Imaginary Eigenvalues Research Articles

Topology and PT symmetry in a non-Hermitian Su–Schrieffer–Heeger chain with periodic hopping modulation

We study the effect of periodic hopping modulation in a Su-Schrieffer-Heeger (SSH) chain with an additional onsite staggered imaginary potential (of strengthγ). Such dissipative, non-Hermitian (NH) extension amply modifies the features of the topological trivial phase (TTP) and the topological nontrivial phase (TNP) of the SSH chain, more so with the periodic hopping distribution. Generally a weak NH potential can respect the parity-time (PT) symmetry keeping the energy eigenvalues real, while a strong potential breaksPTconservation leading to imaginary edge state and complex bulk state energies in the system. We find that the non-zero energy in-gap states, that appear due to periodic hopping modulations even in theγ = 0 limit, take purely real or purely imaginary eigenvalues depending on the strength of bothγand Δ (dimerization parameter). The localization of topological edge states (in-gap states) at the boundaries are investigated that reveals extended nature not only near topological transitions (further away from|Δ/t|=1) but also near the unmodulated limit ofΔ=0. Moreover, localization of the bulk states is observed at the maximally dimerized limit of|Δ/t|=1, which increases further withγ. These dissipative features can offer additional tunability in modulating the gain-loss contrast in optical systems or in designing various quantum information processing and storage devices.

Rapid computation of Hopf bifurcation points of continuous and discrete systems through minimization

For an autonomous nonlinear system, the Hopf bifurcation point along the equilibrium path is a critical feature that indicates whether the values of the parameters change from exhibiting fixed-point behavior to having a periodic orbit. To solve these problems, we developed a method of transforming an eigenvalue problem based on the Jacobian matrix at equilibrium into a minimization problem, enabling the rapid identification of a solution. Specifically, this generalized eigenvalue problem is solved by identifying the vector variable after reducing the number of eigenequations by one in the nonhomogeneous linear system. This can be achieved by normalizing the value of a selected nonzero component of the eigenvector and then moving the column containing this component to the other side of the equation. An appropriate merit function was established in terms of the Euclidean norm of the eigenequation, and this merit function was minimized using the golden section search algorithm to determine the eigenparameters of the bifurcation point. The accuracy of the method for identifying the parameter values and the corresponding imaginary eigenvalues at the Hopf bifurcation points was evaluated for numerous examples for both the continuous and discrete systems. The method was both fast and accurate. Moreover, its stability in the presence of noise was investigated, and the method was robust.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Stability analysis of a multilayer diffusion-reaction heat transfer problem with a very large number of layers

The stability of thermal conduction problems is of much practical interest in the context of Li-ion batteries, power electronic devices, chemical reactors and other engineering systems. While past work in predicting the thermal stability of a multilayer diffusion-reaction problem has been carried out using pole analysis or by determining imaginary eigenvalues of the problem, such techniques are impractical when the number of layers in the problem is very large. This work presents an exact analysis of the stability of a multilayer diffusion-reaction problem, in which the number of layers may be very large. Using homogenization and Heaviside functions based representation of thermophysical properties and parameters, it is shown that the stability of such a system may be easily predicted by computing the determinant of a matrix containing contributions from each layer/interface. Results from the present work are shown to correctly reduce to results from past work for pure-diffusion and single-layer special cases, as well as agree well with numerical simulations. Threshold for thermal stability of a representative Li-ion battery pack containing 1000 Li-ion cells is determined. It is shown that stability of the system is governed by a balance between temperature-dependent heat generation, diffusion and heat removal from the boundary. A key result of practical interest is that even small improvement in thermal diffusivity of Li-ion cells may greatly help reduce the risk of thermal instability. Additionally, practical guidelines for the maximum permissible operating time of a battery pack to avoid thermal runaway are discussed. This work advances the theoretical understanding of multilayer diffusion-reaction problems, and helps understand the thermal stability of Li-ion battery packs containing a very large number of cells, which is very difficult to analyze using traditional techniques.

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.

