Pair Of Eigenvalues Research Articles

Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability

Abstract An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.

Geometric invariance of determining and resonating centers: Odd- and any-number limitations of Pyragas control.

In the spirit of the well-known odd-number limitation, we study the failure of Pyragas control of periodic orbits and equilibria. Addressing the periodic orbits first, we derive a fundamental observation on the invariance of the geometric multiplicity of the trivial Floquet multiplier. This observation leads to a clear and unifying understanding of the odd-number limitation, both in the autonomous and the non-autonomous setting. Since the presence of the trivial Floquet multiplier governs the possibility of successful stabilization, we refer to this multiplier as the determining center. The geometric invariance of the determining center also leads to a necessary condition on the gain matrix for the control to be successful. In particular, we exclude scalar gains. The application of Pyragas control on equilibria does not only imply a geometric invariance of the determining center but surprisingly also on centers that resonate with the time delay. Consequently, we formulate odd- and any-number limitations both for real eigenvalues together with an arbitrary time delay as well as for complex conjugated eigenvalue pairs together with a resonating time delay. The very general nature of our results allows for various applications.

A note on the state space representation of aeroelastic systems for time domain analysis

The time domain state space model for describing an aeroelastic system has usually been obtained by using rational function approximations to write unsteady aerodynamics in the time domain. This strategy has been employed by many researchers because the standard form of unsteady aerodynamics is typically computed in the reduced frequency domain with explicit or implied transcendental functions, which do not readily admit to applying frequency to time domain transforms. However, this article demonstrates that the unsteady aerodynamics do not need to be explicitly written in the time domain when the purpose is to obtain a state space model for aeroelastic analysis. A basic and classical result from the linear algebra is employed for constructing a constant dynamic matrix by using the pair of eigenvalues and eigenvectors obtained iteratively from the pk method. The results show that the new approach provides accurate results for linear dynamic aeroelastic models described in either physical or generalized coordinates.

A paradigmatic system for non-classic interactive buckling

A non-classic double bifurcation of discrete static systems, occurring along a non trivial equilibrium path, is studied. It occurs when, by varying a parameter, a pair of eigenvalues of the tangent stiffness matrix first merge and then disappear. A paradigmatic 2 degrees of freedom system with some symmetry, consisting of an inverted extensible pendulum, so far studied in literature in the linear range only, is proposed. Exact nonlinear solutions are derived, showing the mechanism which leads two distinct bifurcated paths to locally touch each other at the double bifurcation point, and then to detach from the fundamental path. The detrimental effects of the interaction on stability are discussed. An asymptotic bifurcation analysis is also carried out, able to explain how a unique buckling mode existing at the double bifurcation is able to generate, in the perturbation process, two distinct deflections on close bifurcated paths.

An update on coherent scattering from complex non-PT-symmetric Scarf II potential with new analytic forms

The versatile and exactly solvable Scarf II potential has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector, it has gone underutilised. Here, we present the most simple analytic forms for the scattering coefficients \((T(k),R(k),|\det S(k)|)\). On the one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand, they make new predictions – all simple and analytical. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of spectral singularity (SS) into a finite number of complex conjugate pairs of eigenvalues (CCPEs). NSDSS behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising coexistence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, so far, the complex Scarf II potential is not known to be pseudo-Hermitian (\(\eta ^{-1} H\eta =H^\dagger \)) under a metric of the type \(\eta (x)\).

Stochastic asset flow equations: Interdependence of trend and volatility

The complex interaction between the volatility and the price trend is examined. We model stochastic asset prices using the asset flow model with randomness arising directly from supply and demand. We show that the volatility is smallest at the extrema of the expected logarithm of the price. Linearizing the stochastic differential equation (SDE) about equilibrium, we obtain an exact relation for the autocovariance function, and relate it to the (3 by 3) Jacobian of the linearized SDE. In particular, we find the conditions under which one has a pair of complex conjugate eigenvalues of the Jacobian resulting in oscillations. The frequency of the oscillations depends only on the imaginary part of the complex pair, while the decay rate depends only on the real eigenvalue and the real parts of the complex pair. For the deterministic system, oscillations typically decay rapidly. However, randomness induces oscillations to continue indefinitely with a frequency that depends on the parameters of the deterministic system. The computations and analytical results presented here demonstrate that volatility increases when traders place greater emphasis on trend, confirming a generally held belief among practitioners.

