Periodic orbit analysis of a system with continuous symmetry--A tutorial.

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.

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.

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.

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.

We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincare section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry properties of the Hamiltonian system is described, and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid.

Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not captured well by current order reduction techniques, and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices [C. W. Rowley and J. E. Marsden, Phys. D, 142 (2000), pp. 1--19; S. Froehlich and P. Cvitanović, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), pp. 2074--2084], a symmetry reduction tool, and (ii) any standard order reduction technique, resulting in efficient mixed symmetry-dimensionality reduction schemes. In particular, using the dynamically orthogonal (DO) equations [T. P. Sapsis and P. F. J. Lermusiaux, Phys. D, 238 (2009), pp. 2347--2360] in the second step, we obtain a novel nonlinear symmetry-reduced dynamically orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the one-dimensional Korteweg--de Vries and two-dimensional Navier--Stokes equations.

Invariant solutions in a channel flow using a minimal restricted nonlinear model

Recent advances in the development of techniques for controlling chaos have motivated efforts to implement secure communication systems which exploit the properties of chaotic dynamical systems [1]. An efficient algorithm for improving the locking rate between receiver and transmitter in such systems has been reported recently [2]. An essential ingredient in controlling chaotic dynamics is the procedure for targetting the desired periodic or unstable orbit which will be utilised to carry the desired information. In earlier work [3] it has been shown that periodic orbits can be targeted using loss modulation. In the present work it is demonstrated that gain switching of external cavity laser diodes can be used for targeting periodic orbits.

This article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that, there exist infinitely many normalised relative periodic orbits on a dense subset of all energy levels, which are neither fixed points nor relative equilibria.

When analyzing concurrent software applications, symmetry reduction techniques dramatically narrow the size of the state space search by identifying computations that, because of symmetries in the system, are redundant. While analysis algorithms exploiting symmetry reduction are well developed, little has been done in discovering the nature of the symmetries of a system. What is even less researched is discovering symmetries that are particular to a temporal property. This paper proposes a general framework for discovering symmetries in systems that exhibit absolute or relative symmetries depending on the property of interest. Our work extends previous symmetry reduction techniques by making advances in automating generalized model automorphism discovery. Generalized model automorphisms capture exact abstractions and therefore preserve both the validity and the violation of any property of the analyzed system while achieving dramatic state space reduction.

We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse-Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point-charge magnetic field model, we consider the problem of separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

Analysis of dynamic respiratory system mechanics is generally based on a resistance-compliance model in which nonlinearities of the respiratory mechanics indices are not considered. The recently developed SLICE method analyzing consecutive volume slices of the tidal volume was used for determination of non-linear volume-dependent respiratory system mechanics. Volume-dependent compliance C(Slice) and resistance R(Slice) were compared with C(MLR) and R(MLR) obtained by standard multiple linear regression analysis (MLR). Prospective observational study. Pediatric intensive care unit in a university hospital. Fifteen pediatric patients, aged 24 days to 9.6 yrs, weighing 3-67.5 kg. None. With respect to their pulmonary status, the patients were grouped into three clinical groups: patients with no lung diseases, patients with restrictive lung diseases, and patients with obstructive lung diseases. All patients were mechanically ventilated via a cuffed endotracheal tube in the pressure-controlled mode. Flow and airway pressure were measured at the proximal end of the tube and tracheal pressure was continuously calculated. Respiratory mechanics were determined either with the SLICE method or, as reference, by using standard MLR. In most patients, the pressure-volume relationship was nonlinear, particularly in patients with restrictive and obstructive lung diseases. In the presence of considerable nonlinearity, the volume-dependent respiratory mechanics indices obtained by the SLICE method showed better agreement between recalculated and original pressure-volume loops compared with the MLR results. Furthermore, signs of overdistension of the patient's lung became obvious when using the SLICE method, whereas they were undetected by MLR. The SLICE method is well suited for the analysis of nonlinear volume-dependent respiratory system mechanics in pediatric patients. The SLICE method may be used as a first step toward an adaptation of ventilator settings with respect to the actual mechanical status of the patient's respiratory system, and, to prevent pulmonary overdistension.

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.

Symmetry reduction techniques exploit symmetries that occur during the execution of a system, in order to minimize its state space for efficient verification of temporal logic properties. This paper presents a framework for concisely defining and evaluating two symmetry reductions currently used in software model checking, involving heap objects and, respectively, processes. An on-the-fly state space exploration algorithm combining both techniques is also presented. Second, the relation between symmetry and partial order reductions is investigated, showing how one’s strengths can be used to compensate for the other’s weaknesses. The symmetry reductions presented here were implemented in the dSPIN model checking tool. We performed a number of experiments that show significant progress in reducing the cost of finite state software verification.

We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p ≤ 16 of the Poincaré map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincaré map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences.