Phase Space Of System Research Articles

A Graphical Calculus for Lagrangian Relations

Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.

Integration of Nonlinear Dynamics and Machine learning for Diagnostics of a Single-Stage Gear Box


 
 
 The current study concerns diagnostics of a one-stage gear- box based on the integration of physics and machine learn- ing. A physics-based model of this system is developed, then a nonlinear dynamic analysis is performed. The accuracy of the model is validated by comparing fundamental phenomena observed in synthetic and experimental data. To address the diagnostics problem synthetic data are generated for faulty and healthy conditions. Further, physics-informed features are extracted from the phase space of the dynamic system. It is shown that these features are highly informative about the health condition of the system. Also, their advantages over purely statistical features are demonstrated by a feature ranking technique. Subsequently, they are used as inputs in a machine learning model that is developed and optimized for fault diagnostics. The performance of the proposed method is investigated from different aspects, e.g., the accuracy of fault classification, robustness to noise, and generalization to unseen scenarios.
 
 

Chaos-assisted many-body tunnelling.

We study the interplay of chaos and tunneling between two weakly coupled Bose-Josephson junctions. The classical phase space of the composite system has a mixed structure including quasi-integrable self-trapping islands for particles and excitations, separated by a chaotic sea. We show that the many-body dynamical tunneling gap between macroscopic Schrödinger cat states supported by these islands is chaos-enhanced. The many-body tunneling rate fluctuates over several orders of magnitude with small variations of the system parameters or the particle number.

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Abstract This paper deals with the control problems governed by fractional impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder’s fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution, phase space, and approximate controllability of fractional order impulsive systems in Banach spaces. Finally, we investigate the approximate controllability of the fractional order heat equation with non-instantaneous impulses and delay by using the developed results.

Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. We defined conditions for discovering new invariant subsets in the state space of a dynamical system starting from an invariant subset based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subsets in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subsets of the dynamical system and time-series data from any one of the invariant subsets. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subsets can be combined to approximate the global Koopman operator.

Caustics in turbulent aerosols: an excitable system approach

Aerosol particles are inertial particles. They cannot follow the surrounding fluid in a region of high vorticity. These encounters render the particle velocity field ${{\boldsymbol {v}}}$ locally compressible. Caustics occur if the trace of the gradient matrix $\textsf{$\boldsymbol{\sigma}$}$ of the field ${{\boldsymbol {v}}}$ locally diverges. For three-dimensional isotropic and homogeneous turbulence, the dynamics of the gradient matrix can be expressed in terms of three geometric invariants. In the present paper we establish a parametrisation of this problem where the dynamics takes the form of an excitable stochastic dynamical system with a three-dimensional phase space. The deterministic part of the dynamics is solved analytically. We show that the deterministic system has a globally attractive stable fixed point. Small noise induces excursions from the fixed point that typically relax straight back towards the fixed point. Caustics emerge as non-trivial return to a global fixed point when noise excites a trajectory across a stability threshold. The relaxation to the global fixed point will then involve at least one, and it may involve two or even three divergences of $\boldsymbol {Tr}\,\textsf{$\boldsymbol{\sigma}$}$ . Based on a combination of analytical insights and numerical analysis, we determine the rate of occurrence, duration and relative observation probability of caustic events in turbulent aerosols. Our analysis reveals that each approach towards a divergence proceeds along a straight line in the phase space of the dynamical system, which can help identify caustics. Moreover, there are infinite ways in which caustics can arise, namely whenever $\boldsymbol {Tr}\,\textsf{$\boldsymbol{\sigma}$}$ tends to $-\infty$ , so that no two caustics look the same.

Revisiting Kepler: New Symmetries of an Old Problem

The Kepler orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits \(\mathrm { PSL}_2(\mathbb {R})\) as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre \(y^{\prime \prime }= \omega (x, y, y^{\prime })\), vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new.

