Three-dimensional Phase Space Research Articles

Orbit tomography in constants-of-motion phase-space

Abstract Tomographic reconstructions of a 3D fast-ion constants-of-motion phase-space distribution function are computed by inverting synthetic signals based on projected velocities of the fast ions along the diagnostic lines of sight. The fast-ion distribution functions are parameterised by three constants of motion, the kinetic energy, the magnetic moment and the toroidal canonical angular momentum. The reconstructions are computed using both zeroth-order and firstorder Tikhonov regularisation expressed in terms of Bayesian inference to allow uncertainty quantification. In addition to this, a discontinuity appears to be present in the solution across the trapped passing boundary surface in the three-dimensional phase space due to a singularity in the Jacobian of the transformation from position and velocity space to phase space. A method to allow for this apparent discontinuity while simultaneously penalising large gradients in the solution is demonstrated. Finally, we use our new methods to optimise the diagnostic performance of a set of six fans of sightlines by finding where the detectors contribute most complementary diagnostic information for the future COMPASS-Upgrade tokamak.

Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

AbstractA dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set $$\Lambda $$ Λ with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in $$\Lambda $$ Λ weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps $$\mathcal {H}$$ H on $$\mathbb {R}^3$$ R 3 , which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of $$\mathcal {H}$$ H over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of $$\mathcal {H}$$ H .

Correction to: Generalization of Hamiltonian mechanics to a three-dimensional phase space

Abstract In a recent paper [N. Sato, Prog. Theor. Exp. Phys. 2021, 6, 063A01 (2021)] we introduced a generalization of Hamiltonian mechanics to three-dimensional phase spaces in terms of closed 3-forms. This correction addresses an error in the proof of theorem 3, which concerns the existence of a coordinate change transforming a closed 3-form into a constant form. Indeed, invertibility of a 3-form is not sufficient to ensure the existence of a solution Xt to eq. (77) when n > 3. The theorem can be corrected by restricting the class of 3-forms to those that are relevant to generalized Hamiltonian mechanics. Although the new theorem requires a stronger hypothesis, the formulation of dynamical systems with 2 invariants in terms of closed 3-forms, which is the key contribution of the paper, holds.

Slow Invariant Manifolds of Memristor-Based Chaotic Circuits

This work presents an efficient approach for computing the slow invariant manifold of the fourth-order canonical memristor-based Chua circuits using the flow curvature method. First, the magnetic-flux and charge characteristic curve is generated from the classical circuit with a piecewise-linear function. Then, the characteristic curve is generated from the circuit with the piecewise-linear function replaced by a cubic function. Further, the duality principle is applied to studying such memristor-based circuits in the three-dimensional flux-linkage and charge phase space and then in the four-dimensional current–voltage phase space. It is demonstrated that the slow invariant manifolds of these fourth-order memristor-based chaotic circuits can be more directly computed for the first case than the second.

Bicolor Graph of Morse-Smale Cascades on Manifolds of Dimension Three

The purpose of this study is to single out a class of Morse-Smale cascades (diffeomorphisms) with a three-dimensional phase space that allow topological classification using combinatorial invariants. In the general case, an obstacle to such a classification is the possibility of wild embedding of separatrix closures in the ambient manifold, which leads to a countable set of topologically nonequivalent systems. To solve the problem, we study the orbit space of a cascade. The ambient manifold of a diffeomorphism can be represented as a union of three pairwise disjoint sets: a connected attractor and a repeller whose dimension does not exceed one, and their complement consisting of wandering points of a cascade called the characteristic set. It is known that the topology of the orbit space of the restriction of the Morse-Smale diffeomorphism to the characteristic set and the embedding of the projections of two-dimensional separatrices into it is a complete topological invariant for Morse-Smale cascades on three-dimensional manifolds. Moreover, a criterion for the inclusion of Morse-Smale cascades in the topological flow was obtained earlier.These results are used in this paper to show that the topological conjugacy classes of Morse-Smale cascades that are included in a topological flow and do not have heteroclinic curves admit a combinatorial description. More exactly, the class of Morse-Smale diffeomorphisms without heteroclinic intersections, defined on closed three-dimensional manifolds included in topological flows and not having heteroclinic curves, is considered. Each cascade from this class is associated with a two-color graph describing the mutual arrangement of two-dimensional separatrices of saddle periodic points. It is proved that the existence of an isomorphism of two-color graphs that preserves the color of edges is a necessary and sufficient condition for the topological conjugacy of cascades. It is shown that the speed of the algorithm that distinguishes two-color graphs depends polynomially on the number of its vertices. An algorithm for constructing a representative of each topological conjugacy class is described.

