借鉴的方法.     

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.     

Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling

An important issue in resonant vibration energy harvesters is that the best performance of the device is limited to a very narrow bandwidth around the fundamental resonance frequency. If the excitation frequency deviates slightly from the resonance condition, the power out is drastically reduced. In order to overcome this issue of the conventional resonant cantilever configuration, a non-resonant piezomagnetoelastic energy harvester has been introduced by the authors. This paper presents theoretical and experimental investigations of high-energy orbits in the piezomagnetoelastic energy harvester over a range of excitation frequencies. Lumped-parameter nonlinear equations (electromechanical form of the bistable Duffing oscillator with piezoelectric coupling) can successfully describe the large-amplitude broadband voltage response of the piezomagnetoelastic configuration. Following the comparison of the electromechanical trajectories obtained from the theory, it is experimentally verified that the piezomagnetoelastic configuration can generate an order of magnitude larger power compared to the commonly employed piezoelastic counterpart at several frequencies. Chaotic response of the piezomagnetoelastic configuration is also compared against the periodic response of the piezoelastic configuration theoretically and experimentally. Overcoming the bias caused by the gravity in vertical excitation of the piezomagnetoelastic energy harvester is discussed and utilization of high-energy orbits in the bistable structural configuration for electrostatic, electromagnetic and magnetostrictive transduction mechanisms is summarized.     

Hybrid stable-chaotic states in coupled quantum well stadia

We use tunnel current spectroscopy to investigate the quantum states of two GaAs quantum wells coupled by a low (100 meV) (AlGa)As tunnel barrier. A high tilted magnetic field is used to generate strongly chaotic electron motion in the two wells which act as coupled chaotic ‘stadia'. The effect of the tunnel barrier on the dynamics of the system depends on the magnitude of the applied bias voltage V. For V375 mV, the central potential barrier acts as a perturbation which modifies the trajectories of selected periodic orbits in the quantum well. Scattering off the central barrier also generates new periodic orbits involving multiple collisions on all three barriers. These orbits ‘scar' distinct sets of eigenstates which generate periodic resonant peaks in the current–voltage characteristics of the device. When the device is biased such that the injected electrons just surmount the central barrier, our calculations reveal novel hybrid scarred states with both stable and chaotic characteristics.     

Coherent Stochastic Resonance in One Dimensional Diffusion with One Reflecting and One Absorbing Boundaries

It is shown that the single-step periodic signal (periodic telegraph signal) can not produce coherent stochastic resonance for diffusion on a segment with one absorbing and one reflecting end points while the multi-step periodic signal does. The general features of this process are exihibited. The resonant frequency is found to decrease and the mean first passage time at resonant frequency increases linearly, as we increase the length of the medium. The cycle variable is shown to be the proper argument to express the first passage probability at resonance. A formula for first passage probability at resonance is derived in terms of two universal functions, which clearly isolates its dependence on the length of the medium.