Синтез управління стабілізацією геостаціонарного супутника у точці стояння

Розроблено методи та алгоритми балістичного забезпечення управління стабілізацією довготи точки стояння геостаціонарного супутника (ГСС). Еволюція поточних координат ГСС моделювалася диференціальними рівняннями руху матеріальної точки у центральному гравітаційному полі з урахуванням обмежених структурно-параметричних збурень відомої інтенсивності, викликаних загальним впливом Сонця, Місяця, відмінністю гравітаційного поля Землі від центрального тощо. Конкретне математичне моделювання цих збурень не розглядалося, використовувалися лише властивості їх загальної обмеженості за нормою. Для синтезу управління застосовувався відомий в теорії керування орієнтацією (кутовим положенням космічного апарата) аналог — метод декомпозиції загальної задачі синтезу на кінематичну і динамічну. Для їх вирішення використовувалися відомі узагальнення прямого методу Ляпунова дослідження стійкості розвʼязків диференціальних рівнянь на дослідження стійкості замкнених обмежених множин (конкретно багатовимірних еліпсоїдів) у їх фазових просторах. Поряд із методом синтезу управління стабілізацією довготи точки стояння ГСС запропоновано методи еліпсоїдального оцінювання, викликані впливом згаданих збурень помилок стабілізації, що встановилися. Методи засновані на побудові зовнішніх еліпсоїдальних оцінок множин досяжнос­ті та граничних множин у фазових просторах динамічних систем з обмеженими структурно-параметричними збуреннями. Запропоновано також метод отримання гарантованих інтервальних оцінок помилок стабілізації, що встановилися, по кожній з координат. Ефективність запропонованих методів керування ілюструється результатами компʼютер­ного моделювання.

Elliptic Flowers: New Types of Dynamics to Study Classical and Quantum Chaos.

We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong to a new class of dynamical systems called elliptic flowers billiards. Such systems demonstrate a variety of new behaviors which have never been observed or predicted to exist. Elliptic flowers billiards, where a chaotic (non-chaotic) core coexists with the same (chaotic/non-chaotic) type of dynamics in flows were recently constructed. Therefore, all four possible types of coexisting dynamics in the core and tracks are detected. However, it is just the beginning of studies of elliptic flowers billiards, which have already extended the imagination of what may happen in phase spaces of nonlinear systems. We outline some further directions of investigation of elliptic flowers billiards, which may bring new insights into our understanding of classical and quantum dynamics in nonlinear systems.

Suppression of Chaotic Oscillations in Small Energy Systems

The paper considers the suppression of chaotic oscillations in small energy systems that occur in emergency modes and lead to the phenomenon of voltage collapse, which corresponds to the process of voltage drop in the network, which can be accompanied by a complete shutdown of the affected area. The paper also presents a method that has been developed and that allows changing the spectrum of Lyapunov’s characteristic indicators and converting chaotic oscillations in a small power system to regular dynamic modes. The method of synthesis of control actions is based on the theorem of topological equivalence of hyperbolic nonlinear systems and their linearized models as well as on and the use of numerical integration of nonlinear differential equations describing the behavior of power systems in order to construct a phase portrait and calculate Lyapunov’s characteristic exponents. The results of the work consist in the synthesis of feedback, which ensures the formation of a spectrum of Lyapunov’s characteristic indicators with negative values. The suppression of chaotic regimes occurs by forming a spectrum of negative Lyapunov’s characteristic indicators in a closed system. The parameters of the regulator in the feedback circuit are determined using the modal control method based on the solution of the matrix algebraic Sylvester equation. The solution of the problem of transition from a chaotic regime to a regular movement in a small power system is considered. To test the operability of the proposed method of chaos suppression, the spectrum of Lyapunov’s characteristic indicators is calculated and trajectories in the phase space of the initial nonlinear system and the system with control action are constructed. For energy systems with chaotic dynamics, synthesized feedback makes it possible to suppress chaotic fluctuations and switch to regular modes, thereby preventing the occurrence of emergency modes.

Discontinuous dynamics of a class of 3-DOF friction impact oscillatory systems with rigid frame and moving jaws

In this paper, the discontinuous dynamics of a class of 3-DOF (three-degree-of-freedom) friction impact oscillatory systems with rigid frame and moving jaws is investigated by using the high-dimensional flow switchability theory. For the close relationship between any two objects of the 3-DOF system, we directly investigate the dynamics of whole system rather than respective dynamics of each object separately. The flow barriers in such a system are also considered based on the inequalities of coefficients for dynamic friction and static friction. Based on discontinuities/nonsmoothness of the oscillatory system, the six-dimensional phase space of whole system is divided into different domains and boundaries/edges in relative coordinates, and the local singularity and switchability of flows to high-dimensional boundaries and edges are fully investigated. In order to better understand the complexity of dynamics of the 3-DOF system, several typical motions are numerically simulated, such as passable, sliding, grazing, stick, impact and periodic motions. For partial numerical results, a comparison between simulation data obtained in MATLAB and MSC Adams has been performed. The study of such a 3-DOF model enriches the high-dimensional flow switchability theory, and provides a theoretical basis for optimization design and parameter selection of mechanical mechanism with jaws.

