Periodic orbits in tall laterally heated rectangular cavities.

We study the mechanism of bifurcation of three-dimensional periodic orbits from the plane of motion of the primaries. The branchings of simple and double periodic three-dimensional orbits occur at the plane periodic orbits of indifferent vertical stability. These vertical-critical plane orbits are doubly important. Not only are they necessary in order to specify the vertically stable segments of the families of plane periodic orbits, but they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely, the simple and double periodic. We describe here suitable numerical techniques for their accurate determination and for their continuation through the range of the mass parameter μ of the problem. We use these techniques to determine the first few vertical-critical orbits of the basic families of periodic orbits of the problem, namely, families l, m, i, h, a, b and c, for the entire range of μ. The initial conditions of over three hundred vertical-critical orbits are given, classified in twenty-one series. Their horizontal stability is also examined and this leads to predictions of where in the range of μ stable three-dimensional periodic orbits may not branch from these basic families.

This paper studies complete stable and unstable periodic solutions for n-dimensional nonlinear discrete dynamical systems. The positive and negative iterative mappings of discrete systems are used to develop mapping structures of the stable and unstable periodic solutions. The complete bifurcation and stability analysis are presented for the stable and unstable periodic solutions which are based on the positive and negative mapping structures. A comprehensive investigation on the Henon map is carried out for a better understanding of complexity in nonlinear discrete systems. Given is the bifurcation scenario based on positive and negative mappings of the Henon map, and the analytical predictions of the corresponding periodic solutions are achieved. The corresponding eigenvalue analysis of the periodic solutions is presented. The Poincare mapping sections relative to the Neimark bifurcations of periodic solutions are presented. A parameter map for periodic and chaotic solutions is developed. The complete unstable and stable periodic solutions in nonlinear discrete systems are presented for the first time. The results presented in this paper provide a new idea for one to rethink the current existing theories.

Bifurcations analysis of oscillating hypercycles

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to $50^\circ$ - $60^\circ$, which may be related with the existence of real planetary systems.

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.

I N RECENT years, many proposed satellite missions have undergone a paradigm shift away from large, expensive spacecraft toward the use of smaller satellites flying in formation. Although formation flying offers a number of advantages, such as robustness of the mission to failure and potential cost savings, the benefits are partially offset by the increased complexity of the guidance, navigation, and control systems and by V-limited mission durations. The development of advantageous relative dynamics between the constituent satellites could help extend mission durations by minimizing fuel usage. In particular, periodic or quasiperiodic relative orbits would reduce the control effort necessary to maintain the formation and significantly prolong the mission’s lifetime. Attempts to identify naturally periodic relative orbits can be found throughout the current literature. Vaddi et al. [1] modified initial conditions from the Hill–Clohessy–Wiltshire (HCW) equations to enforce bounded relative motion in the presence of small eccentricity and second-order differential gravity terms. Kasdin and Kolemen [2] solved the Hamilton–Jacobi equation in terms of epicyclic orbital elements to derive bounded, periodic orbits in the presence of higher-order gravity and certain perturbations. Schaub and Alfriend [3] identified the conditions for J2-invariant relative orbits bymatching the secular drift rates of themean orbital elements of the chief and deputy satellites. In most of these analytical cases, however, the authors established precise periodic motion using approximated dynamical models. When higher fidelity dynamics and realistic orbital parameters are used, these methods break down and exhibit relative orbit drift or a high sensitivity to initial condition errors. Several numerical approaches to the problem of relative orbit periodicity have also been attempted. Sabatini et al. [4] and Izzo and Sabatini [5] used a refined genetic algorithm to optimize relative trajectories for maximum periodicity in the presence of J2 perturbations. Quasi-periodic orbits were obtained for two sets of “magic” inclinations (49.11 and 63.43 deg), at which resonance between the formation’s inand out-of-plane motion results in projected circular orbit (PCO) formations with very small orbital drift. Damaren [6] formulated an iterative shooting approach based on the Newton method to close the relative orbit and achieve “almost” periodic initial conditions. This technique requires a low-level state-feedback control loop to track a trajectory based on the HCW equations, but is robust to initial condition errors and exhibits very low drift characteristics in circular orbits. A similar approach was taken by Becerra et al. [7], who applied a nonlinear Hamiltonian model to the Newton method to obtain quasi-periodic relative initial conditions of the deputy in the presence of the J2 perturbation. A linear quadratic regulator (LQR) controller was used to track reference trajectories that were developed from sinusoidal functions of time and based on a Keplerian orbit. As yet, a method of numerically designing reference trajectories that match or closely match the natural perturbed relative motion of the deputy satellite in a generic orbit, and that are robust to initial condition errors, has not been developed. Such trajectories, with a properly designed low-authority controller, could be tracked for minimal V requirements, thus enabling formation flying missions to be significantly extended. The intent of this study is to design such trajectories by continuing the development of the methods used to search for almost-periodic orbits, begun in [6], and applying them to the J2-invariant trajectories in [3]. The resulting relative orbits will approach true periodicity by capitalizing on advantageous initial conditions, numerically designed reference trajectories, and optimal control strategies.

