Pair Of Complex Eigenvalues Research Articles

Eigenvalues bifurcating from the continuum in two-dimensional potentials generating non-Hermitian gauge fields

It has been recently shown that complex two-dimensional (2D) potentials Vɛ(x,y)=V(y+iɛη(x)) can be used to emulate non-Hermitian matrix gauge fields in optical waveguides. Here x and y are the transverse coordinates, V(y) and η(x) are real functions, ɛ>0 is a small parameter, and i is the imaginary unit. The real potential V(y) is required to have at least two discrete eigenvalues in the corresponding 1D Schrödinger operator. When both transverse directions are taken into account, these eigenvalues become thresholds embedded in the continuous spectrum of the 2D operator. Small nonzero ɛ corresponds to a non-Hermitian perturbation which can result in a bifurcation of each threshold into an eigenvalue. Accurate analysis of these eigenvalues is important for understanding the behavior and stability of optical waves propagating in the artificial non-Hermitian gauge potential. Bifurcations of complex eigenvalues out of the continuum is the main object of the present study. Using recent mathematical results from the rigorous analysis of elliptic operators, we obtain simple asymptotic expansions in ɛ that describe the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate into a single eigenvalue, while every other threshold can bifurcate into a pair of complex eigenvalues. These bifurcations can be controlled by the Fourier transform of function η(x) evaluated at certain isolated points of the reciprocal space. When the bifurcation does not occur, the continuous spectrum of 2D operator contains a quasi-bound-state which is characterized by a strongly localized central peak coupled to small-amplitude but nondecaying tails. The analysis is applied to the case examples of parabolic and double-well potentials V(y). In the latter case, the bifurcation of complex eigenvalues can be dampened if the two wells are widely separated.

Bound states in Bose-Einstein condensates with radially-periodic spin-orbit coupling

We consider Bose-Einstein condensate (BEC) subject to the action of spin-orbit-coupling (SOC) periodically modulated in the radial direction. In contrast to the commonly known principle that periodic potentials do not create bound states, the binary BEC maintains multiple localized modes in the linear limit, with their chemical potential falling into spectral gaps of the (numerically found) radial band structure induced by the spatial modulation of the SOC. In the presence of the repulsive nonlinearity, the SOC modulation supports fundamental gap solitons of the semi-vortex types, as well as higher-order vortex gap solitons. The localization degree and stability of the gap solitons strongly depend on the location of their chemical potential in the gap. Stability intervals for vortex gap solitons in a broad range of the intrinsic vorticity, from −2 to 3, are identified. Thus, the analysis reveals the previously unexplored mechanism of linear and nonlinear localization provided by the spatially periodic modulation of SOC, which may be extended to other settings, such as those for optical beams and polaritons. Unlike the commonly known quartets of eigenvalues for small perturbations, in the present system the instability is accounted for by shifted complex eigenvalue pairs.

Analisis Dinamik Model Matematika Penyebaran COVID-19 Pada Populasi SEIR

This study discusses the dynamic analysis of the mathematical model of the spread of COVID-19 using daily cases in Indonesia which are classified into four variables, namely, Susceptible (S), Exposed (E), Infected (I), and Recovered (R) which are then analyzed dynamically by calculate the equilibrium point and look for stability properties. The two equilibrium points of this model are the disease-free equilibrium point and the endemic equilibrium point . Then, it is linearized around the equilibrium point using the given parameters. Linearization around the equilibrium point produces four eigenvalues, one of which is positive. Linearization around the equilibrium point yields two negative real eigenvalues and a pair of complex eigenvalues with negative real parts. Phase portraits and numerical simulations have shown that all variables S, E, I, and R will be asymptotically stable locally towards the equilibrium point, namely the endemic equilibrium point . Thus, based on the dynamic analysis obtained, it is shown that the disease-free equilibrium point is unstable and the endemic equilibrium point is locally asymptotically stable.