Study on the Reduction of the Resonant Stress of Turbine Blades Caused by the Stage Interaction Force (Simultaneous Optimization of Blade Resonant Stress and Amount of Unbalance)

Abstract Although the bladed disks of turbomachinery are nominally designed to be cyclically symmetric (tuned system), the vibration characteristics of individual blades on a disk vary slightly owing to manufacturing tolerances, deviations in material properties, wear during operation, etc. These small variations break the cyclic symmetry and split eigenvalue pairs. Actual bladed disks with small variations are called mistuned systems. The resonant stress on a mistuned bladed disk may become large and cause a blade failure due to high cycle fatigue. Especially for a bladed disk with a free-standing blade structure, it is essential to use a sufficiently large safety factor at the design stage, considering the mistuning effect. Traditionally, blade designers have adopted various countermeasures to reduce the resonant stress at the design stage. Besides the countermeasures adopted in blade design, for a variable speed engine, which cannot avoid resonance, it is thought that a practical optimization method for a mistuned system (bladed disk with a free-standing blade structure) is to sort the blades so that the resonant stress is minimized. In this study, a simultaneous method for optimizing the blade resonant stress and the amount of unbalance causing rotor vibration is proposed. In this method, first, the natural frequencies and weights of all blades on a disk are measured. Then, a mistuned system is assembled and the analysis model is generated. Next, the resonant stress and the amount of unbalance in the mistuned system are analyzed. To reduce the computation time, the reduced-order model known as fundamental mistuning model (FMM) is used to calculate the resonant stress in the mistuned system. The analyses of the resonant stress and the amount of unbalance are carried out repeatedly, sorting the blades on the disk, and the optimal solution is explored using Monte Carlo simulations (MCS) or discrete differential evolution (DDE). As an example, a mistuned bladed disk of an aero-engine was analyzed, and the validity of the proposed method was verified.

Dynamic Stability Analysis of a Hydrofoiling Sailing Boat using CFD

Abstract. This paper describes the study of the dynamic stability of a hydrofoiling sailing boat called the “Goodall Design Foiling Viper”. The goal of Goodall Design is to make hydrofoiling accessible to a wider public, whereas it was previously reserved for professional sailors at the highest level of the sport. To allow for safe operation, stability is an essential characteristic of the boat. The aim of this work is to find a strategy to perform a dynamic stability analysis using computational fluid dynamics (CFD), which can be used in a preliminary design stage. This paper starts by establishing a theoretical framework to perform the dynamic stability analysis. A stability analysis has to be performed around an equilibrium state which depends on operating parameters such as speed, centre of gravity, etc. . A fluid-structure interaction strategy is applied to determine these equilibrium states. The last part discusses the stability characteristics of the Viper. The framework managed to assess the dynamic stability of the Viper and found 5 longitudinal eigenmodes: two complex conjugated pairs of eigenvalues and one real eigenvalue. It can be concluded that the boat was both statically and dynamically stable.

Long-Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions in the Presence of a Discrete Spectrum

The long-time asymptotic behavior of solutions to the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the xt-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

Smale-Williams solenoids in autonomous system with saddle equilibrium.

We construct an autonomous low-dimensional system of differential equations by replacement of real-valued variables with complex-valued variables in a self-oscillating system with homoclinic loops of a saddle. We provide analytical and numerical indications and argue that the emerging chaotic attractor is a uniformly hyperbolic chaotic attractor of Smale-Williams type. The four-dimensional phase space of the flow consists of two parts: a vicinity of a saddle equilibrium with two pairs of equal eigenvalues where the angular variable undergoes a Bernoulli map, and a region which ensures that the trajectories return to the origin without angular variable changing. The trajectories of the flow approach and leave the vicinity of the saddle equilibrium with the arguments of complex variables undergoing a Bernoulli map on each return. This makes possible the formation of the attractor of a Smale-Williams type in Poincaré cross section. In essence, our model resembles complex amplitude equations governing the dynamics of wave envelops or spatial Fourier modes. We discuss the roughness and generality of our scheme.

Flutter analysis of mistuned bladed disks (Effect of bladed disk structure on flutter characteristics)

Although bladed disks are nominally designed to be cyclically symmetric (tuned system), the vibration characteristics of all the blades on a disk are slightly different due to the manufacturing tolerance, deviations in the material properties, and wear during operation. These small variations break the cyclic symmetry and split the eigenvalue pairs. Bladed disks with small variations are referred to as a mistuned system. In the forced response of a mistuned bladed disk, the responses of all the blades become different, and the response of a certain blade may become extremely large due to splitting of the duplicated eigenvalues and distortion of the vibration modes. On the other hand, many researchers suggest that mistuning suppresses blade flutter, because the complete traveling wave mode is not formed in a disk. In the previous paper, authors studied the forced response of the mistuned bladed disk, using the reduced order model SNM (Subset of Nominal system Modes), and clarified that the effect of the bladed disk structure on the amplification factor. This work is a follow-up study on the previous paper. The stability analyses of mistuned bladed disks with the free-standing blade structure and the continuous ring-blade structure are carried out extensively, using the reduced order model SNM. Comparing the results of the stability analyses of mistuned bladed disks, the effect of the bladed disk structure on the blade flutter is clarified.

Double gaussianization of graph spectra

The graph spectrum is the set of eigenvalues of a simple graph with n vertices. Here we fold this graph spectrum at a given pair of reference eigenvalues and then exponentiate the resulting folded graph spectrum. This process produces double Gaussianized functions of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidences from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. In the examples that we have examined, they enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove here several results for the trace of the double Gaussianized adjacency matrix of simple graphs–the double Gaussianized Estrada index. Finally we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or inactive ones. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. Such indices reflect the electron population of the HOMO/LUMO and eigenvalues close to them, in the so-called K and L regions of PAHs.