Bifurcations of periodic orbits and band zones of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is first considered without any dissipation or external forcing. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in the in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling simultaneously occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution and to detect the bifurcation location. Although the trajectory of the period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced away from the bifurcation point. Pre-compression is found to affect the periodic orbit frequencies, which in turn result in changes in the wave propagation band zones. These changes are illustrated by studying vibration transmission through a granular chain at different frequencies to ascertain the band zone limits.

Global bifurcations in Rayleigh-Bénard convection. Experiments, empirical maps and numerical bifurcation analysis

Global bifurcations in dynamical systems are of considerable interest because they can lead to the creation of chaotic behaviour [Hilborn, 1994]. Global bifurcations are to be distinguished from local bifurcations around an unstable periodic solution. Typically, they occur when a homoclinic point is created. A homoclinic point is an intersection point between the stable and the unstable manifold of a steady state saddle point p on the Poincare section of a, at least, 3D flow. The presence of a homoclinic point implies a complicated geometrical structure of both the stable and the unstable manifolds usally referred to as a homoclinic tangle. When a homoclinic tangle has developed, a trajfectory that comes close to the saddle point behaves in an erratic way, showing sensitivity to initial conditions.

The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity $e'=1$, but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter $\mu=0.5$ (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke's computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to $\mu$ and $e'<1$. Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.

In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincar\'e surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.

Ensuring reliable operation of renewable energy sources requires robust grid-connected converters. Depending on the choice of synchronization methods, grid-connected converters may exhibit distinct nonlinear behavior that plays a vital role in determining their transient stability. Recently, the grid-voltage-modulated direct power control (DPC) has been proposed as an alternative to the conventional phase-locked loop (PLL) to enhance the dynamic response of the grid-following converter (GFLC). However, existing studies have primarily treated the DPC-based GFLC as a linear system. In this paper, we investigate the nonlinear behavior of this converter under weak grids using a large-signal model based on double reference frames. Our findings reveal that the DPC-based GFLC demonstrates sustained oscillation. Interestingly, the stable periodic orbit observed does not arise from a Hopf bifurcation but rather a saddle-node bifurcation of periodic orbits. This critical bifurcation is characterized by the coexistence of a stable periodic orbit and a stable equilibrium point, resulting in a sudden contraction of the converter's stability region. Furthermore, we provide a comparison between the nonlinear behavior of PLL-based GFLCs and DPC-based GFLCs. To validate our findings, we present full-circuit simulations and laboratory experiments.

Acontrol lawisderivedandanalyzed thatstabilizesa classof unstableperiodicorbitsin theHillrestrictedthreebody problem. The control law is derived by stabilizing the short-time dynamics of motion about a trajectory by the use of a feedback law specie ed by the instantaneous eigenvalue and eigenvector structure of the trajectory. This law naturally generalizes to a continuous control law along an orbit. By applying the control to an unstable periodic orbit, we can explicitly compute the stability of the control over long periods of time by computing the monodromy matrix of the periodic orbit with its neighborhood modie ed via the control law. For the case of an unstable halo periodic orbit in the Hill restricted three-body problem, we e nd that the entire periodic orbit can be stabilized. The resulting stable periodic orbits have three distinct oscillation modes in their center manifold. We discuss how this control can be applied to formation e ight about a halo orbit. Some practical implementation issues of the control are also considered. We show that the control acceleration can be provided by a low-thrust engine and that the total fuel cost of the control can be quite reasonable. I. Introduction T HIS paper studies the stabilization of an unstable periodic orbit in the Hill problem, which can serve as a general model for motion in the Earth‐ sun system. Results of this study will be relevant to the dynamics and control of a constellation of spacecraft in an unstable orbitalenvironment such as found near the Earth‐ sun libration points. It will also shed light on the practical control and computation of a single spacecraft trajectory in an unstable orbital environment over long time spans. We investigate the application of feedback control laws to stabilize a periodic orbit in the sense of Lyapunov (see Ref. 1) (note, not asymptotic stability). Thus, the stabilized trajectory will consist of oscillatory motions about the nominal trajectory, which in this context can be interpreted as motions in the center manifold of the stabilized periodic orbit. We show that an entire class of such control laws can be dee ned and their stability analyzed as a timeperiodic linear system. The fuel expenditure for such control laws is often quite small and scales with the mean distance between the controlled motion and the nominal trajectory (which is a periodic orbit in this application). The problem of spacecraft control in unstable orbits is not new. (See Refs. 2 and 3 for reviews.) However, these previous studies have focused on stationkeeping control for a single spacecraft and have not considered how the relative motion of a formation of spacecraft could be stabilized and their dynamics modie ed, which is what we consider here. Proposed space observatories of the future include ambitious interferometricimagersthatusebaselinesofhundredsorthousandsof kilometers between spacecraft to attain sufe ciently high resolutions to image planets around distant stars. To carry out these imaging procedures requires that the relative motion between spacecraft be known extremely accurately and that the spacecraft “ e ll in” an effective image e eld as they move relative to each other. A periodic

In this work we study a wide class of symmetric control systems that has the Chua's circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-three global bifurcations that act as organizing centres of the complex dynamics Chua's circuit exhibits in the parameter range considered. A detailed (although partial) bifurcation set in a three'parameter space is presented in this paper. We show relations between several high-codimension bifurcations of equilibria, periodic orbits and global connections. Some of the global bifurcations found have been neither analytically nor numerically treated in the literature.