Properties Of Dynamical Systems Research Articles

Effects of autocorrelated disorder on the dynamics in the vicinity of the many-body localization transition

The presence of frozen uncorrelated random on-site potential in interacting quantum systems can induce a transition from an ergodic phase to a localized one, the so-called many-body localization. Here we numerically study the effects of autocorrelated disorder on the static and dynamical properties of a one-dimensional many-body quantum system which exhibits many-body localization. Specifically, by means of some standard measures of energy level repulsion and localization of energy eigenstates, we show that a strong degree of correlations between the on-site potentials in the one-dimensional spin-1/2 Heisenberg model leads to suppression of the many-body localization phase, while level repulsion is mitigated for small disorder strengths, although energy eigenstates remain well extended. Our findings are also remarkably manifested in the time domain, on which we put the main emphasis, as shown by the time evolution of experimentally relevant observables, like the return probability and the spin autocorrelation function.

Variational dynamics as a ground-state problem on a quantum computer

We propose a variational quantum algorithm to study the real time dynamics of quantum systems as a ground-state problem. The method is based on the original proposal of Feynman and Kitaev to encode time into a register of auxiliary qubits. We prepare the Feynman-Kitaev Hamiltonian acting on the composed system as a qubit operator and find an approximate ground state using the Variational Quantum Eigensolver. We apply the algorithm to the study of the dynamics of a transverse field Ising chain with an increasing number of spins and time steps, proving a favorable scaling in terms of the number of two qubit gates. Through numerical experiments, we investigate its robustness against noise, showing that the method can be use to evaluate dynamical properties of quantum systems and detect the presence of dynamical quantum phase transitions by measuring Loschmidt echoes.

Detecting physical laws from data of stochastic dynamical systems perturbed by non-Gaussian α-stable Lévy noise

Massive data from observations, experiments and simulations of dynamical models in scientific and engineering fields make it desirable for data-driven methods to extract basic laws of these models. We present a novel method to identify such high dimensional stochastic dynamical systems that are perturbed by a non-Gaussian α-stable Lévy noise. More explicitly, firstly a machine learning framework to solve the sparse regression problem is established to grasp the drift terms through one of nonlocal Kramers–Moyal formulas. Then the jump measure and intensity of the noise are disposed by the relationship with statistical characteristics of the process. Three examples are then given to demonstrate the feasibility. This approach proposes an effective way to understand the complex phenomena of systems under non-Gaussian fluctuations and illuminates some insights into the exploration for further typical dynamical indicators such as the maximum likelihood transition path or mean exit time of these stochastic systems.

Dynamic properties analysis of ball screw feed system based on improved hybrid model

Ball screws are widely used as feed systems for machine tools due to their high transmission efficiency and long service life. However, the large inertia force generated by the ball screw in high-speed motion will excite its flexible mode and cause system vibration. In the actual working condition, the dynamics of the system will also change with the nut position and workpiece mass. In order to further study the modeling method and dynamic properties of the ball screw feed system, this paper firstly improves the traditional hybrid model by introducing an energy loss factor to modify the hysteresis damping of the model. Further, the rationality of the improved model is verified by hammer test. Finally, the model was used to analyze the dynamic properties of the system and to obtain the influence law of workpiece mass and nut position on the system natural frequency.

Mixing and rigidity along asymptotically linearly independent sequences

AbstractWe use Gaussian measure-preserving systems to prove the existence and genericity of Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let $\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in [0,1]$ and let $\phi _1,\ldots ,\phi _N:\mathbb N\rightarrow \mathbb Z$ be asymptotically linearly independent (that is, for any $(a_1,\ldots ,a_N)\in \mathbb Z^N\setminus \{\vec 0\}$ , $\lim _{k\rightarrow \infty }|\sum _{j=1}^Na_j\phi _j(k)|=\infty $ ). Then the class of invertible Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ for which there exists a sequence $(n_k)_{k\in \mathbb {N}}$ in $\mathbb {N}$ with for any measurable $A,B\subseteq [0,1]$ and any $j\in \{1,\ldots ,N\}$ , is generic. This result is a refinement of a result due to Stëpin (Theorem 2 in [Spectral properties of generic dynamical systems. Math. USSR-Izv.29(1) (1987), 159–192]) and a generalization of a result due to Bergelson, Kasjan, and Lemańczyk (Corollary F in [Polynomial actions of unitary operators and idempotent ultrafilters. Preprint, 2014, arXiv:1401.7869]).