Stochastic Asset Flow Equations: Interdependence of Trend and Volatility

The interaction between the volatility and price dynamics is explored. We model stochastic asset prices using the asset flow model with randomness arising directly from supply and demand. We show that the volatility is smallest at the extrema of the price. Linearizing the stochastic differential equation (SDE) about equilibrium, we obtain an exact relation for the autocovariance function, and relate it to the (3 by 3) Jacobian of the linearized SDE. In particular, we find the conditions under which one has a pair of complex conjugate eigenvalues of the Jacobian resulting in oscillations. The frequency of the oscillations depends only on the imaginary part of the complex pair, while the decay rate depends only on the real eigenvalue and the real parts of the complex pair. For the deterministic system, oscillations typically decay rapidly. However, randomness induces oscillations to continue indefinitely with a frequency that depends on the parameters of the deterministic system. The computations and analytical results presented here demonstrate that volatility increases when traders place greater emphasis on trend, confirming a generally held belief among practitioners.

Global stability analysis of a 90°-bend pipe flow

The present work investigates the stability properties of the flow in a 90°-bend pipe with curvature δ=R/Rc=1/3, with R being the radius of the cross-section of the pipe and Rc the radius of curvature at the pipe centreline. Direct numerical simulations (DNS) for values of the bulk Reynolds number Reb=UbD/ν between 2000 and 3000 are performed. The bulk Reynolds number is based on the bulk velocity Ub, the pipe diameter D, and the kinematic viscosity ν. The flow is found to be steady for Reb⩽2500, with two main pairs of symmetric, counter-rotating vortices in the section of the pipe downstream of the bend. The presence of two recirculation regions is detected inside the bend: one on the outer wall and the other on the inner side. For Reb⩾2550, the flow exhibits a periodic behaviour, oscillating with a fundamental non-dimensional frequency St=fD/Ub=0.23. A global stability analysis is performed in order to determine the cause of the transition from the steady to the periodic regime. The spectrum of the linearised Navier-Stokes operator reveals a pair of complex conjugate eigenvalues with positive real part, hence the transition is ascribed to a Hopf bifurcation occurring at Reb,cr≈2531, a value much lower than the critical Reynolds number for the flow in a torus with the same curvature. The velocity components of the unstable direct and adjoint eigenmodes are investigated, and they display a large spatial separation, most likely due to the non-normality of the linearised Navier-Stokes operator. Thus, the core of the instability, also known in the literature as the wavemaker, is sought performing an analysis of the structural sensitivity of the unstable eigenmode to spatially localised feedbacks. The region located 15° downstream of the bend inlet, on the outer wall, is the most receptive to this kind of perturbations, and thus corresponds to where the instability originates. Since this region coincides with the outer-wall separation bubble, it is concluded that the instability is linked to the strong shear by the backflow phenomena. The present results are relevant for technical applications where bent pipes are frequently used, and their stability properties have hitherto not been studied.

Eigenvalues of uncorrelated, density-difference matrices and the interpretation of Δ-self-consistent-field calculations.

Two theorems on the eigenvalues of differences of idempotent matrices determine the natural occupation numbers and orbitals of electronic detachment, attachment, or excitation that pertain to transitions between wavefunctions that each consist of a single Slater determinant. They are also applicable to spin density matrices associated with Slater determinants. When the ranks of the matrices differ, unit eigenvalues occur. In addition, there are ±w pairs of eigenvalues where |w| ≤ 1, whose values are related to overlaps, t, between the corresponding orbitals of Amos and Hall, and Löwdin by the formula w=±1-t2 12. Generalized overlap amplitudes, including Dyson orbitals and their probability factors, may be inferred from these eigenvalues, which provide numerical criteria for: classifying transitions according to the number of holes and particles in final states with respect to initial states, identifying the most important effects of orbital relaxation produced by self-consistent fields, and the analysis of Fukui functions. Two similar theorems that apply to sums of idempotent matrices regenerate formulae for the natural orbitals and occupation numbers of an unrestricted Slater determinant that were published first by Amos and Hall.

New families of non-parity-time-symmetric complex potentials with all-real spectra

New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials, which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions, which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.

Multipulse Homoclinic Orbits and Chaotic Dynamics of a Reinforced Composite Plate with Carbon Nanotubes

The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite plate.

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

<p style='text-indent:20px;'>The article aims to examine the dynamic transition of the reaction-diffusion Fitzhugh-Nagumo system defined on a thin spherical shell and a 2D-rectangular domain. The mathematical tool employed is the theory of phase transition dynamics established for dissipative dynamical systems. The main results in this paper include two parts. First, for the system on a thin spherical shell, we only focus on the transition from a real simple eigenvalue. More precisely, if the first eigenspace is three–dimensional, the system undergoes either a continuous transition or a jump transition. Besides, a mix transition is also allowed if the first eigenspace is one–dimensional. Second, for the system on a rectangular domain, both the transitions from a simple real eigenvalue and a pair of simple complex eigenvalues are considered. Our results imply that two steady-state solutions bifurcate, which are either attractors or saddle points, and a Hopf bifurcation is also possible in the system on the rectangular domain.

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, known as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger.The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrates the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter.