Eigenvalues Of System Research Articles

Dynamics of a periodic benthic-drift model for two species competition

This paper is devoted to the study of a time-periodic model for two species competition in river environments, where the populations reproduce and compete in the benthic zone and diffuse in the drifting water zone. We obtain a complete threshold result on the persistence and extinction of two species in terms of principal eigenvalues for degenerate periodic reaction-diffusion systems, and establish the coexistence of two competing populations under persistence conditions.

Control Oriented Modeling, Experimentation, and Stability Analysis of an Autorotating Samara

Abstract This paper presents a control-oriented model for describing the steady-state and dynamic behavior of a single-winged samara seed-pod in autorotative descent. A negligible lateral center of mass motion and constant, prescribed roll-angle to develop a simplified and compact model. Spanwise aerodynamic dependence is exchanged for an independent blade element representation with two tuned parameters to account for the effects of leading-edge vortex phenomena. The resulting model is a fourth-order nonlinear dynamical system. The accuracy of this model is established by validating it against our own experimental data as well as against those reported in the literature by other researchers. The validation exercise reveals that zero roll-angle is a viable assumption that significantly reduces model complexity while retaining accuracy. A necessary condition is derived for the existence of steady autorotation of the samara under free descent. Furthermore, a stability analysis is conducted suggesting that the eigenvalues of the fourth-order system, linearized about the autorotational equilibrium, can be well-represented by those of two decoupled two-dimensional systems. The analysis reveals the critical parameters that determine stability of sustained autorotation. Such stability analysis provides a platform for similar analytical exploration of future model improvements. The validity of this compact model suggests the plausibility of designing and controlling simple autorotative mechanisms based on these dynamics.

Thermodynamic properties and superstatistics analysis of particle in the non-relativistic system under the influence of trigonometric scarf potential using Dong proper quantization

There has been an intense study on the eigenvalue and eigenfunction of the nonrelativistic quantum system. The Schrodinger equation is important in the nonrelativistic quantum system that is used to analyze the quantum information of the system, so it attached the interest of the researchers to conduct the exploration on it. In most of the studies, the researchers expand the study of the solution of the Schrodinger equation to the physical properties of the system. The thermodynamical property is one of the physical properties and key to analyzing the characteristic of materials. It needs more investigation since it has the potential to form a new design of the material. In this study, the Schrodinger equation with trigonometric Scarf potential was solved using the Supersymmetry WKB quantization condition scheme, where the variable of trigonometric Scarf potential was transformed into the Poschl-Teller potential. The energy spectra were obtained by comparing the constant parameters between Trigonometric Scarf potential and Poschl-Teller potential and by using Dong proper quantization condition of transformed Supersymmetry WKB for trigonometric Scarf potential. The result shows that the energy spectra are influenced by the radial quantum number and the potential width, and the energy increase by the increase of the radial quantum number and the potential width. The thermodynamic properties of the system were obtained approximately using the energy spectra equation and were expressed in the erf function. The superstatistics mechanics was approximately obtained by using modified Delta Dirac function distribution for the Boltzmann factor. These properties are important to analyze the characteristic of the designed solids.

Analysis of Communication Simulation Model of Urban Comprehensive Energy Network Based on the Internet of Things

In urban construction, all energy infrastructures have been realized by the information structure. However, due to the non-uniform structure of each energy network, the data cannot be more compatible, and the communication guarantee based on the Internet of Things is one of the most urgent problems to be solved. In this paper, a reasonable network computing offloading model is designed in the edge computing layer, and the computing offloading algorithm is introduced. Considering the dependency between subtasks, an optimization problem is established, and a central scheduling algorithm is proposed to solve it. An optimized distributed unloading algorithm is proposed according to the solution idea of the classical distributed unloading algorithm. Simulation results show that the optimized distributed unloading algorithm proposed in this paper has good solving efficiency and performance, and reduces the task processing delay. At the same time, the experimental simulation proves that the efficiency of multiple edge servers in processing subtasks is higher than that of a single server. It is established that changing the convergence speed and communication delay stability margin of the system by changing the system eigenvalue is beneficial to design a better communication network structure to keep the system stable.

