Non-generic spectral statistics in the quantized stadium billiard

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

Every chaotic attractor is thought to be composed of the collection of unstable manifolds of all its unstable periodic orbits (UPOs). These UPOs can be used for communication by amplitude or phase modulation just as one would modulate stable periodic orbits (resonant frequencies of a linear system or limit cycles-in a nonlinear system). Using UPOs for communications has a variety of benefits. Since a chaotic attractor contains an infinite number of UPOs, a single time series can carry many different messages by placing each one on a different UPO. Because UPOs are topological properties of a chaotic attractor, the signal has a natural noise immunity, and receipt of a signal should be robust to moderate differences between the transmitter and receiver parameters. >

Deterministic chaotic systems give rise to very complex dynamical behavior in spite of the fact that they are usually of very low dimension. The exceptionally high sensitivity to changes in initial conditions makes dealing with these systems non-trivial due to inherent measurement errors and limited numerical precision. Within a chaotic attractor it is usually possible to identify unstable point attractors and unstable periodic orbits. One of the widely accepted techniques for stabilizing unstable point attractors and periodic orbits is the so called OGY controller method developed by Ott, Grebogi, and Yorke. In the case of stabilizing an unstable point attractor within the chaotic map, the OGY controller effectively forces the system to move on the corresponding stable manifold in the state-space. On the other hand, it can be shown that this strategy coincides with the design of a full state feedback controller that places the unstable eigenvalues of the state matrix of the locally linearized system at zero while keeping the location of the stable eigenvalues unchanged. When the stabilization of the original, open-loop system is the only goal, the OGY approach does not offer any advantage over other stabilizing control actions such as pole placement to arbitrary but stable eigenvalue locations. In addition, neither OGY nor arbitrary pole placement explicitly deal with the problem of noise sensitivity. Hence, when the original chaotic system is noise corrupted, it is not clear if the OGY approach is even close to being optimal with respect to noise cancelation with "acceptable" control action. Although the answer to this question is negative in the general case, there are situations where the optimal system behavior is achieved. The contribution of this paper is twofold. The first contribution is the rigorous discussion of the relationship between the OGY approach and the pole placement and the extension of the OGY method to stabilization of systems with more than one unstable eigenvalue. The second contribution of the paper is the proof that the OGY control approach to the equilibrium point stabilization in the presence of persistent, magnitude bounded process noise is actually optimal when the system performance is measured by the l∞-norm of the control signal. The proof is based on properties of the optimal l1-norm controller design and on the parameterization of all stabilizing controllers for a given linear system.

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

Quantum scars correspond to enhanced probability densities along unstable classical periodic orbits. In recent years, research on quantum scars has extended to various systems including the many-body regime. In this work we focus on the formation, prevalence, and stability of linear bouncing-ball (BB) scars in two-dimensional (2D) quantum wells. These scars have relevance as effective and controllable channels in quantum transport. We utilize imaginary time propagation to solve the 2D Schr\"odinger equation within an arbitrary external confining potential, specifically the quantum well model with external perturbations. We show how BB scars begin to emerge with a single perturbative peak, such as a repulsive bump or attractive dip that simulates the effect of a charged nanotip in the system. We then identify the optimal size of the perturbative peak to maximize the prevalence of these scars. Finally, we investigate the stability of BB scars against external noise and find that some of them are remarkably robust. This suggests promising opportunities for further applications of BB scars in quantum transport.

Unstable periodic orbits (UPOs) extracted from experimental electrocardiograph signals are reported for normal and pathological human cardiac rhythms. The periodicity and distribution of the orbits on the chaotic attractor are found to be indicative of the state of health of the cardiac system. The normal cardiac system is characterized by three to four UPOs with typical periodicities and intensities. However, pathological conditions such as premature ventricular contraction, atrio ventricular block, ventricular tachy arrhythmia, and ventricular fibrillation have UPOs whose periodicity and intensity distribution are quite distinct from those of the healthy cases and are characteristic of the pathological conditions. Eigenvalues and the largest positive Lyapunov exponent value for the UPOs are also reported. The UPOs are shown to be insensitive to the embedding dimension and the present UPO analysis is demonstrated to be reliable by the method of surrogate analysis.