Dynamic magnetic properties and phase diagrams of Fe4N system

The dynamic magnetic properties of the nonequilibrium Fe4N system are calculated on the basis of correlated effective-field theory (EFT). The dynamic phase diagrams are plotted for different values of the oscillating magnetic field and the crystal field. The reentrant behavior, the dynamic tricritical point and the dynamic critical end point observed in certain case by using EFT are established to emerge as the generic nonequilibrium features of the Fe4N system. Moreover, we compare our results with those produced by mean-field theory (MFT), and find that the first-order phase transition and the reentrant phenomena are significantly reduced due to the consideration of spin–spin thermal fluctuations in EFT approximations. Our findings lead us to conclude that the thermal fluctuations considered in EFT are a critical factor for the dynamic magnetic properties of the nonequilibrium Fe4N system.

On Evolving Natural Curvature for an Inextensible, Unshearable, Viscoelastic Rod

We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an infinite dimensional dynamical system, are globally well-posed. For every value of the terminal thrust, these equations contain a smooth embedded curve of static solutions (equilibrium points). We characterize the spectrum of the linearized equations about an arbitrary equilibrium point, and using this information and a convergence result for dynamical systems due to Brunovský and Polácik, we prove that every solution to the quasistatic equations of motion converges to an equilibrium point as time goes to infinity.

How Does Noise Induce Order?

In this paper we present a general result with an easily checkable condition that ensures a transition from chaotic regime to regular regime in random dynamical systems with additive noise. We show how this result applies to a prototypical family of nonuniformly expanding one dimensional dynamical systems, showing the main mathematical phenomenon behind Noise Induced Order. Accepted at Journal of Statistical Physics

Magnetic fluxes of solar active regions of different magneto-morphological classes – I. Cyclic variations

ABSTRACT Data for 3046 solar active regions (ARs) observed since 1996 May 12 to 2021 December 27 were utilized to explore how the magnetic fluxes from ARs of different complexity follow the solar cycle. Magnetograms from the Michelson Doppler Imager instrument on the Solar and Heliospheric Observatory and from the Helioseismic and Magnetic Imager instrument on the Solar Dynamics Observatory were utilized. Each AR was classified as a regular bipolar AR (classes A1 or A2), or as an irregular bipolar AR (class B1), or as a multipolar AR (classes B2 or B3). Unipolar ARs were segregated into a specific class U. We found the following results. Unsigned magnetic fluxes from ARs of different classes evolve synchronously following the cycle, the correlation coefficient between the flux curves varies in a range of 0.70–0.99. The deepest solar minimum is observed simultaneously for all classes. Only the most simple ARs were observed during a deepest minimum: A1- and B1-class ARs. The overall shape of a cycle is governed by the regular ARs, whereas the fine structure of a solar maximum is determined by the most complex irregular ARs. Approximately equal amount of flux (45–50 per cent of the total flux) is contributed by the A-class and B-class ARs during a solar maximum. Thus, observations allow us to conclude that the appearance of ARs with the magnetic flux above 1021 Mx is caused by the solar dynamo that operates as a unique process displaying the properties of a non-linear dynamical dissipative system with a cyclic behaviour and unavoidable fluctuations.

Sudden quench of harmonically trapped mass-imbalanced fermions

Dynamical properties of two-component mass-imbalanced few-fermion systems confined in a one-dimensional harmonic trap following a sudden quench of interactions are studied. It is assumed that initially the system is prepared in the non-interacting ground state and then, after a sudden quench of interactions, the unitary evolution is governed by interacting many-body Hamiltonian. By careful analysis of the evolution of the Loschmidt echo, density distributions of the components, and entanglement entropy between them, the role of mass imbalance and particle number imbalance on the system’s evolution stability are investigated. All the quantities studied manifest a dramatic dependence on the number of heavy and lighter fermions in each component at a given quench strength. The results may have implications for upcoming experiments on fermionic mixtures with a well-defined and small number of particles.

Recurrence properties for linear dynamical systems: An approach via invariant measures

We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector x0, for an adjoint operator T on a separable dual Banach space X, one can construct a T-invariant probability measure which contains x0 in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.