Multiple phase transitions and anomalous non-Hermitian skin effect

Non-Hermiticity is revolutionizing the common understanding in the Hermitian scope by interfacing with topological physics. A single non-Hermitian band can host nontrivial topology that is defined over the nonzero windings of the system's eigenvalues, but not the eigenvectors, as opposed to the Hermitian topological systems. This unique non-Hermitian band topology is responsible for the intriguing non-Hermitian skin effect. Here, we show that the band topology of non-Hermitian one-band models can experience multiple phase transitions. Each transition is identified by a change of the eigenvalue winding. Such an interesting phenomenon is enabled by strategically manipulating the long-range couplings. Associated with the multiple phase transitions, the non-Hermitian skin effect exhibits anomalous behaviors, i.e., in different phases, the skin modes take on different forms of wave localizations. At the transition points, the skin modes evolve into Bloch-wavelike extended states, as if the Hermiticity is restored. For demonstration, we consider a modified Hatano-Nelson model with nonreciprocal next-nearest neighbor couplings. Therein, two phase transitions are identified, corresponding to a stationary and a dynamic one (with respect to the reciprocal nearest-neighbor coupling). Exact analytical derivations are performed to analyze the phase transitions and the associated anomalous non-Hermitian skin effect, which show consistency with the numerical results based on the generalized Brillouin zone method. Our work reveals rich non-Hermitian topological physics in one-band systems, which are considered to be topologically trivial in Hermitian systems. By further showing the anomalous non-Hermitian skin effect, intriguing wave manipulations may inspire experiments in classical realms such as photonics and phononics where versatile non-Hermitian controls are feasible.

Microscopic Quantum Jump: An Interpretation of Measurement Problem

Nearly a century has passed, since the birth of quantum mechanics, and yet the measurement problem has not been solved. We investigate the measurement problem from two aspects. First we scrutinize the basic postulates adopted by existing theories and identify the postulate of classicality of apparatus (PCA) to be the origin of the trouble. Second, we analyze the simplest possible experimental setup, a single photon particle as an observed system S and a detector as an apparatus A, and we find that a quantum jump occurs as a microscopic interaction between S and a single particle in A. We call this a microscopic quantum jump (MIJ). The MIJ selects system eigenvalues (SEVs) such as a two-dimensional position and arrival time for an incident photon. The MIJ outputs a microscopic particle (MIP), which carries the information of the SEVs potentially. In the apparatus A, the MIP triggers amplification cascade of secondary particles, which we call the intermediate particles (IMPs). The IMPs are initially a few, but become plenty after the amplification. The output of the amplification is a macroscopic observable (MAO) such as a current pulse, which carries the information of the SEVs in actuality. The measurement is complete when the MAO is obtained. By adopting the postulate of the MIJ and by discarding the PCA, we have constructed a measurement theory, which is consistent with standard quantum mechanics.

Study on the dynamic characteristics of the suspender with additional dampers and a frequency-based multiple parameter identification method for the system

The measurement of tensile forces inside suspenders plays an important role in safety inspection and monitoring during construction and operation of large structural engineering projects, including arch bridges and suspension bridges. The frequency-based method is one of the most widely used methods for tension force measurement of such structure components, but it is affected by many factors, such as the additional damping from a damper and various boundary conditions. In this study, the equation of damped motion for a suspender-damper system is derived, a numerical solution method based on the finite difference scheme is given, and a frequency-based multiple parameter identification method is proposed based on the derivative of the system eigenvalue. The study shows that the position and damping coefficient of the damper exert an important effect on the frequency and the plural mode of the suspender-damper system. The influence mechanism is discussed in the study. The numerical examples indicate that the proposed parameter identification algorithm can identify multiple system parameters of the suspender with a high precision and the maximum error is no more than 1%. The influence of the errors in the frequencies is also discussed in this study.

