Imaginary Pair Research Articles

Amplitude equations for wave bifurcations in reaction–diffusion systems

A wave bifurcation is the counterpart to a Turing instability in reaction–diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary n-component systems ( n⩾3 ) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction–diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

General multi-soliton and higher-order soliton solutions for a novel nonlocal Lakshmanan–Porsezian–Daniel equation

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, a novel integrable nonlocal LPD equation corresponding to a \(3\times 3\) Lax pair is proposed. Secondly, the inverse scattering process with a novel left-right \(3\times 3\) matrix Riemann–Hilbert(RH) problem is constructed. The analytical properties and symmetry relations for the Jost functions and scattering data are considerably different from the local ones. Due to the special symmetry properties for the nonlocal LPD equation, the zeros of the RHP problem are purely imaginary or occur in pairs. With different types and configuration of zeros, the soliton formula is provided and the rich dynamical behaviors for the three kinds of multi-solitons for the novel nonlocal LPD equation are demonstrated. Third, by a technique of adding perturbed parameters and limiting process, the formula of higher-order solitons for the nonlocal LPD equation is exhibited. Lastly, the plots of diverse higher-order solitons and various solutions corresponding to different combinations of the following zeros: purely imaginary higher-order zeros, purely imaginary simple zeros, pairs of non-purely imaginary simple zeros and pairs of non-purely imaginary higher-order zeros are displayed.

A Nightmare

A Nightmare Talia Neffson (bio) She knew what she'd told Lizzie was true. Head injuries do produce an alarming and seemingly unaccountable amount of blood. Mira had never seen so much blood in her life. When she was done crying, she washed her face in the downstairs half-bath and dried it on a burgundy hand towel with a big S monogrammed on it. She tried to remember if Lizzie had changed her last name. It seemed strange and embarrassing that she couldn't remember. Then she remembered. The living room was strewn with toys, board books, little and big shoes, and fossilized Cheerios, just as it had been every time Mira visited, which wasn't often. The only time she could remember its being visibly tidied was for Little Man's first birthday party. She was surprised that Lizzie hadn't tidied this time, with the other couples coming over, though she wasn't sure if she was surprised in a good way or a bad way. Mira couldn't remember where she'd learned about head injuries bleeding heavily but it made sense—so many blood vessels, and of course from an evolutionary standpoint the priority had to be protecting the all-important brain. She had asked the EMT guy if wounds like this always bleed a lot and he'd said, "Sometimes." Otherwise he'd spoken only to Ben, who had been seated on the floor in front of the sink, holding the wadded-up beach towel against his head. Ben's usual serene manner had seemed transformed by the circumstances into a stunned silence that might've been calm or shock. He'd barely said a word. Mira hadn't seen the exact moment his complexion turned to a brown-tinged, pallid gray from its usual warm, light bronze ("like Lenny Kravitz," said Lizzie, when they'd first started [End Page 489] dating). But it was a noticeable change and almost as unnerving as the blood. There was a name for that color. Taupe? Ecru? It made Mira think of the monochrome del Sarto paintings Marco had tried to convince her to like. "Well, you know I am just a dumb American. I'm probably not sophisticated enough to appreciate it," she'd said. "But you are not dumb," Marco had responded seriously, continuing to examine the painting. "You are a very intelligent woman." Humor, especially sarcasm, had always been a problem with Marco. At first Mira had assumed it was a language problem. Later she realized it was not just a language problem. Three years and eight months. Lizzie refused to refer to him as anything other than "that fucking Italian." They'd played the name game after dinner while the two couples with the husbands Ben knew from work were still there. Lizzie and Mira's team went first. Lizzie put down the first scrap of paper and, without pausing, pulled her cardigan off the back of her chair, draped it over her shoulders and tied the sleeves in a loose knot at her chest. Then she leaned forward and lifted an imaginary pair of glasses to her forehead, as if scrutinizing something up close. "The chiaroscuro!" she said with a rolling, extravagant accent, "Molto bellissimo! You see?" "That fucking Italian," Mira answered flatly. The other couples laughed the resentful, strained laughter of people witnessing an inside joke from the outside. Lizzie's motion for "Polo" looked more like vacuuming than swinging a mallet from horseback, but Mira still got it. Lizzie looked at the next paper and, again without pausing, said in the same exaggerated Canarsie accent she often used to imitate their mothers, "Imagine you're a deer …" "Marisa Tomei," said Mira. They went on like that until Lizzie picked up a paper that made her pause. She put bunny ears behind her head. "Hugh Hefner?" guessed Mira. "Mel Blanc?" The other couples laughed unrestrainedly. Lizzie huffed, then their time was up. They won anyway. They always won when they paired up for the name game. [End Page 490] Ben had offered to sit the game out so they wouldn't have an uneven number. Then he'd gone upstairs to...

TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations.

TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chemical simulations of molecules, clusters, periodic systems, and solutions. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy–cost ratio, such as density functional theory including local hybrids and the random phase approximation (RPA), GW-Bethe–Salpeter methods, second-order Møller–Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resolution-of-the-identity approximation, imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent additions to TURBOMOLE’s functionality, including excited-state methods, RPA and Green’s function methods, relativistic approaches, high-order molecular properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE’s current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE’s development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted.

New R-matrices with non-additive spectral parameters and integrable models of strongly correlated fermions

We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two one-parameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra Uq(gl(2|1)), we obtain the explicit expression for the Uq(gl(2|1))− invariant R-matrix which is of non-difference form in spectral parameters. Using this R­ matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms.

New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system

This note revisits a 4-D quadratic autonomous hyper-chaotic system in Zarei and Tavakoli (2016) and mainly considers some of its rich dynamics not yet investigated: global boundedness, invariant algebraic surface, singularly degenerate heteroclinic cycle and limit cycle. The main contributions of the work are summarized as follows: Firstly, we prove that for 4a ≥ c > 2a > 0, d > 0 and e > 0 the solutions of that system are globally bounded by constructing a suitable Lyapunov function. Secondly, Q=x3−12ax12=0 is found to be one of invariant algebraic surfaces with the cofactor -4a for the model. Thirdly, numerical simulations for c=0 not only illustrate different types of infinitely many singularly degenerate heteroclinic cycles near which chaotic attractors or limit cycles generate, but also that some of more degenerate (in term of a pure imaginary pair, one zero and one negative eigenvalue) or stable (in sense of three negative eigenvalues and one null eigenvalue) non-isolated equilibria (0,0,x3,0)(x3∈R) directly change into the limit cycles or chaotic attractors with a small perturbation of c > 0, which is in the absence of singularly degenerate heteroclinic cycles and degenerate pitchfork bifurcation at the non-isolated equilibria. In particular, some kind of forming mechanism of Lorenz attractor and the hyper-chaotic attractor of that system with (a,b,c,d,e)=(10,28,83,1,16) is revealed, which are collapses of singularly degenerate heteroclinic cycles and explosions of stable non-isolated equilibria. Finally, circuit experiment implements the aforementioned hyper-chaotic attractor, showing very good agreement with the simulation results.

Discrete Eigenvalues of Case Spectrum with Anisotropic Scattering

Discrete eigenvalues of a one-speed linear transport equation with anisotropic scattering are studied. It is shown that there is only one pair of real discrete eigenvalues for linear, quadratic, or triplet scattering for a nonmultiplicative medium. For a multiplicative medium there is one imaginary pair of eigenvalues or at most four eigenvalues. These can form one real and one imaginary pair, two imaginary pairs, or a quartet. The range of parameters for these different situations is derived analytically. These are then supported by numerical results that are tabulated in tables for each type of scattering.

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.

Cumulative Angular Distortion Curve of Multi-Pass Welding at Thick Plate of Offshore Structures

In the fabrication of offshore oil and gas facilities, the significance of dimension control is growing continuously. But, it is difficult to determine the deformation of the structure during fabrication by simple lab tests due to the large size and the complicated shape. Strain-boundary method (a kind of shrinkage method) based on the shell element was proposed to predict the welding distortion of a structure effectively. Modeling of weld geometry in shell element is still a difficult task. In this paper, a concept of imaginary temperature pair is introduced to handle the effect of geometric factors such as groove shape, plate thickness and pass number, etc. Single pass imaginary temperature pair formula is derived from the relation between the groove area and the FE mesh size. By considering the contribution of each weld layer to the whole weldment, multi-pass imaginary temperature is also derived. Since the temperature difference represents the distortion increment, cumulative distortion curve can be drawn by integrating the temperature difference. This curve will be a useful solution when engineers meet some problems occurred in the shipyard. A typical example is shown about utilization of this curve. Several verifications are conducted to examine the validity of the proposed methodology. The applicability of the model is also demonstrated by applying it to the fabrication process of the heavy ship block. It is expected that the imaginary temperature model can effectively solve the modeling problem in shell element. It is also expected that the cumulative distortion curve derived from the imaginary temperature can offer useful qualitative information about angular distortion without FE analysis.