An optimal gains matrix for time-delay feedback control

To unravel the temporal features of the peripheral tissue injury induced persistent nociceptive discharge, single wide dynamic range (WDR) unit activity was recorded extracellularly in lumbar dorsal horn of anesthetized rats and interspike interval (ISI) series were obtained. Subcutaneous (s.c.) bee venom (BV) injection induced persistent discharge of spinal WDR neurons and has been well established to be a good model in evaluation of tissue injury induced pain. By applying a more novel approach, i.e., the unstable periodic orbit (UPO) identification method, we detected a family of significant separate UPOs (period-1, 2 and 3 orbits) within the ISI series of BV-induced nociceptive discharge, but not spontaneous background activity of spinal WDR neuron. Furthermore, temporally dynamic changes of UPOs at lower period-1, 2 and 3 for 4 successive time segments within 1 h time course of WDR unit firing showed temporally dynamic changes, i.e., new orbits with longer ISIs emerged and those with shorter ISIs vanished with time change. By using this method we suggest that BV-induced nociceptive discharge of spinal WDR neuron be a kind of deterministic activity and various UPOs may play some role in temporal coding of sensory information.

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragas. A recent paper by Fiedler et al. Phys. Rev. Lett. 98, 114101 (2007) uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al. for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragas-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragas control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al. example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.

We consider a dissipative version of the standard nontwist map. It is known that nontwist systems may present a robust transport barrier, called shearless curve, that gives rise to an attractor that retains some of its properties when dissipation is introduced. This attractor is known as shearless attractor, and it may be quasiperiodic or chaotic depending on the control parameters. We describe a route for the destruction and resurgence of the quasiperiodic shearless attractor by analyzing the manifolds of the unstable periodic orbits (UPOs) which are fixed points of the map. We show that the shearless attractor is destroyed by a collision with the UPOs and it resurges after the reconnection of the unstable manifolds of different UPOs.

A period-one unstable periodic orbit (UPO) is identified in the inter--edge-localized-mode (ELM) intervals of a high-performance quasisteady state discharge of JT-60U using UPO identification based on their recurrence properties and the so-called fixed point transform. This result is tested by use of surrogate data sets. This diagnostics invariant interpretation of the ELM confinement state gives detailed and applicable quantitative information since the UPOs in general form the skeleton of dynamical invariants and have in numerous systems been shown to be controllable by weak parameter perturbations.

Time-delayed feedback can be used to stabilize unstable periodic orbits (UPOs) in dynamical systems. The technique involves feedback of signals delayed by the orbit's period so that the feedback signals vanish on the target orbit and hence the UPO becomes a solution of the original system. An obvious advantage of this method is that it requires only the knowledge of the period of the target UPO. In this brief, a pulsewidth-modulated current-mode H-bridge inverter serves as a subject of investigation. Under certain parameter ranges, it exhibits higher periodic solutions and chaos. A simple time-delay feedback control method is applied to stabilize the UPOs. By using the time-delayed feedback control method, the stability limit and gain range of the system are increased.

Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in part of the chaotic set contained in that region. This result has been rigorously shown to be valid for hyperbolic chaotic systems. Chaotic sets encountered in most physical situations, however, are typically nonhyperbolic. The purpose of this paper is to test the goodness of the unstable periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems. We first directly compare the natural measure from a typical trajectory on the chaotic set with that evaluated from unstable periodic orbits embedded in the set. As an indirect check, we then compute the difference between the long-time average values of physical quantities evaluated with respect to a typical trajectory and those computed from unstable periodic orbits. Results with the H\'enon map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic systems.

In an elliptic billiard, we find avoided level crossings taking place over wide ranges, which are of a Demkov type for generations of eigenfunctions localized on an islands chain and its pair unstable periodic orbit. For a proof of the existence of avoided level crossings, first, we show that the quantized eigenvalue of the unstable periodic orbit, obtained by the Einstein-Brillouin-Keller quantization rule, passes the eigenvalues of bouncing-ball modes localized on the unstable periodic orbit after Demkov type avoided level crossings so that pairs of bouncing-ball modes are sequentially generated. Next, by using a perturbed Hamiltonian, we show that off-diagonal elements in Hamiltonian are nonzero, which give rise to an interaction between two eigenfunctions. Last, we verify that the observed phenomenon is Fermi resonance: that is, the quantum number difference of two normal modes equals the periodic orbits, where eigenfunctions are localized after an avoided level crossing.