An explicit phase field material point method for modeling dynamic fracture problems

AbstractA novel explicit phase field material point method (ex‐PFMPM) is proposed for modeling dynamic fracture problems. The rate‐dependent phase field governing equation is discretized by a set of particles, and the phase field is updated by the explicit forward‐difference time integration. Furthermore, the stability of the ex‐PFMPM is studied. A novel explicit critical time step formula is obtained based on the system eigenvalues in one dimension and then extended to two and three dimensions. The critical time step formula takes the effect of particle position and neighboring cell interaction into consideration, and can also be used in an explicit phase field finite element method. Several numerical examples, including a dynamic crack branching, a plate with pre‐existing crack under velocity boundary conditions and a three point bending problem are studied to verify the proposed ex‐PFMPM. The use of the history field in the explicit method is studied, which shows that it will lead to fake phase field update and overestimation of the fracture energy in the unloading case. All of the numerical results show that the proposed ex‐PFMPM has the capacity of modeling the crack initiation and propagation problems with both accuracy and efficiency.

Bragg spectrum, K-theory, and gap labeling of aperiodic solids

The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute the topological Bragg spectrum. We relate the topological Bragg spectrum to topological invariants (Chern numbers) of the solid and to the gap-labeling group, which is the group of possible gap labels for the spectrum of a Schrödinger operator describing the electronic motion in the solid.

Drude weights in one-dimensional systems with a single defect

Ballistic transport of a quantum system can be characterized by Drude weight, which quantifies the response of the system to a uniform electric field in the infinitely long timescale. The Drude weight is often discussed in terms of the Kohn formula, which gives the Drude weight by the derivative of the energy eigenvalue of a finite-size system with the periodic boundary condition in terms of the Aharonov-Bohm flux. Recently, the Kohn formula is generalized to nonlinear responses. However, the nonlinear Drude weight determined by the Kohn formula often diverges in the thermodynamic limit. In order to elucidate the issue, in this work we examine a simple example of a one-dimensional tight-binding model in the presence of a single defect at zero temperature. We find that its linear and non-linear Drude weights given by the Kohn formula (i) depend on the Aharonov-Bohm flux and (ii) diverge proportionally to a power of the system size. We argue that the problem can be attributed to different order of limits. The Drude weight according to the Kohn formula (``Kohn--Drude weight'') indicates the response of a finite-size system to an adiabatic insertion of the Aharonov-Bohm flux. While it is a well-defined physical quantity for a finite-size system, its thermodynamic limit does not always describe the ballistic transport of the bulk. The latter should be rather characterized by a ``bulk Drude weight'' defined by taking the thermodynamic limit first before the zero-frequency limit. While the potential issue of the order of limits has been sometimes discussed within the linear response, the discrepancy between the two limits is amplified in nonlinear Drude weights. We demonstrate the importance of the low-energy excitations of $O(1/L)$, which are excluded from the Kohn--Drude weight, in regularizing the bulk Drude weight.

Dependence of eigenvalues of Dirac system on the parameters

AbstractThe dependence of eigenvalues of Dirac system with general boundary conditions is studied. It is shown that the eigenvalues of Dirac operators depend not only continuously but also smoothly on the coefficients, the boundary conditions, and the endpoints of the problem. Furthermore, the differential expressions of the eigenvalues as regards these parameters are given. The results obtained in this paper would provide theoretical support for the numerical calculations of eigenvalues of the corresponding problems.

An Efficient High-to-Low Iterative Method for Light Water Reactor Analysis Based on NEAMS Tools

The so-called two-step method involving the consecutive lattice physics and core simulation has been successfully and widely used in large-scale nuclear reactor calculations thanks to its superior computational efficiency and a satisfactory level of accuracy. However, its performance is challenged by the ever-increasing level of heterogeneity in core designs due to the use of infinite lattice approximation in the lattice calculation and its inability to update cross-section sets according to the core environment change. This paper introduces an alternative approach for light water reactor steady-state core analysis. During the core calculation process, iterations between the local lattice transport calculation and the global core nodal calculation are conducted. These iterations continuously update the boundary condition applied to the lattice model and generate updated cross-section sets. This is done through the iteration between the local lattice transport calculation and the global core nodal simulation. The neutronics high-to-low (Hi2Lo) scheme was formulated using Nuclear Energy Advanced Modeling and Simulation or NEAMS codes, in particular, with the modified PROTEUS-MOC and PROTEUS-NODAL serving as the transport lattice solver and full-core nodal solver, respectively. The verification of the implemented Hi2Lo iterative scheme on the two-dimensional C5G7-TD benchmark problem shows that the Hi2Lo scheme outperforms the two-step approach in terms of prediction accuracy for the key responses of interest (e.g., the system eigenvalue and power distribution) at a computational cost lower than that of the direct full-core transport calculation. To further improve its efficiency, an acceleration method has been developed and implemented for the Hi2Lo approach, and the results indicate that the acceleration method can significantly reduce the run time of a full-core transport solution by a factor of 14 while generating solutions with comparable accuracy.

Reconfigurable enhancement of actuation forces by engineered losses in non-Hermitian metamaterials

While boosting signals with amplification mechanisms is a well-established approach, attenuation mechanisms are typically considered an anathema because they degrade the efficiency of the structures employed to perform useful operations on these signals. An emerging alternate viewpoint promotes losses as a novel design element by utilizing the notion of exceptional point degeneracies (EPDs)—points in parameter space where the eigenvalues of the underlying system and the associated eigenvectors simultaneously coalesce. Here, we demonstrate a direct consequence of such eigenbasis collapse in elastodynamics—an unusual enhancement of actuation force by a judiciously designed non-Hermitian metamaterial supporting an EPD that is coupled to an actuation source. Intriguingly, the EPD enables this enhancement while maintaining a constant signal quality. Our work constitutes a proof-of-principle design which can promote a new class of reconfigurable nano-indenters and robotic-actuators. Importantly, it reveals the ramifications of non-Hermiticity in boosting the Purcell emissivity enhancement factor beyond its expected value, which can guide the design of metamaterials with enhanced emission that does not deteriorate signal quality for mechanical, acoustic, optical, and photonic applications.

Propagation dynamics of a nonlocal spatial Lyme disease model in a time–space periodic habitat

This paper is concerned with the investigation of Lyme disease spread via a time–space periodic nonlocal spatial model in an unbounded domain. We first study the spatial periodic initial problem of the model system and discuss the existence of principal eigenvalue of a linear system with the spatial nonlocality induced by time delay under a smooth assumption. Then we establish the existence of the spreading speeds, and show its coincidence with the minimal wave speed. We further perform a perturbation argument to remove this aforementioned assumption and provide an estimation of the spreading speeds in terms of the spectral radius. Simulations are presented to illustrate our analytic results.

Bounds on the eigenvalues of systems with delay

We review some known bounds for eigenvalues of matrices and use similar techniques to derive bounds for nonlinear eigen problems and the eigenvalues for LTI systems with delays. Two classes of results are presented. The first are based on Hermitian decompositions, the second on Gershgorin's theorem. The bounds are easily computable. We reflect on implications for stability theory which may be contrasted with bounds that have been obtained via Riccati stability based on Lyapunov-Krasovskii theory.