Inverse modeling of wind turbine drivetrain from numerical data using Bayesian inference

Wind turbine drivetrain systems experience occasional failure in their operational life-time. Dynamic loads in these systems often differ from those predicted in the design phase, which may lead to incipient faults and damage eventually resulting to unpredicted failures. Faults in a system cause changes in the system dynamic properties. Therefore, a calibrated dynamic model can be used for fault diagnosis of the drivetrain system. The current paper implements a Bayesian inference method for physics-based model updating for the first time to estimate the parameters of a physics-based wind turbine drivetrain model as well as the unknown input loads using measurement data. An information-theoretic approach is adopted to study the identifiability of different model parameters given different subsets of drivetrain measurements. The 5 MW NREL wind turbine is used as a case study, and its drivetrain is modeled by a 5-mass torsional model. Torsional shaft stiffnesses and rotor mass moment of inertia are the parameters reflecting dynamics of the drivetrain and are assumed unknown. Moreover, the aerodynamics input torque applied to the drivetrain system is assumed as unknown. The Bayesian inference method is used to estimate jointly the unknown model parameters and aerodynamic input torque using numerically simulated measurement data. The case studies show that the proposed model updating approach can accurately estimate the low-speed shaft stiffness, rotor mass moment of inertia, and the time history of aerodynamic torque jointly using only measurements from the generator including generator torque and rotational speed.

Adaptive variable structure observer for system states and disturbances estimation with application to building climate control system in a smart grid

In order to reach the ambitious net-zero emission target by 2050, various technological solutions need to be developed to ensure efficient utilisation of energy. Commercial and residential buildings are a big source of greenhouse gas emissions, where efficient utilisation of energy can play a major role towards decarbonisation of the buildings sector. Heat pumps have recently emerged as an effective solution for space heating applications in buildings. Energy-efficient operation of heat pumps will make a significant contribution toward making buildings energy-efficient. In this context, heat pump control systems have a major role. Some of the existing literature on the heat pump control systems assume that various system states are available to measure. This may not always be true and/or economical to measure all the states. Moreover, the system is subject to various disturbances which cannot be directly measured. To reduce the number of sensors in heat pump control systems, an adaptive observer is developed in this paper to estimate inaccessible system states and disturbances simultaneously. An advantage of the proposed approach is that it does not require any bound on the disturbance itself, however, only assumes that the rate of change of disturbance is bounded. This is always the case in practice. In the developed method, adaptive control techniques and variable structure control techniques are combined to implement the proposed observer. In order to estimate the unknown disturbance, an augmented systems model is considered. Globally uniformly ultimately bounded property of the error dynamical systems is established by suitably designing the adaptive laws. The developed method is applied to a model of the heat dynamics of a house floor heating system connected to a ground source-based heat pump. Different disturbance signals formats and amplitudes are considered to show the effectiveness of the proposed technique. Simulation results are given to demonstrate the suitability of the proposed method.

Inducing multistability in discrete chaotic systems using numerical integration with variable symmetry

Multistability is an inherent property of many nonlinear dynamical systems. However, finding exact conditions where a certain nonlinear system is multistable, is a complex problem. In this study, we propose a novel approach for inducing multistability in a discrete system by applying the numerical integration method with variable symmetry to a continuous monostable system and finding a range of the symmetry coefficients for which multiple attractors coexist in its discrete model. We consider the well-known Chen system as an example, showing that multistability can be induced by discretization with variable symmetry. A special two-stage algorithm is proposed for a fast search for hidden attractors. Using the proposed algorithm, we found several multistable versions of the discrete Chen system and investigated their attractors and basins of attraction. By applying numerical backward error analysis, we discovered a generalized continuous Chen system with additional terms which were not present in the original system and have shown its approximate equivalence to the obtained discrete system. The results of this study can be used for inducing artificial multistability in a wide range of chaotic systems with possible applications in chaotic cryptography, communication, and chaos-based sensing.

The large key space image encryption algorithm based on modulus synchronization between real and complex fractional-order dynamical systems.

