Pure Imaginary Eigenvalues Research Articles

Time Crystal in a Single-Mode Nonlinear Cavity.

Time crystal is a class of nonequilibrium phases with broken time-translational symmetry. Here, we demonstrate the time crystal in a single-mode nonlinear cavity. The time crystal originates from the self-oscillation induced by a linear gain and is stabilized by a nonlinear damping. We show in the time crystal phase there are sharp dissipative gap closing and pure imaginary eigenvalues of the Liouvillian spectrum in the thermodynamic limit. Dynamically, we observe a metastable regime with the emergence of quantum oscillation, followed by a dissipative evolution with a timescale much longer than the oscillating period. Moreover, we show there is a dissipative phase transition at the Hopf bifurcation, which can be characterized by the photon number fluctuation in the steady state. These results pave a new promising way for further experiments and deepen our understanding of time crystals.

Computation of Normal Form and Unfolding of Codimension-3 Zero-Hopf–Hopf Bifurcation

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.

General soliton solutions for the complex reverse space-time nonlocal mKdV equation on a finite background

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the PT symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or PT symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process.

MULTIPERIODIC OSCILLATIONS IN SECOND-ORDER LYAPUNOV'S SYSTEMS WITH A DIFFERENTIATION OPERATOR IN THE DIRECTION OF A QUASI-PERIODIC VECTOR FIELD

Abstract. The work is devoted to studying oscillatory solutions of Hamiltonian systems of two equations with a differentiation operator in the directions of a vector field with a quasi-periodically varying velocity, in which the frequencies satisfy the condition of strong incommensurability. The system is quasi–linear, has a pair of conjugate pure imaginary eigenvalues, admits the first integral, and is real analytical in time variables in the neighbourhood of the real axis of the complex plane, and space variables on the real plane. It can be called the Lyapunov system. Each equation of a vector field is represented by an independent quasi-periodic scalar function. According to one study by J. Moser the integrals of such equations, under the condition of strong incommensurability, are described by the sum of linear and quasi-periodic functions. To investigate the existence of oscillatory solutions in the system under consideration, Lyapunov's method is extended to this system, considering the result of J. Moser. The differentiation operator with a quasi-periodic property is considered for the first time, and periodic or constant coefficients are often considered in connection with various issues of the theory of multi-frequency oscillations. Thus, the result of the conducted research is an analogue of the result on Lyapunov's periodic systems in the case of quasi-periodic systems with a special differentiation operator.

Numerically stable iterative methods for computing matrix sign function

The main objective of this study is to develop two novel iterative schemes for computing the sign of a matrix with no pure imaginary eigenvalues. Detailed convergence analysis and asymptotical stability of the proposed methods are discussed. It is shown that the new schemes converge globally by drawing attraction basins. In addition, we extend the obtained results to compute nontrivial solutions of the Yang–Baxter‐like equation, provided the given matrix has no eigenvalues on the imaginary axis. To illustrate the effectiveness of theoretical developments, a class of numerical examples for matrices of various dimensions, including eigenvalue clustering, are worked out.

Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system∂tu=JLu\partial _{t}u=JLuin a Hilbert spaceXX. We assume that L:X→X∗\ L:X\rightarrow X^{\ast }induces a bounded and symmetric bi-linear form⟨L⋅,⋅⟩\left \langle L\cdot ,\cdot \right \rangleonXX, which has only finitely many negative dimensionsn−(L)n^{-}(L). There is no restriction on the anti-self-dual operatorJ:X∗⊃D(J)→XJ:X^{\ast }\supset D(J)\rightarrow X. We first obtain a structural decomposition ofXXinto the direct sum of several closed subspaces so thatLLis blockwise diagonalized andJLJLis of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy ofetJLe^{tJL}. In particular,etJLe^{tJL}has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relaten−(L)n^{-}\left ( L\right )and the dimensions of generalized eigenspaces of eigenvalues of JL\ JL, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostlyJJwas assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Stability switching and its directions in cournot duopoly game with three delays

<p style='text-indent:20px;'>A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.</p>

Localized photonic states and dynamic process in nonreciprocal coupled Su-Schrieffer-Heeger chain.

We investigate the localized photonic states and dynamic process in one-dimensional nonreciprocal coupled Su-Schrieffer-Heeger chain. Through numerical calculation of energy eigenvalue spectrum and state distributions of the system, we find that different localized photonic states with special energy eigenvalues can be induced by the nonreciprocal coupling, such as zero-energy edge states, interface states and bound states with pure imaginary energy eigenvalues. Moreover, we analyze the dynamic process of photonic states in such non-Hermitian system. Interestingly, it is shown that the nonreciprocal coupling has an evident gathering effect on the photons, which also break the trapping effect of topologically protected edge states. In addition, we consider the impacts of on-site defect potentials on the dynamic process of photonic states for the system. It is indicated that the photons go around the defect lattice site and still present the gathering effect, and different forms of laser pulses can be induced with the on-site defect potentials in different lattice sites. Furthermore, we present the method for the quantum simulation of current model based on the circuit quantum electrodynamic lattice.

NLS and KdV Hamiltonian linearized operators: A priori bounds on the spectrum and optimal [formula omitted] estimates for the semigroups

Motivated by the NLS and KdV linearizations near traveling waves, we study general forms of such operators. We prove a priori bounds on the unstable spectrum, by showing that if any unstable spectrum exists, it is contained in a strip around the real axis, with an explicit estimate of its width in terms of the potentials. To the best of our knowledge, this is the first result of this nature in the literature. We show that all sufficiently large (relative to the potential) pure imaginary eigenvalues are necessarily simple. In the case of spectral stability, we show optimal, at most polynomial in time, L2 bounds for the associated semigroups generated such linearized operators. As it is for finite matrices, the power rate matches the maximal size of any Jordan block minus one.