Can the photon velocity be derived from the Klein–Gordon equation?

In their most recent article, Grado-Caffaro et al. have addressed the question of the ‘photon velocity’. They have expressed the photon velocity in terms of the wavefunctions of the Klein–Gordon equation (Grado-Caffaro and Grado-Caffaro [4]). In this note, we closely follow their work and explicitly obtain the photon velocity using the free solutions of the Klein–Gordon equation. It is shown that the plane wave solutions give rise to six possible values of the photon velocity. Two of these solutions are the most expected ( v = ± c ). The remaining four solutions, the real pair ±0.786 c and the imaginary pair ±1.272i c are difficult to comprehend.

A formulation of the Cauchy method suitable for the synthesis of lossless circuit models of microwave filters from lossy measurements

In this letter, a novel formulation of the Cauchy method is presented, to be employed in the extraction of circuit models from measured(lossy) response of microwave filters. In particular, it is shown how to generate,from lossy measurements, the characteristic polynomials which allow synthesis of a lossless low-pass prototype (and the evaluation of the coupling matrix). To this purpose, suitable constraints are introduced in order to have the transmission zeros generated with the Cauchy method strictly compatible with the low pass synthesis requirements (pure imaginary or complex pair with opposite real part). The application of the proposed method has been verified employing measurements from a test filter for GSM base stations. This method can find application in computer-aided tuning procedures.

The simplest normal form and its application to bifurcation control

This paper is concerned with the computation of the simplest normal forms with perturbation parameters, associated with codimension-one singularities, and applications to control systems. First, an efficient method is presented to compute the normal forms for general semi-simple cases, which combines center manifold theory and normal form theory in one unified procedure. The efficient approach is then applied to find the explicit simplest normal forms of general n-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain either single zero or a purely imaginary pair. In addition to near-identity nonlinear transformation, time and parameter rescalings are used to obtain the simplest normal forms. It is shown that, unlike the classical normal forms, the simplest normal forms for single zero and Hopf singularities are finite up to an arbitrary order, which greatly simplify stability and bifurcation analysis. The new method is applied to consider controlling bifurcations of the Lorenz system and a nonlinear electrical circuit. Symbolic programs have been developed using Maple, which greatly facilitates applications.

On the Stability Degree

Let A be a nondegenerate operator: |A| 6 = 0 (This is equivalent to the condition |B| 6 = 0.) Then the eigenvalues of A can be of three types: real pairs ±a, pure imaginary pairs ±ib, and quadruples ±a± ib. It was shown in [1] that the linear system (2) with nondegenerate operator A that admits an integral in the form of a nondegenerate quadratic form (1) can always be reduced to a system of Hamiltonian equations. Therefore, the theory developed below applies to this (formally, more general) case. The instability degree u of system (2) is defined as the number of roots (counted according to their multiplicities) of the characteristic equation of A in the right half-plane, and the stability degree s is defined as the number of pure imaginary roots (counted according to their multiplicities) of the characteristic equation of A. By i and i− we denote the positive and negative indices of inertia of the quadratic form (1). By the Thomson theorem [2], u ≡ i−mod 2. (3) In particular, if i− (or i) is odd, then so is u. Consequently, the characteristic polynomial of A necessarily has a positive real root in this case; therefore, x = 0 is an unstable equilibrium. More precisely, the classical Thomson theorem holds for linear mechanical systems subjected to additional gyroscopic and dissipative forces. The congruence (3) was proved in [3] for systems of general form (with Ḣ ≤ 0). If Ḣ = 0, then the Thomson theorem is equivalent to the following assertion on the stability degree: s is even if and only if

Numerical solution of a quadratic eigenvalue problem*1

We consider the quadratic eigenvalue problem (QEP) ( λ 2 M + λG + K ) x =0, where M = M T is positive definite, K = K T is negative definite, and G =− G T . The eigenvalues of the QEP occur in quadruplets (λ, λ ,−λ,− λ ) or in real or purely imaginary pairs ( λ ,− λ ). We show that all eigenvalues of the QEP can be found efficiently and with the correct symmetry, by finding a proper solvent X of the matrix equation MX 2 + GX + K =0, as long as the QEP has no eigenvalues on the imaginary axis. This solvent approach works well also for some cases where the QEP has eigenvalues on the imaginary axis.