Self-oscillating systems with controlled phase of external force

The purpose of this work is to study self-oscillatory systems under adaptive external action. This refers to the situation when the phase of the external action additionally depends on the dynamical variable of the oscillator. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with three-dimensional phase space. Methods. Methods of charts of dynamical regimes and charts of Lyapunov exponents are used, as well as the construction of phase portraits and stroboscopic sections. Results. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with a three-dimensional phase space. The pictures of characteristic dynamical regimes are described. Scenarios for the development of multidimensional chaos are described. Illustrations are given of the influence of the control parameter, which is responsible for the degree of dependence of the phase on the oscillator variable, on the dynamics of the system at different frequencies of action. Conclusion. The taling into account of the dependence of the phase on a dynamical variable leads to an extension of the tongues of subharmonic resonances, which are weakly expressed in the classical van der Pol oscillator. This is especially noticeable for even resonances of periods 2 and 4. For the generator of quasi-periodic oscillations in the non-autonomous case, three-frequency tori are observed, their regions begin to dominate with an increase in the adaptivity parameter, displacing the tongues of resonant two-frequency tori. A variety of multidimensional chaos characterized by an additional Lyapunov exponent close to zero is discovered, the possibility of developing hyperchaos as a result of destruction is shown.

The problem of acceleration in the dynamics of a double-link wheeled vehicle with arbitrarily directed periodic excitation

This study investigates the motion of a nonholonomic mechanical system that consists of two wheeled carriages articulated by a rigid frame. There is a point mass which oscillates at a given angle ?? to the main axis of one of the carriages. As a result, periodic excitation occurs in the system. The equations of motion in quasi-velocities are obtained. Eventually, the dynamics of a double-link wheeled vehicle is modeled by a system that defines a nonautonomous flow on a three-dimensional phase space. The behavior of integral curves at large velocities depending on the angle ?? is investigated. We use the generalized Poincar?e transformation and reduce the original problem to the stability problem for the system with a degenerate linear part. The proof of stability uses the restriction of the system to the central manifold and averaging by normal forms up to order 4. The range of values of ?? for which one of the velocity components increases indefinitely is found and asymptotics for the solutions of the initial dynamical system is determined.

A New Model to Describe Quarkonium Systems under Modified Cornell Potential at Finite Temperature in pNRQCD

In the present work, the three-dimensional modified radial Schrödinger equation is analytically solved. The nonrelativistic interactions under new modified Cornell potential (NMCP, in short) at finite temperature, are extended to the symmetries of nonrelativistic noncommutative space phase (NRNSP, in short), using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). W generalize this process by adding multi-variable coupling potentials , and together with the modified Cornell potential model in three-dimensional nonrelativistic quantum mechanics noncommutative phase space (3D-NCSP, in short). The new energy eigenvalues and the corresponding Hamiltonian operator are calculated in 3D-NCSP symmetries instead of solving the modified Schrödinger equation with the Weyl Moyal star product. The present results, in (3D-NCSP), are applied to the charmonium and bottomonium masses at finite temperature. The present approach successfully generalizes the energy eigenvalues at finite temperature in 3D-NCSP symmetries. It is found that the perturbative solutions of the discrete spectrum and quarkonium mass can be expressed by the Gamma function, the discreet atomic quantum numbers of the state and the potential parameters ( ), in addition to noncommutativity parameters ( and ). The total complete degeneracy of new energy levels of NMCP changed to become equals to the value instead the values in ordinary quantum mechanics. Our obtained results are in good agreement with the already existing literature in NCSP. Keywords: Schrödinger Equation, Heavy Quarkonium System, Cornell Potential, Noncommutative Space Phase, Bopp’s Shift Method. Subject Classification Numbers: 03.65.-w; 03.65.Ge; 03.65. Fd; 03.65.Ca

Data-driven stochastic simulation leading to the allometric scaling laws in complex systems.

We propose a data-driven stochastic method that allows the simulation of a complex system's long-term evolution. Given a large amount of historical data on trajectories in a multi-dimensional phase space, our method simulates the time evolution of a system based on a random selection of partial trajectories in the data without detailed knowledge of the system dynamics. We apply this method to a large data set of time evolution of approximately one million business firms for a quarter century. Accordingly, from simulations starting from arbitrary initial conditions, we obtain a stationary distribution in three-dimensional log-size phase space, which satisfies the allometric scaling laws of three variables. Furthermore, universal distributions of fluctuation around the scaling relations are consistent with the empirical data.

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.

Myocardial infarction detection using ITD, DWT and deterministic learning based on ECG signals.

Nowadays, cardiovascular diseases (CVD) is one of the prime causes of human mortality, which has received tremendous and elaborative research interests regarding the prevention issue. Myocardial ischemia is a kind of CVD which will lead to myocardial infarction (MI). The diagnostic criterion of MI is supplemented with clinical judgement and several electrocardiographic (ECG) or vectorcardiographic (VCG) programs. However the visual inspection of ECG or VCG signals by cardiologists is tedious, laborious and subjective. To overcome such disadvantages, numerous MI detection techniques including signal processing and artificial intelligence tools have been developed. In this study, we propose a novel technique for automatic detection of MI based on disparity of cardiac system dynamics and synthesis of the standard 12-lead and Frank XYZ leads. First, 12-lead ECG signals are synthesized with Frank XYZ leads to build a hybrid 4-dimensional cardiac vector, which is decomposed into a series of proper rotation components (PRCs) by using the intrinsic time-scale decomposition (ITD) method. The novel cardiac vector may fully reflect the pathological alterations provoked by MI and may be correlated to the disparity of cardiac system dynamics between healthy and MI subjects. ITD is employed to measure the variability of cardiac vector and the first PRCs are extracted as predominant PRCs which contain most of the cardiac vector's energy. Second, four levels discrete wavelet transform with third-order Daubechies (db3) wavelet function is employed to decompose the predominant PRCs into different frequency bands, which combines with three-dimensional phase space reconstruction to derive features. The properties associated with the cardiac system dynamics are preserved. Since the frequency components above 40 Hz are lack of use in ECG analysis, in order to reduce the feature dimension, the advisable sub-band (D4) is selected for feature acquisition. Third, neural networks are then used to model, identify and classify cardiac system dynamics between normal (healthy) and MI cardiac vector signals. The difference of cardiac system dynamics between healthy control and MI cardiac vector is computed and used for the detection of MI based on a bank of estimators. Finally, experiments are carried out on the PhysioNet PTB database to assess the effectiveness of the proposed method, in which conventional 12-lead and Frank XYZ leads ECG signal fragments from 148 patients with MI and 52 healthy controls were extracted. By using the tenfold cross-validation style, the achieved average classification accuracy is reported to be 98.20%. Results verify the effectiveness of the proposed method which can serve as a potential candidate for the automatic detection of MI in the clinical application.

The Impact of deformed space-phase parameters into HAs and HLM systems with the improved Hulthen plus Hellmann potentials model in the presence of temperature-dependent confined Coulomb potential within the framework of DSE

In this work, the improved Hulthen plus Hellmann potentials model in the presence of temperature-dependent confined Coulomb potential ´ IHHPTd model is adopted as the quark-antiquark interaction potential for studying the mass spectra of heavy mesons in the three-dimensional nonrelativistic quantum mechanics noncommutative phase space (3DNRQm-NCSP) symmetries. In addition, we found another application for this potential through its description for hydrogen atoms He+, Li+2 and Be+. We solved the deformed Schrödinger equation analytically using the generalized Bopp’s shift method and standard perturbation theory. The new energy eigenvalues E (u,d)hy nc−n and (E hlm n−g, E hlm n−mand E hlm n−l ) for hydrogen atoms and heavy mesons such as charmonium cc and bottomonium cb and corresponding deformed Hamiltonian operators H nc hhp(r, Θ, θ, λ, λ, σ, σ, ²) and H nc hhp(r, Θ, θ, λ, λ, σ, σ, gs) were obtained, respectively. The present results are applied for calculating the new mass of heavy mesons. Four special cases were considered when some of the improved potential parameters were set to zero, resulting into improved Hellmann potential, improved Yukawa potential, improved Coulomb potential, and improved Hulthen potential, in ´ (3DNRQm-NCSP) symmetries. The limiting cases are analyzed for Θ, σ and χ −→ 0 are compared with those of literature.

Attractor Reconstruction for Quantifying the Arterial Pulse Wave Morphology During Device-Guided Slow Breathing

Attractor reconstruction is a new way to quantify the morphology of a cardiovascular waveform by plotting all data points in the three-dimensional phase space, generating a trajectory of overlapping loops. The aim of this study is to show the feasibility of an automatic approach to quantify pulse wave attractors from a device-guided breathing study, and to link attractor features to pulse waveform features. The recently developed feature extraction technique is applied to arterial pulse waveform data of 30 patients with treated hypertension. The patients performed a device-guided slow breathing exercise. The generated attractors were categorized into three different shapes: triangular attractors, bent attractors and attractors with overlapping arms. The average height of the attractors continuously and progressively dropped from 41.8 [35.4, 55.1] AU to 34.5 [25.4, 47.3] AU (p < 0.001) during the breathing exercise. We have shown that the novel approach to quantify pulse wave attractors is feasible and can be used to detect changes in the morphology of arterial pulse waveforms. Device-guided slow breathing exercise has a shrinking effect on the average height of the attractors, which may indicate a reduction in blood pressure.

Euler integral as a source of chaos in the three–body problem

In this paper we address, from a purely numerical point of view, the question, raised in Pinzari (2019), Pinzari (2020), and partly considered in Pinzari (2020), Di Ruzza et al. (2020), Chen and Pinzari (2021), whether a certain function, referred to as “Euler Integral”, is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the “unperturbed separatrix”, which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in Gierzkiewicz and Zgliczyński (2019), Zgliczynski and Gidea (2004), Wilczak and Zgliczynski (2003).

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, {hat{u}}(hat{{mathbf {x}}},{hat{t}}) that is coupled to the concentration of an immobile extracellular substrate, {hat{s}}(hat{{mathbf {x}}},{hat{t}}). Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to {hat{s}}, while a logistic growth term models cell proliferation. The extracellular substrate {hat{s}} is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, c = c_{mathrm{min}}, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, c > c_{mathrm{min}}. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.

Generalization of Hamiltonian mechanics to a three-dimensional phase space

Abstract Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic $2$-form, a closed second-order differential form. Due to closure, the symplectic $2$-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie–Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic $2$-form is replaced by a symplectic $3$-form. We show that, using the closure of the symplectic $3$-form, a result analogous to the classical Lie–Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic $3$-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third-order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic $3$-form.

Automatic identification of epileptic seizures using volume of phase space representation.

Epilepsy is a neurological disorder that affects people of any age, which can be detected by Electroencephalogram (EEG) signals. This paper proposes a novel method called Volume of Phase Space Representation (VOPSR) to classify seizure and seizure-free EEG signals automatically. Primarily, the recorded EEG signal is disintegrated into several Intrinsic Mode Functions (IMFs) using the Empirical Mode Decomposition (EMD) method and the three-dimensional phase space have been reconstructed for the obtained IMFs. The volume is measured for the obtained 3D-PSR for different IMFs called VOPSR, which is used as a feature set for the classification of Epileptic seizure EEG signals. Support vector machine (SVM) is used as a classifier for the classification of epileptic and epileptic-free EEG signals. The classification performance of the proposed method is evaluated under different kernels such as Linear, Polynomial and Radial Basis Function (RBF) kernels. Finally, the proposed method outperforms noteworthy state-of-the-art classification methods in the context of epileptic EEG signals, achieving 99.13% accuracy (average) with the Linear, Polynomial, and RBF kernels. The proposed technique can be used to detect epilepsy from the EEG signals automatically without human intervention.

Stability analysis for design improvement of bio-inspired flapping wings by energy method

This study attempts to reach a broad understanding of the stability properties of nonlinear time-periodic flapping wing structures. Two bio-system models, Hummingbird (6DOF) and Hawkmoth (3DOF) are developed for this purpose. Initial analysis on the Hummingbird model, which is based on the Floquet theory, kinetic energy integration, and phase portrait technique, indicates lack of stability in hover flight. Kinetic energy integration is carried out on the extended model of the Hawkmoth to find the domain of attraction and increase the level of stability by varying the design parameters. Here, the hinge location of the wing, flapping amplitude, flapping frequency, and mean angle of attack are adjusted to achieve steady hovering trim condition. The results will be used for hovering control design that is a problem with no clear solution and has been widely studied based on linear analysis and averaging theory regardless of the stability properties. In addition, by averaging theory and linearization technique, the range of mean angle of attack that is associated with the pitch angle is also determined by using eigenvalues to achieve dynamic stability for further nonlinear analysis. Sensitivity analysis is then performed with the phase portrait and energy approach on the equivalent dynamics to reach the main cause of instability. Improved parameters are applied to the non-autonomous dynamics, and for the first time, the creation of a strange attractor is observed in three-dimensional phase space by Floquet analysis.

Generalizing Poisson Algebra with Geometry

Using a differential geometric interpretation of Hamiltonian mechanics, a generalized Poisson bracket formulation is developed for a three-dimensional phase space characterized by a triplet of canonical variables.

A Vector Angle Method of Rolling Bearing Fault Classification by Phase-Space Reconstruction Technique

Abstract A fault classification method is developed by the phase-space reconstruction technique for a rotor system with a rolling bearing fault. Based on the nonlinear time series analysis, proper choices on time delay and embedding dimension are firstly discussed to accomplish the phase-space reconstruction from an arbitrarily one-dimensional time series, then the vector angle calculation is derived for each point in the reconstructed trajectory of an illustrating artificial signal, from which the vector angles of the points are composed of 90° and non-90° ones. To perform the fault feature analysis of the rolling bearing, an experimental rig of the rotor-bearing system and dynamical model of the system are established to collect the one-dimensional acceleration signals of three types of rolling bearing faults, and the topology of the reconstructed trajectory in three-dimensional phase space is characterized. It is interesting to find that the effects of various rolling bearing faults on the reconstructed trajectories are different, and the fault features can be extracted successfully by the distribution percentage of 90° and non-90° vector angles of the points in the reconstructed trajectory in three-dimensional phase space, which cannot be identified by the traditional attractor reconstruction method. Moreover, the higher the fault frequency, the lower proportion the healthy points.