Constraining work fluctuations of non-Hermitian dynamics across the exceptional point of a superconducting qubit

Thermodynamics constrains changes to the energy of a system, both deliberate and random, via its first and second laws. When the system is not in equilibrium, fluctuation theorems such as the Jarzynski equality further restrict the distributions of deliberate work done. Such fluctuation theorems have been experimentally verified in small, nonequilibrium quantum systems undergoing unitary or decohering dynamics. Yet, their validity in systems governed by a non-Hermitian Hamiltonian has long been contentious due to the false premise of the Hamiltonian's dual and equivalent roles in dynamics and energetics. Here we show that work fluctuations in a non-Hermitian qubit obey the Jarzynski equality even if its Hamiltonian has complex or purely imaginary eigenvalues. With postselection on a dissipative superconducting circuit undergoing a cyclic parameter sweep, we experimentally quantify the work distribution using projective energy measurements and show that the fate of the Jarzynski equality is determined by the parity-time symmetry of, and the energetics that result from, the corresponding non-Hermitian, Floquet Hamiltonian. By distinguishing the energetics from non-Hermitian dynamics, our results provide the recipe for investigating the nonequilibrium quantum thermodynamics of such open systems. Published by the American Physical Society 2024

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.

Thermodynamic bound on spectral perturbations, with applications to oscillations and relaxation dynamics

In discrete-state Markovian systems, many important properties of correlation functions and relaxation dynamics depend on the spectrum of the rate matrix. Here we demonstrate the existence of a universal trade-off between thermodynamic and spectral properties. We show that the entropy production rate, the fundamental thermodynamic cost of a nonequilibrium steady state, bounds the difference between the eigenvalues of a nonequilibrium rate matrix and a reference equilibrium rate matrix. Using this result, we derive thermodynamic bounds on the spectral gap, which governs autocorrelation times and the speed of relaxation to a steady state. We also derive the thermodynamic bounds on the imaginary eigenvalues, which govern the speed of oscillations. We illustrate our approach using a simple model of biomolecular sensing. Published by the American Physical Society 2024

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.

Normal forms for retarded functional differential equations associated with zero-double-Hopf singularity with 1 : 1 resonance

This manuscript introduces a framework that focuses on the singularity of a zero-double-Hopf system with 1 : 1 resonance in general retarded differential equations (RDDs). Initially, practical algorithms are proposed to identify the zero-double-Hopf singularity and the associated generalized eigenspace that corresponds to zero and two pairs of purely imaginary eigenvalues. Subsequently, by utilizing center manifold reduction and normal form techniques, we derive a reduced form of parameterized retarded differential systems up to third-order terms.

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.

Management of the discrete eigenvalues of the nonlinear Schrödinger equation using periodic dispersion perturbation

The effect of the periodic oscillation of the coefficient at the second-order derivative of the nonlinear Schrödinger equation on the soliton content of propagating pulses is studied numerically. Resonant fission of multisoliton pulses and soliton fusion are described in terms of eigenvalues of the Zakharov–Shabat spectral problem. Resonant conditions are connected with internal periodicity of solitons and breathers. The resonance can be changed by variation of initial pulse parameters. When the propagation distance does not exceeds a few oscillation periods of the coefficient, a wide resonances can be obtained. With a large number of the oscillation periods, the output pulse shapes and corresponding set of the complex eigenvalues of the Zakharov–Shabat spectral problem become extremely sensitive to the initial pulse parameters. The change in the pulse propagation process is described as a bifurcation. Two types of bifurcations were are found. The first bifurcation type is connected with the join of two imaginary eigenvalues in the single point and then the split of these eigenvalues in real parts symmetrically, so their imaginary part becomes the same. The second one is described by the joining of two eigenvalues with symmetrical real parts in the single complex point and their subsequent splitting. After splitting, the eigenvalues have zero real parts and different imaginary parts. I have found that transformations of the initial pulses are described by pairwise interaction of the eigenvalues. The eigenvalue bifurcations under effect of the change of the initial pulse amplitude and separation between pulses are studied. A periodically modulated media can be used both to control soliton pulse shapes and eigenvalues of the Zakharov–Shabat spectral problem.