Equations of Motion of Dynamical Systems From Kinematic Characteristics of the Phase Space

Abstract In the analysis of most engineering dynamical systems, relativistic considerations are unnecessary, allowing an absolute time and simultaneity of events to be assumed. In this article, it is first established that this simultaneity links all generalized coordinates and velocities via simple kinematic relations in the phase space. Subsequently, it is shown that equations of motion of dynamical systems can be derived by imposing these kinematic relations on a generating equation, which is a generalized form of the Jacobi’s integral. A specific process is presented for combining the kinematic relations with the generating equation to yield correct equations of motion. The process is validated by using it to prove the Lagrange’s equation. Examples are provided to demonstrate the approach. The aforementioned kinematic relations are fundamental characteristics of the phase space of general dynamical systems. They provide a novel perspective on equations of motion in analytical dynamics, which leads to a new method of deriving them.

Unpredictability in Hamiltonian systems with a hierarchical phase space

One of the main consequences of the complex hierarchical structure of chaotic regions and stability islands in the phase space of a typical nonlinear Hamiltonian system is the phenomenon of stickiness. The chaotic orbits that approach an island are trapped in its neighborhood for arbitrarily long times, in which the orbits behave similarly as quasiperiodic orbits. In this paper, we characterize the boundary between chaos and regular motion in the phase space of the standard map for distinct parameter values. The orbits are distinguished between regular and chaotic employing a recently proposed method of weighted Birkhoff averages. We quantify the dimension of the boundaries of the islands using the uncertainty exponent. In our simulations, we show that the dimension of the island's boundary depends on the scale of the initial condition uncertainty and the level of the hierarchical structure. We also show that the trapping in the vicinity of the islands causes an obstruction in the predictability of the final state of an orbit. We present how this loss of predictability results in larger dimensions at the inner levels of the islands.

Classification of radial Kerr geodesic motion

We classify radial timelike geodesic motion of the exterior non-extremal Kerr spacetime by performing a taxonomy of inequivalent root structures of the first order radial geodesic equation using a novel compact notation and by implementing the constraints from polar, time and azimuthal motion. Four generic root structures with only simple roots give rise to eight non-generic root structures when either one root becomes coincident with the horizon, one root vanishes or two roots becomes coincident. We derive the explicit phase space of all such root systems in the basis of energy, angular momentum and Carter's constant and classify whether each corresponding radial geodesic motion is allowed or disallowed from existence of polar, time and azimuthal motion. The classification of radial motion within the ergoregion for both positive and negative energies leads to 6 distinguished values of the Kerr angular momentum. The classification of null radial motion and near-horizon extremal Kerr radial motion are obtained as limiting cases and compared with the literature. We explicitly parametrize the separatrix describing root systems with double roots as the union of the following three regions that are described by the same quartic respectively obtained when (1) the pericenter of bound motion becomes a double root; (2) the eccentricity of bound motion becomes zero; (3) the turning point of unbound motion becomes a double root.

Detection of a chaotic signal with noise by quantization over several amplitude levels in the model of a radio engineering chaos generator

A method for detecting a continuous chaotic signal with a Gaussian noise by quantizing over four amplitudes and comparing the parameters of the obtained discrete sequence with the parameters of a similar sequence for a non-noisy chaotic signal is proposed. A quantization scheme is developed. It is determined by the structure of the phase space of a dynamic system that generates a chaotic signal. The source of the chaotic signal is a model of a radio engineering generator. The identified chaotic signal has a positive senior Lyapunov exponent and a continuous power spectrum. Keywords: chaotic signals, detection of chaotic signals, non-coherent reception of chaotic signals, quantization of chaotic oscillations.