On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods

Abstract In this short note, we define an s × s matrix Ks constructed from the Hilbert matrix H s = ( 1 i + j - 1 ) i , j = 1 s {H_s} = \left( {{1 \over {i + j - 1}}} \right)_{i,j = 1}^s and prove that it has at least one pair of complex eigenvalues when s ≥ 2. Ks is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, and our result gives a matrix-theoretical proof that the method does not have large-grain parallelism when its order is larger than or equal to 4.

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.

Noise amplification in frictional systems: Oscillatory instabilities.

It was discovered recently that frictional granular materials can exhibit an important mechanism for instabilities, i.e., the appearance of pairs of complex eigenvalues in their stability matrix. The consequence is an oscillatory exponential growth of small perturbations which are tamed by dynamical nonlinearities. The amplification can be giant, many orders of magnitude, and it ends with a catastrophic system-spanning plastic event. Here we follow up on this discovery, explore the scaling laws characterizing the onset of the instability, the scenarios of the development of the instability with and without damping, and the nature of the eventual system-spanning events. The possible relevance to earthquake physics and to the transition from static to dynamic friction is discussed.

Oscillatory Instabilities in Frictional Granular Matter.

Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to the vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e., the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations that are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.

Homoclinic saddle to saddle-focus transitions in 4D systems

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model.

Identificación de una Bifurcación de Hopf con o sin Parámetros

En este trabajo se presenta los lineamientos teóricos que identifica la existencia de una bifurcación genérica de Hopf, con o sin parámetros, a través de un punto o una línea de puntos de equilibrio para un sistema dinámico continúo. Con estos lineamientos definidos, se analiza la bifurcación de Hopf sin parámetros a través del estudio de contornos viscosos para un sistema de ecuaciones diferenciales parciales conformado por un término de reacción - difusión; con condición inicial constante a trozos y una línea de puntos de equilibrio que presenta un par de valores propios complejos conjugados con parte real nula en su linealización a medida que se aproxima al origen. En condiciones adecuadas, se distinguen dos casos para este tipo de bifurcación genérica de Hopf sin parámetros: hiperbólica y elíptica, genéricas en el sentido que es una bifurcación controlada en una línea de puntos de equilibrio.

On the existence of homoclinic orbits in some class of three-dimensional piecewise affine systems

In this paper, we investigate a class of three-dimensional piecewise affine systems with the matrix in each subsystem processing a pair of complex eigenvalues and a real eigenvalue. Furthermore, we obtain some sufficient and necessary conditions for the existence of homoclinic orbits under suitable assumptions. Finally, some concrete examples are presented to illustrate our results.

Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions.

Paraxial linear propagation of light in an optical waveguide with material gain and loss is governed by a Schrödinger equation with a complex potential. In this Letter, new classes of non-parity-time (PT)-symmetric complex potentials featuring conjugate-pair eigenvalue symmetry are constructed by operator symmetry methods. Due to this eigenvalue symmetry, it is shown that the spectrum of these complex potentials is often all-real. Under parameter tuning in these potentials, a phase transition can also occur, where pairs of complex eigenvalues appear in the spectrum. A distinctive feature of the phase transition here is that the complex eigenvalues may bifurcate out from an interior continuous eigenvalue inside the continuous spectrum; hence, a phase transition takes place without going through an exceptional point. In one spatial dimension, this class of non-PT-symmetric complex potentials is of the form V(x)=h'(x)-h2(x), where h(x) is an arbitrary PT-symmetric complex function. These potentials in two spatial dimensions are also derived. Diffraction patterns in these complex potentials are further examined, and unidirectional propagation behaviors are demonstrated.

Pseudo-Hermitian Hamiltonians generating waveguide mode evolution

We study the properties of Hamiltonians defined as the generators of transfer matrices in quasi- one-dimensional waveguides. For single- or multi-mode waveguides obeying flux conservation and time-reversal invariance, the Hamiltonians defined in this way are non-Hermitian, but satisfy symmetry properties that have previously been identified in the literature as "pseudo Hermiticity" and "anti-PT symmetry". We show how simple one-channel and two-channel models exhibit transitions between real, imaginary, and complex eigenvalue pairs.