Special modes induced by inter-chain coupling in a non-Hermitian ladder system

We explore some interesting phenomena in a simple non-Hermitian ladder system. Special modes with energy eigenvalues closely related to the inter-chain-coupling strength appear in the non-Hermitian ladder system. We show that a phase transition occurs whereby special modes with pure real eigenvalues can switch to special modes with pure imaginary eigenvalues, when the inter-chain-coupling strength changes from symmetric to asymmetric. We find that the density profiles of all the special modes are completely identical under certain conditions, even if the inter-chain-coupling strength is added into the non-Hermitian ladder system in different ways. Moreover, we also demonstrate that the different inter-chain couplings are fundamentally equivalent to adding different on-site potential energies into the non-Hermitian ladder system.

Coherent structures for breather–soliton molecules and breather molecules of the modified KdV equation

The real modified Korteweg–de Vries equation governs the modulation of weakly nonlinear waves. We first review the multiple soliton solutions to the mKdV equation by means of the inverse scattering method in detail. It is found the soliton solutions are related to pure imaginary discrete eigenvalues, while the breathers are derived from complex eigenvalues. A novel expression for the mulitple soliton solution is presented which is used to construct the soliton and breather solutions. By introducing resonance condition for solitons and breathers, some resonant structures for breathers and solitons, or soliton bound states are first constructed for the real mKdV equation, such as breather molecules, breather–soliton molecules. Our work demonstrates the interactions among breather molecules and breather–soliton molecules are nonelastic by the meaning the breathers and solitons change their sizes.

Inverse Problem for a Stieltjes String Damped at an Interior Point

In this paper, we consider the spectral problem for a Stieltjes string with both ends fixed and with one-dimensional damping at an interior point. We solve an inverse problem of recovering parameters of the Stieltjes string using partial information on the string and partial information on the spectrum. First we prove uniqueness and give an algorithm of recovering the unknown parameters of the string by virtue of a finite number of distinct not pure imaginary complex eigenvalues and a certain even number of real eigenvalues. For the case where only complex eigenvalues are used we solve the existence problem, i.e. propose conditions on a set of complex numbers necessary and sufficient for these numbers to be a subset of the spectrum of the problem.

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.

Stability analysis and numerical simulation of a composite laminated piezoelectric rectangular plate system

This paper investigates the stability behavior in three-degree-freedom composite laminated piezoelectric rectangular plate system based on the Routh-Hurwitz criterion and the normal form theory. Firstly, the stability behavior of the composite laminated piezoelectric rectangular plate is determined by applying the Routh-Hurwitz criterion. Then, the normal forms of the composite laminated piezoelectric rectangular plate systems for the case of one non-semi-simple double zero and two pairs of pure imaginary eigenvalues are obtained by using the normal form theory. Finally, numerical simulations are carried out to illustrate the main results.

Anomalous localization enhancement in one-dimensional non-Hermitian disordered lattices

We study numerically the localization properties of eigenstates in a one-dimensional random lattice described by a non-Hermitian disordered Hamiltonian, where both the disorder and the non-Hermiticity are inserted simultaneously in the on-site potential. We calculate the averaged participation number, the Shannon entropy and the structural entropy as a function of other parameters. We show that, in the presence of an imaginary random potential, all eigenstates are localized in the thermodynamic limit and strong anomalous Anderson localization occurs at the band center. In contrast to the usual localization anomalies where a weaker localization is observed, the localization of the eigenstates near the band center is strongly enhanced in the present non-Hermitian model. This phenomenon is associated with the occurrence of a large number of strongly-localized states with pure imaginary energy eigenvalues.

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous n -dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold \mathcal{Z}\subset\mathbb{R}^n of periodic solutions satisfying \dim(\mathcal{Z}) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x'=Mx , when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system: x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2 F_2^n(x), in \mathbb{R}^{d+2} , where \varepsilon is a small parameter, M is a (d+2)\times(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d-m non-zero real eigenvalues.

Bifurcation analysis for a functionally graded materials plate: two pairs of pure imaginary eigenvalues

In this paper, the dynamics in the vicinity of a compound critical point which is characterized by two pairs of pure imaginary eigenvalues are studied for a simply supported functionally graded materials rectangular plate. The solutions for equilibria and quasi-periodic motions are obtained and stability conditions of these solutions are presented. The numerical simulations confirm the analytical predictions.

Hopf Bifurcations in a Predator–Prey Model with an Omnivore

We study limit cycle bifurcations in a three-dimensional Lotka–Volterra system which models a food web of three species, one of which is an omnivore. First, using a new approach based on the elimination theory of the computational algebra we find necessary and sufficient conditions for existence of a pair of pure imaginary eigenvalues for the Jacobian of the system at the stationary point with positive coordinates. Then it is shown that the system can have two small limit cycles bifurcating from the singular point.

A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.

A fourth-order method for computing the sign function of a matrix with application in the Yang–Baxter-like matrix equation

In this study, first we derive a novel iteration scheme for the sign of a matrix with no pure imaginary eigenvalues. The fourth-order convergence speed of this scheme is given in detail. Secondly, we extend the obtained results so as to calculate the solution of the Yang–Baxter-like equation for the matrix A with no pure imaginary eigenvalues. Some numerical tests are also furnished to manifest the applicability of our method.