Infernal and exceptional edge modes: non-Hermitian topology beyond the skin effect

The classification of point gap topology in all local non-Hermitian (NH) symmetry classes has been recently established. However, many entries in the resulting periodic table have only been discussed in a formal setting and still lack a physical interpretation in terms of their bulk-boundary correspondence. Here, we derive the edge signatures of all two-dimensional phases with intrinsic point gap topology. While in one dimension point gap topology invariably leads to the NH skin effect, NH boundary physics is significantly richer in two dimensions. We find two broad classes of non-Hermitian edge states: (1) infernal points, where a skin effect occurs only at a single edge momentum, while all other edge momenta are devoid of edge states. Under semi-infinite boundary conditions, the point gap thereby closes completely, but only at a single edge momentum. (2) NH exceptional point dispersions, where edge states persist at all edge momenta and furnish an anomalous number of symmetry-protected exceptional points. Surprisingly, the latter class of systems allows for a finite, non-extensive number of edge states with a well defined dispersion along all generic edge terminations. Concomitantly, the point gap only closes along the real and imaginary eigenvalue axes, realizing a novel form of NH spectral flow.

(Invited) Multiscale Diffusion-Reaction Heat Transfer Analysis for Predicting Thermal Runaway in Li-Ion Cells and Battery Packs

Thermal runaway in Li-ion cells and battery packs remains a topic of much importance for ensuring the safety of electrochemical energy conversion and storage systems. Thermal runaway also indirectly affects performance, as conservative overdesign to reduce the risk of thermal runaway affects system-level performance metrics such as energy storage density. Given the considerable challenges in experimental investigation of thermal runaway, theoretical models to predict thermal runaway characteristics of a cell or battery pack are critically needed. Such models can help guide the design of experiments, and ensure that practical systems are reasonably safe without being needlessly overdesigned.This talk will summarize ongoing work on theoretical analysis of multilayer diffusion-reaction heat transfer problems for predicting thermal runaway in Li-ion cells and battery packs. Following a linearization of heat generation terms, the governing energy conservation equation is solved in a multilayer geometry representative of a Li-ion cell comprising multiple electrode and separator layers, or of a battery pack containing several cells. It is mathematically shown that under certain conditions, this problem may admit a finite number of imaginary eigenvalues, which are shown to cause thermal runaway. It is shown that the occurrence of imaginary eigenvalues is governed by a delicate balance between heat generation, thermal conduction within the multilayer body and heat removal from the boundaries, based on which, an explicit threshold condition for thermal runaway is derived. This threshold condition is interpreted in the context of parameter values relevant to Li-ion cells. For example, it is shown that boundary cooling is severely limited in its role to prevent thermal runaway, whereas improving thermal transport within the cell is likely to be a lot more effective. This theoretical model is also extended to a two-layer problem comprising a finite layer adjoined by a semi-infinite body that may model immersion cooling of a Li-ion cell or battery pack. It is shown that while this diffusion-reaction problem is, in principle, unconditionally unstable, the finite time period of any discharge process may allow the temperature field to still remain within a safe envelope, if designed appropriately. Based on this mathematical model, scenarios related to immersion cooling of pouch cells are investigated.The theoretical work described in this talk is broadly relevant to several diffusion-reaction systems. In the particular context of Li-ion cells, these results may form the foundation of techniques to proactively predict and prevent thermal runaway in Li-ion cells and battery packs.

Benjamin–Feir Instability of Stokes Waves in Finite Depth

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth {mathtt h} is larger than a critical threshold texttt{h}_{scriptscriptstyle {textsc {WB}}}approx 1.363 . In this paper, we completely describe, for any finite value of mathtt h >0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth texttt{h}_{scriptscriptstyle {textsc {WB}}}, which coincides with the one predicted by Whitham and Benjamin, such that, for any 0< mathtt h < texttt{h}_{scriptscriptstyle {textsc {WB}}}, the eigenvalues close to zero are purely imaginary and, for any mathtt h > texttt{h}_{scriptscriptstyle {textsc {WB}}}, a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As {mathtt h} rightarrow texttt{h}_{scriptscriptstyle {textsc {WB}}}^{, +} the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits texttt{h}rightarrow +infty (deep water) and mu rightarrow 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size mathcal {O}(mu ), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth texttt{h} is carefully tracked.

Decentralized False-Data Injection Attacks Against State Omniscience: Existence and Security Analysis

This paper focuses on how false-data injection (FDI) attacks compromise state omniscience, which needs each node in a jointly detectable sensor network to estimate the entire plant state through distributed observers. To reveal the global vulnerability of state omniscience, we investigate decentralized false-data injection (DFDI) attacks that destabilize the estimation error dynamics but eliminate their influences on the residual in each sensor node. Firstly, the sufficiency and necessity for the existence of such attacks are studied from system eigenvalues and attackable sensors. Secondly, the self-generated DFDI attack sequences independent of system real-time data are designed to achieve the attack objective with elaborate parameters. Especially, the DFDI attack sequences are improved to maintain real values even if the system matrix only has unstable imaginary eigenvalues. Finally, we analyze the secure range for observer interaction weights and the sensor protection scheme to guarantee the security of state omniscience under DFDI attacks. The theoretical results for DFDI attacks are demonstrated with the linearized discrete-time model of an aircraft system.

