Bifurcation and branching of equilibria in reversible-equivariant vector fields

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.

The techniques from singularity theory are applied to the multiparameter bifurcation problem. The classification of (D4, S1)-equivariant bifurcation problems with topological codimension less than or equal to 2 is given. The corresponding recognition conditions are set up.

T HE relative motion of a satellite (follower) with respect to the reference satellite (leader) in a circular orbit is described by autonomous nonlinear differential equations. The linearized equations at the origin are known as Hill–Clohessy–Wiltshire (HCW) equations [1]. The in-plane motion and the out-of plane motion are independent. The latter is a simple sinusoidal motion. The former has also periodic solutions under the so-called Clohessy–Wiltshire condition. Periodic solutions are given by sinusoidal functions, and the relative orbits are ellipses. These orbits are parametrized by two parameters representing size and phase. Because of this simple nature, relative periodic orbits are used for formation flying [2]. Periodic orbits of the nonlinear relative dynamics are also characterized by the parameters of the follower’s elliptic orbit and its inclination angle in an inertial reference frame, and initial conditions for periodic orbits are explicitly given [3]. If the reference orbit is elliptic, the equations of relative motion involve the true anomaly and the radius of the orbit, which are periodic functions. The linearized equations of motion at the origin are known as Tschauner–Hempel (TH) equations. The in-plane motion and the out-of plane motion remain independent. However, the derivation of the state transition matrix in this case is not immediate. In earlier studies [4–6], the true anomaly is used as independent variable, because this simplifies the resulting equations and the transition matrix can be obtained for them. Rendezvous problems have been extensively studied using this representation [7,8]. The out-of-plane motion is a sinusoidal motion, and the inplane motion has periodic solutions [2,9]. The condition, which generalizes the CW condition, for periodic solutions is given in [9]. Periodic orbits of the TH equations are also used for formation flying [2,9,10]. However, they are more complicated compared to ellipses of the HCW equations, and their shape changes with eccentricity. Some of them are very much distorted. Hence, it is not intuitively clear how to design relative orbits. The direct study of the relative dynamics using time as independent variable is given in [11], and a series expansion of the state transition matrix in the eccentricity is obtained in [12]. A closed form of the state transition matrix is recently derived in [13],where a brief history of elliptic rendezvous is also found. The characterization of periodic orbits of the nonlinear relative dynamics as functions of time is extended to the elliptic case, and initial conditions for periodic orbits are explicitly given [14]. The general solution of the relative dynamics based on the orbital elements is obtained in [15], and the relative motion invariant manifold, where the solution lies, is determined. This technical note is concernedwith the generation of all periodic solutions of the in-plane motion of the TH equations. For this purpose, the state transition matrix of [16], which is given as a function of the true anomaly, is used. The solution is parametrized by four constants, and the condition for periodic solution follows from this. As in the case of the HCWequations, size and phase parameters are introduced. The main contributions of this note are summarized as follows. First, the region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Given size and phase parameters, a graphical method to determine the initial point of a periodic orbit at a given true anomaly is proposed. Varying the value of the true anomaly, the whole family of periodic orbits is generated. The time domain versions of the region and periodic orbits are determined by multiplying the radius of the leader orbit. This clarifies how the shape of a periodic orbit changes with eccentricity and why the shapes of some orbits are distorted. As the eccentricity goes to zero, the region of periodic motion shrinks to a single ellipse, which is a periodic orbit of the HCW equations. The size parameter gives the distance between the leader and the follower, and the phase determines the position of the follower. As in [15], the maximum and minimum distances to the leader can be calculated. Hence, our method is useful to design relative orbits with size and position specifications. This note is organized as follows. Section II reviews the equations of relativemotion along an elliptic orbit. Section III gives a geometric method to draw periodic orbits of the TH equations. Finally, Section IV gives conclusions.

Steady state bifurcations and exact multiplicity conditions via Carleman linearization

In this study, we study the two-dimensional Navier–Stokes flows with time-periodic external forces. Invariant solutions, including periodic orbits and relative periodic orbits, are extracted with the recurrent flow analysis, while low-dimensional projections based on the dynamic mode decomposition algorithm are used to reduce the cost of searching nearly recurrences. When the period of forces gets a constant increase, the flows change from the stable time-periodic state to oscillate and even turbulent flows. In all cases, one periodic orbit is identified near the initial stage. This orbit represents the stable/unstable base state, and the trajectories of vorticity fields are trapped inside it or escape away from it leading to oscillating/turbulent motions. For the oscillating flows, periodic orbits without any symmetries play the role that the flows visit them and then move away from them to other orbits. In addition, for a moderate period of forces, a bursting phenomenon occurs and the state of oscillating flows turns to turbulent flows with the rapid increase in energy. For the turbulent motions, one unstable periodic, which qualitatively represents the shapes of a large vortex dipole that exists in the turbulent motions, is obtained. Its statistical significance is shown by the frequency of that flows visit it.

Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems

4.1 Example manifolds and their representations 4.2 Supercritical boundary value problems 4.2.1 A weak substitute for the vector-field x 4.2.2 Critical points of a free functional 4.2.3 Critical points of constrained functionals 4.2.4 Applications 4.3 Harmonic maps 4.4 Supercritical boundary value problems: revisited 4.4.1 A better substitute for the vector-field x? 4.4.2 Conformal vector-fields and conformal maps 4.4.3 Yamabe’s equation 4.4.4 Yamabe’s equation with boundary terms 4.4.5 Conformal vector fields on conformally flat manifolds 4.4.6 The bifurcation problem on \(\mathbb {R}^n\), \(\mathbb {S}^n\), \(\mathbb {H}^n\) 4.4.7 The bifurcation problem on rotation surfaces 4.5 Harmonic maps into conformally flat manifolds

The steady state bifurcations near a double zero eigenvalue of the reaction diffusion equation associated with a tri-molecular chemical reaction (the Brusselator) are analysed. Special emphasis is put on three degeneracies where previous results of Schaeffer and Golubitsky do not apply. For these degeneracies it is shown by means of a LiapunovSchmidt reduction that the steady state bifurcations are determined by codimension-three normal forms. They are of types (9)31, (8)221 and (6a)ρ,κ in a recent classification of Z(2)-equivariant imperfect bifurcations with corank two. Each normal form couples an ordinary corank-1 bifurcation in the sense of Golubitsky and Schaeffer to a degenerate Z(2)-equivariant corank-1 bifurcation of Golubitsky and Langford in a specific way.

The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for C ∞ C^{\infty } systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a C ∞ C^{\infty } foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

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.

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. The system's relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the "method of slices," which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the "slice" can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.

The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to a spatially periodic domain play a prominent role in shaping the invariant sets of its chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (traveling wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetries lead to existence of equilibrium and periodic orbit solutions, induce decomposition of state space into invariant subspaces, and enforce certain structurally stable heteroclinic connections between equilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, how the geometry of its dynamical state space is organized by a rigid `cage' built by heteroclinic connections between equilibria, and demonstrate the preponderance of unstable relative periodic orbits and their likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuous symmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space flow through projections onto low-dimensional, PDE representation independent, dynamically invariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energy transfer rates.

Analytic and simulation studies for the steady-state equilibria and bifurcations of coupled microlaser arrays are described. Lateral cavity interactions affect the gain in each cavity, leading to active photonic lattice behavior, equivalent to a nonlinear coupled oscillator lattice. The coupled-cavity rate equations are employed to follow the coherent photon and carrier population in each lattice site. Fixed-point-type steady states, of constant lattice phase shift, result for low coupling strengths; the radiation envelope for these states conforms with a periodic Bloch state over the array. Bifurcations to limit cycles of increasing complexity occur at higher coupling via period doubling sequences. The associated spatial patterns of photon and carrier lattice distribution resemble photonic convection cells. Limit cycles of different periods, emanating mathematically from different original fixed points, coexist at high strengths, each one accessible from different initial conditions. The multiplicity of possible limit cycles in systems with many degrees of freedom (number of lattice sites) combined with changes in their accessibility from initial conditions offers new insights to chaotic transitions, compared to low dimensionality paradigms.

One of the goals of this paper is to describe explicitly the generic movement of eigenvalues through a one-to-one resonance in a linear Hamiltonian system which is equivariant with respect to a symplectic representation of a compact Lie group. The authors classify, this movement, and hence answer the question of when the collisions are 'dangerous' in the sense of Krein (1950) by using a combination of group theory and definiteness properties of the associated quadratic Hamiltonian. For example, for systems with no symmetry or O(2) symmetry generically the eigenvalues split, whereas for systems with S1 symmetry, generically the eigenvalues may split or pass. It is in this last case that one has to use both group theory and energetics to determine the generic eigenvalue movement. The way energetics and group theory are combined is summarized in table 1. The result is to be contrasted with the bifurcation of steady states (zero eigenvalue) where one can use either group theory alone (Golubitsky and Stewart 1987), or definiteness properties of the Hamiltonian (Carton-Oh) to determine whether the eigenvalues split or pass on the imaginary axis.