Discrete‐time global leader‐following consensus of a group of general linear systems using bounded controls

SummaryThis paper deals with the problem of global leader‐following consensus of a group of discrete‐time general linear systems with bounded controls. For each follower agent in the group, we construct both a bounded state feedback control law and a bounded output feedback control law. The feedback laws for each input of an agent use a multi‐hop relay protocol, in which the agent obtains the information of other agents through multi‐hop paths in the communication network. The number of hops each agent uses to obtain its information about other agents for an input is less than or equal to the sum of the number of real eigenvalues on the unit circle and the number of pairs of complex eigenvalues on the unit circle of the subsystem corresponding to the input, and the feedback gains are constructed from the adjacency matrix of the communication network. We show that these control laws achieve global leader‐following consensus when the communication topology among follower agents forms a strongly connected and detailed balanced directed graph and the leader is a neighbor of at least one follower agent. Copyright © 2017 John Wiley & Sons, Ltd.

Direct Numerical Simulation of Turbulent Channel Flow on High-Performance GPU Computing System

The flow of a viscous fluid in a plane channel is simulated numerically following the DNS approach, and using a computational code for the numerical integration of the Navier-Stokes equations implemented on a hybrid CPU/GPU computing architecture (for the meaning of symbols and acronyms used, one can refer to the Nomenclature). Three turbulent-flow databases, each representing the turbulent statistically-steady state of the flow at three different values of the Reynolds number, are built up, and a number of statistical moments of the fluctuating velocity field are computed. For turbulent-flow-structure investigation, the vortex-detection technique of the imaginary part of the complex eigenvalue pair in the velocity-gradient tensor is applied to the fluctuating-velocity fields. As a result, and among other types, hairpin vortical structures are unveiled. The processes of evolution that characterize the hairpin vortices in the near-wall region of the turbulent channel are investigated, in particular at one of the three Reynolds numbers tested, with specific attention given to the relationship that exists between the dynamics of the vortical structures and the occurrence of ejection and sweep quadrant events. Interestingly, it is found that the latter events play a preminent role in the way in which the morphological evolution of a hairpin vortex develops over time, as related in particular to the establishment of symmetric and persistent hairpins. The present results have been obtained from a database that incorporates genuine DNS solutions of the Navier-Stokes equations, without superposition of any synthetic structures in the form of initial and/or boundary conditions for the simulations.

The Field of Flow Structures Generated by a Wave of Viscous Fluid Around Vertical Circular Cylinder Piercing the Free Surface

The diffraction of water waves induced by a large-diameter, surface-piercing, vertical circular cylinder is studied numerically. The Navier-Stokes equations in primitive variables are considered for the simulation of a given wave case, and the technique is followed of the Direct Numerical Simulation (DNS). The criterion of the imaginary part of the complex-eigenvalue pair of the velocity-gradient tensor for the extraction of the flow vortical structures is applied to the computed fields, so unveiling a complex configuration of structures at the free surface, and at the cylinder external walls. The most energetic modes of the flow are further extracted from the DNS-simulated fields by using the Karhunen-Loève decomposition (KL). A “reduced” velocity field is reconstructed using the first three most energetic eigenfunctions of the decomposition, and its evolution is followed through a sequence of time steps.

Triggered Fronts in the Complex Ginzburg Landau Equation

We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg–Landau equation. We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wave trains nucleate. Our results show existence of coherent, “heteroclinic” profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our results also give expansions for the wavenumber of wave trains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading-order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis.

Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction–diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

Stroh formalism for icosahedral quasicrystal and its application

The Stroh formalism for two-dimensional deformations of the icosahedral quasicrystal is studied, for which there are six pairs of complex eigenvalues. The closed-form expressions for the elastic displacement and stress fields induced by a dislocation in an icosahedral quasicrystal are obtained using the extended Stroh formalism. The effect of phonon–phason coupling elastic constant on mechanical behavior are also discussed.

Lyapunov Inverse Iteration for Identifying Hopf Bifurcations in Models of Incompressible Flow

The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [K. Meerbergen and A. Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982--1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.