This paper constructs and analyzes the dynamical properties of a new fractional-order real hyper-chaotic system and its corresponding complex variable system. A thorough analysis was done by employing stability of equilibrium points, phase plots, Lyapunov spectrum, and bifurcation analysis for the consequences of varying fractional-order derivative and parameter values on the system. For the first time, a modulus synchronization scheme is proposed to synchronize real and complex fractional-order dynamical systems. Based on Lyapunov stability theory, non-linear controllers are designed to achieve the proposed modulus synchronization scheme. A new modulus synchronization encryption algorithm with a large key space size for digital images is introduced for the application. The experimental results and analysis validate the desired algorithm. Also, we compare our result of the new encryption algorithm with the previously published literature and verify the efficacy of the considered scheme. Numerical simulations are given to validate the theoretical analysis

Solving Cubic Matrix Equations Arising in Conservative Dynamics

In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the need to solve polynomial matrix equations, a classical and important topic both in theoretical and in applied mathematics. Solving numerically these equations is challenging due to the presence of several conservation laws which our finite models incorporate and which must be retained while integrating the equations of motion. In the last thirty years, the theory of geometric integration has provided a variety of techniques to tackle this problem. These numerical methods require solving both direct and inverse problems in matrix spaces. We present three algorithms to solve a cubic matrix equation arising in the geometric integration of isospectral flows. This type of ODEs includes finite models of ideal hydrodynamics, plasma dynamics, and spin particles, which we use as test problems for our algorithms.

CLASSIFICATION OF ONE DIMENSIONAL DYNAMICAL SYSTEMS BY COUNTABLE STRUCTURES

AbstractWe study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Finite Element Modeling of Dynamic Properties of the Delta Robot with Base Frame.

The paper presents the finite element modeling of the dynamic properties of a delta robot attached to a steel frame. A distinguishing feature of the proposed modeling method is the application of the Guyan reduction method in modeling of frame foundations. The frame in question was analyzed in two variants: (i) without attached robot and (ii) with attached robot. Based on the established model, the dynamic properties (i.e., natural frequencies, mode shapes, and frequency response functions) of the frame in the two variants were analyzed. The obtained results were then experimentally verified and validated. It was found that the developed model showed an average relative error for natural frequencies of 4.3% in the case of the frame and 5.6% in the case of the frame with the robot. The paper demonstrated the validity of the proposed model, allowing accurate and fast determination of robotic system dynamic properties.

A new framework for frequency-dependent polarizable force fields.

A frequency-dependent extension of the polarizable force field "Atom-Condensed Kohn-Sham density functional theory approximated to the second-order" (ACKS2) [Verstraelen et al., J. Chem. Phys. 141, 194114 (2014)] is proposed, referred to as ACKS2ω. The method enables theoretical predictions of dynamical response properties of finite systems after partitioning of the frequency-dependent molecular response function. Parameters in this model are computed simply as expectation values of an electronic wavefunction, and the hardness matrix is entirely reused from ACKS2 as an adiabatic approximation is used. A numerical validation shows that accurate models can already be obtained with atomic monopoles and dipoles. Absorption spectra of 42 organic and inorganic molecular monomers are evaluated using ACKS2ω, and our results agree well with the time-dependent DFT calculations. Also for the calculation of C6 dispersion coefficients, ACKS2ω closely reproduces its TDDFT reference. When parameters for ACKS2ω are derived from a PBE/aug-cc-pVDZ ground state, it reproduces experimental values for 903 organic and inorganic intermolecular pairs with an MAPE of 3.84%. Our results confirm that ACKS2ω offers a solid connection between the quantum-mechanical description of frequency-dependent response and computationally efficient force-field models.

Studying the Influence of Synthetic Inertia on Electric Power System Transient Stability

Development of wind power around the world is a promising solution to the energy crisis. In power systems with a predominant share of wind power plants connected through power converters, their influence on the power system dynamic properties becomes significant. One of the important problems is reduction of the power system total inertia. In view of this, grid operators place demands on connected wind power plants for active power control and participation in frequency control. One possible solution is using synthetic inertia in the wind power plant control system. Research activities in this field place focus on improving the frequency stability and coordinating the synthetic inertia controller with other wind power plant control systems. However, the influence of synthetic inertia and its tuning on the power system stability are still poorly studied. The paper analyzes the effect of synthetic inertia on the power system transient stability. It is shown that by adjusting the synthetic inertia parameters, it is possible not only to obtain the required inertial response, but also to increase the power system transient stability. The influence of the measurement time window and frequency variation rate calculation, which is important in certain modes, is also studied. The study results are obtained for test and real-scale power system configurations.