Nonintegrability of truncated Poincaré-Dulac normal forms of resonance degree two

We give sufficient conditions for three- or four-dimensional truncated Poincaré-Dulac normal forms of resonance degree two to be meromorphically nonintegrable when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of simple planar systems, and use an approach for proving the meromorphic nonintegrability of planar systems, which is similar to but more sophisticated than the previously developed one. Our result also implies that general three- or four-dimensional systems are analytically nonintegrable if they are formally transformed into one of the truncated normal forms satisfying the sufficient conditions.

Direct reduction approach and soliton solutions for the integrable space–time shifted nonlocal Sasa-Satsuma equation

This paper investigates the space–time shifted nonlocal Sasa-Satsuma equation which is constructed by imposing a space–time shifted symmetry constraint on the two-component Sasa-Satsuma system. By applying the direct reduction approach, specific parameter values are determined to obtain N-soliton solutions of the shifted nonlocal equation. The resulting one-soliton and two-soliton solutions are analyzed both theoretically and graphically. Additionally, the asymptotic behavior of two-soliton solution with two purely imaginary eigenvalues is studied. The findings shed light on the mathematical properties and dynamic behaviors of shifted nonlocal integrable systems.

Periodic and torus motions of a two-degree-of-freedom dry friction vibration system

Vibration induced by dry friction is ubiquitous in various engineering fields. To explore the vibration characteristics for further studies and/or controls, it is of great theoretical and practical significances to investigate the non-linear dynamic behaviors of the friction systems. This study considers the slight vibration of a two-degree-of-freedom non-linear dry friction excitation system. The differential equations of system motion are established according to Newton’s law of motion. Moreover, the system’s non-linear dynamic is studied when the block velocity is always less than the friction surface velocity. The results indicate that the linearized matrix of the vibration system has a pair of purely imaginary eigenvalues for some critical values of the relevant parameters. The Poincaré-Birkhoff normal forms are utilized to simplify the motion equation under the non-resonant assumption to obtain a simplified equation with only the resonant terms. Furthermore, the truncated part of the simplified equation is analyzed in the case of only linear terms degeneration. Finally, numerical simulations reflect some qualitative conclusions about the system’s local dynamic properties, including equilibrium point, periodic motion, torus motion, and their stability.

Kuznetsov–Ma rogue wave clusters of the nonlinear Schrödinger equation

In this work, we investigate rogue wave (RW) clusters of different shapes, composed of Kuznetsov–Ma solitons (KMSs) from the nonlinear Schrödinger equation (NLSE) with Kerr nonlinearity. We present three classes of exact higher-order solutions on uniform background that are calculated using the Darboux transformation (DT) scheme with precisely chosen parameters. The first solution class is characterized by strong intensity narrow peaks that are periodic along the evolution x-axis, when the eigenvalues in DT scheme generate KMSs with commensurate frequencies. The second solution class exhibits a form of elliptical rogue wave clusters; it is derived from the first solution class when the first m evolution shifts in the nth-order DT scheme are nonzero and equal. We show that the high-intensity peaks built on KMSs of order n-2m periodically appear along the x-axis. This structure, considered as the central rogue wave, is enclosed by m ellipses consisting of a certain number of the first-order KMSs determined by the ellipse index and the solution order. The third class of KMS clusters is obtained when purely imaginary DT eigenvalues tend to some preset offset value higher than one, while keeping the x-shifts unchanged. We show that the central rogue wave at (0, 0) always retains its n-2m order. The n tails composed of the first-order KMSs are formed above and below the central maximum. When n is even, more complicated patterns are generated, with m and m-1 loops above and below the central RW, respectively. Finally, we compute an additional solution class on a wavy background, defined by the Jacobi elliptic dnoidal function, which displays specific intensity patterns that are consistent with the background wavy perturbation. This work demonstrates an incredible power of the DT scheme in creating new solutions of the NLSE and a tremendous richness in form and function of those solutions.