An Improved Contour-Integral Algorithm for Calculating Critical Eigenvalues of Power Systems Based on Accurate Number Counting

This paper proposes an improved contour-integral algorithm based on exact number counting inside the interesting region to calculate critical eigenvalues of large-scale power systems. Firstly, the difference between the eigenvalues inside and outside the region of the non-hermitian matrix is derived using trapezoidal integral. Based on this difference, a small matrix can be skillfully constructed with reasonable parameters so that the exact number of eigenvalues is precisely equal to the eigenvalue number of the small matrix whose real parts are larger than 0.5. Further, rational parameters can be adaptively adjusted to obtain exact results exploiting the minimum of eigenvalues of the constructed matrix. The exact eigenvalue number can help set the contour-integral algorithm's initial subspace. We automatically expand the space by comparing the results of each iteration with the exact number to ensure that all eigenvalues are obtained. Numerical experiments demonstrate the effectiveness and reliability of the precise number counting algorithm and the improved contour integral algorithm.

High-Order Mean and Moment Exponential Stability Analysis Using Elimination and Duplication Matrices

This study presents a necessary and sufficient condition for high-order moment stability of linear stochastic systems. Such a high-order stability notion is efficient as a metric of both stability and robustness to system randomness. The condition is established using two key matrices called the generalized elimination and duplication matrices, which efficiently represent high-order moment vectors. The high-order vector can be treated as the state variable of another linear deterministic system that involves the generalized matrices. The necessary and sufficient condition for the high-order moment stability is associated with the eigenvalues of the deterministic system. In addition, the high-order moment stability is equivalent to high-order mean stability if the order is even.

Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

<p style='text-indent:20px;'>In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [<xref ref-type="bibr" rid="b12">12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math id="M1">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math id="M2">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math id="M3">\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M4">\begin{document}$ m\rightarrow +\infty $\end{document}</tex-math></inline-formula>. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.</p>

A Bound on the Existence of the Maximum Jointly Invariant Set of Input-Coupled Systems

We present a set-theoretical characterization of a bound on the maximal portion that an agent can cede of its input variable to another agent. By ceding control authority, agents can decompose coupling variables into public and private parts, which is of interest in situations of partial cooperation. In particular, sufficient conditions under which the non-existence of the maximum robust control invariant set is guaranteed are provided, expressed in terms of support functions and the dominant system eigenvalue. Finally, the results are illustrated via stable and unstable example systems with different coupling.

Effect of System Mode and Structural Damping Perturbation on the Mistuned Forced Response and Aerodynamic Damping of Embedded Compressor Rotors

Abstract This paper is a continuation of previous papers by the authors that focuses on predicting the mistuned embedded rotor blade forced response. The compressor under consideration is a part of a 3.5stage rig located at Purdue University. Previously, the current authors have discussed the impact of sideband traveling wave forcing functions on the mistuned response and pinned down the reason for a constant underprediction in the amplification factor. This prompted further research to determine the sensitivity of the response to a known change in the system mode. In the first section of the current paper, the authors perturb the system modes frequencies in a probabilistic manner and compute the influence of the system mode on the mistuning amplification factor. The second part of this study involves determining the impact of a perturbation in the structural damping on the mistuned response. The study is also extended to an aerodynamic dynamic analysis wherein the impact of a perturbation in system mode frequency and structural damping is determined both individually and as a combined influence. Finally, a brief investigation of system eigenvalues and eigenvectors is conducted to understand the impact of mistuning on aerodynamic damping suppression. This analysis is joined with a broad examination of the relationship between damping and the extent of mistuning. The key conclusions from this paper are (1) the mistuned forced response was highly sensitive to the system mode input, i.e., although the input was probabilistic, the output was deterministic, (2) since the aerodynamic damping dominates in the case study, a change in the structural damping parameter has minimal effect on the mistuned response, particularly the mistuning amplification factor, (3) the results of the flutter analysis show that a perturbation in the system mode frequency stabilizes the system much faster than a perturbation in the structural damping, and the latter has minimal influence on the eigenvalues of the solution, (4) under the application of mistuning, compression of the damping components of the complex eigenvalues is dominant over the expansion of the frequency components and is amplified when the structural coupling is not considered, and (5) in cases of both random and near alternate blade mistuning, the relationship between mistuning and damping is positive up to a point, after which further increasing mistuning begins to reduce system damping.