Обнаружение хаотического сигнала на фоне шума путем квантования по нескольким уровням амплитуды в модели радиотехнического генератора хаоса

A method for detection a continuous chaotic signal with a Gaussian noise by quantizing over four amplitudes and comparing the parameters of the obtained discrete sequence with the parameters of a similar sequence for a non-noisy chaotic signal is proposed. A quantization scheme is developed. It is determined by the structure of the phase space of a dynamic system that generates a chaotic signal. The source of the chaotic signal is a model of a radio engineering generator. The identified chaotic signal has a positive senior Lyapunov exponent and a continuous power spectrum.

Analysis and Visualization of High‐Dimensional Dynamical Systems’ Phase Space Using a Network‐Based Approach

The concept of attractors is considered critical in the study of dynamical systems as they represent the set of states that a system gravitates toward. However, it is generally difficult to analyze attractors in complex systems due to multiple reasons including chaos, high‐dimensionality, and stochasticity. This paper explores a novel approach to analyzing attractors in complex systems by utilizing networks to represent phase spaces. We accomplish this by discretizing phase space and defining node associations with attractors by finding sink strongly connected components (SSCCs) within these networks. Moreover, the network representation of phase space facilitates the use of well‐established techniques of network analysis to study the phase space of a complex system. We show the latter by introducing a new node‐based metric called attractivity which can be used in conjunction with the SSCC as they are highly correlated. We demonstrate the proposed method by applying it to several chaotic dynamical systems and a large‐scale agent‐based social simulation model.

Efficient Characterization of Quantum Evolutions via a Recommender System

We demonstrate characterizing quantum evolutions via matrix factorization algorithm, a particular type of the recommender system (RS). A system undergoing a quantum evolution can be characterized in several ways. Here we choose (i) quantum correlations quantified by measures such as entropy, negativity, or discord, and (ii) state-fidelity. Using quantum registers with up to 10 qubits, we demonstrate that an RS can efficiently characterize both unitary and nonunitary evolutions. After carrying out a detailed performance analysis of the RS in two qubits, we show that it can be used to distinguish a clean database of quantum correlations from a noisy or a fake one. Moreover, we find that the RS brings about a significant computational advantage for building a large database of quantum discord, for which no simple closed-form expression exists. Also, RS can efficiently characterize systems undergoing nonunitary evolutions in terms of quantum discord reduction as well as state-fidelity. Finally, we utilize RS for the construction of discord phase space in a nonlinear quantum system.

ICompare: A Package for Automated Comparison of Solar System Integrators*

Abstract We present a tool for the comparison and validation of the integration packages suitable for Solar System dynamics. iCompare, written in Python, compares the ephemeris prediction accuracy of a suite of commonly-used integration packages (JPL/HORIZONS, OpenOrb, OrbFit at present). It integrates a set of test particles with orbits picked to explore both usual and unusual regions in Solar System phase space and compares the computed to reference ephemerides. The results are visualized in an intuitive dashboard. This allows for the assessment of integrator suitability as a function of population, as well as monitoring their performance from version to version (a capability needed for the Rubin Observatory’s software pipeline construction efforts). We provide the code on GitHub with a readily runnable version in Binder (https://github.com/dirac-institute/iCompare).

Two-parameter non-smooth bifurcations of period-one motions in a plastic impacting oscillator

A 2-DOF plastic impacting oscillator is considered. The discontinuity boundaries of the system phase space are defined as well as the local maps and Poincaré map. Four cases of sliding bifurcation of limit cycles in impact system are described schematically. The period-one motion modes and occurrence regions of the system are identified in the two-parameter space by numerical simulation. One-parameter bifurcation diagrams and trajectories are computed to explain the evolutions of adjacent periodic motions and the non-smooth bifurcation behaviors including grazing and four types of sliding bifurcation. Due to the discontinuous grazing bifurcation there appear a narrow hysteresis domain in the two-parameter evolution between neighboring non-impacting and impacting motions. Grazing, switching–sliding and multi-sliding bifurcations of periodic sticking motions are all manifested as rising phenomenon of the time trajectory, but the different locations of the rises in the sticking phase correspond to the different types of non-smooth bifurcations. The grazing induced dynamics and the characteristics of rising events are discussed. Notably, the codimension-2 bifurcation points are detected and investigated by numerical continuation algorithm.