Numerical solution of a quadratic eigenvalue problem

We consider the quadratic eigenvalue problem (QEP) ( λ 2 M+ λG+ K) x=0, where M= M T is positive definite, K= K T is negative definite, and G=− G T. The eigenvalues of the QEP occur in quadruplets (λ, λ,−λ,− λ) or in real or purely imaginary pairs ( λ,− λ). We show that all eigenvalues of the QEP can be found efficiently and with the correct symmetry, by finding a proper solvent X of the matrix equation MX 2+ GX+ K=0, as long as the QEP has no eigenvalues on the imaginary axis. This solvent approach works well also for some cases where the QEP has eigenvalues on the imaginary axis.

One- and two-dimensional solitons in second-harmonic-generating lattices.

In a model of a dynamical lattice with the on-site second-harmonic-generating nonlinearity and harmonic intersite couplings (that may be equal or different for the fundamental and second harmonics), various solitary-wave solutions are considered in one and two dimensions (1D and 2D). Fundamental (single-hump) solitons are identified in either dimension and their stability is examined and compared to previous results as well as to what is known for the model's continuum counterpart. Stability limits in terms of the coupling constants, which depend on the value of the phase-mismatch parameter, are found for solitons of the twisted-mode type in the 1D lattice, and for their counterparts of two different types (one being a discrete vortex) in the 2D lattice. When the twisted-mode soliton is unstable, the instability, which may be either oscillatory or due to imaginary eigenfrequency pairs, transforms the unstable soliton into a stable fundamental one, in both 1D and 2D cases.

On Lyapunov stability and normal forms of nonlinear systems with a nonsemisimple critical mode. II. Imaginary eigenvalues pair

For pt. I see ibid., vol. 47, no. 6 (June 2000). This paper studies nonlinear systems associated with or linked to a nonsemisimple (NSS) pair of imaginary eigenvalues. Stability criteria and Poincare normal forms (PNF's) are derived via explicit construction of Lyapunov functions. It is a counterpart and continuation of Part I, which considered the case with an NSS zero eigenvalue. While originating from stability analysis for a nonsemisimple critical mode, the development extends to coexisting (semi-) simple critical modes and to stable modes, as in Part I. The present NSS imaginary pairs (NSSIP) case is shown to possess certain characteristics common to NSS zero (NSSZ), such as the nonlinear lag and the staircase structure, but retains its own identities as well. These include the less surprising, the submodular variable-pair circles which emerge from the units of the potential generator and the more remarkable stricter specification in the dynamics, which is possibly attributable to the inherited conjugacy. An elementary partial differential equation traversing the development shows potential for nontrivial generalization.

On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an “unstable disk theorem” through a simple energy–Casimir argument [Holm et al., Phys. Rep. 123, 1–116 (1985)]. The eigenvalues of the linear Hamiltonian system are of four types: real pairs (c,−c), purely imaginary pairs (id,−id), quadruples (±c±id), and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincaré-type difference equation, i.e., it can be represented as the spectral equation of an infinite matrix operator, and the infinite matrix operator is a sum of a constant-coefficient infinite matrix operator and a compact infinite matrix operator. We have obtained a complete spectral theory.

Computation of the coefficients associated with the normal form of a resonant five-dimensional system

In this paper, normal forms and the related coefficients associated with a five-dimensional resonant nonlinear system are investigated. It is assumed that the linearized system has two pure imaginary pairs and a zero eigenvalue. A modified normal form method is applied to a resonant model 1 : 2; however, the ideas presented here can be applied to other resonant models, as well. A symbolic computer language, MAPLE, has been used to obtain most of the results.

Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation

This paper considers an unfolding of a degenerate reversible 1–1 resonance (or Hamiltonian–Hopf) bifurcation for four-dimensional systems of time reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient ( q 2=0) that determines whether the bifurcation is super- or subcritical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur. J. Mech. B/Fluids, 15 (1996) 367–393), given a sign of the coefficient of a higher-order term ( q 4<0). Here, their analysis is generalised to include the other sign of q 4, motivated by a fourth-order ODE whose solutions model localised buckling of struts and steady states of the generalised Swift–Hohenberg equation. Numerical experiments are undertaken to determine the global behaviour of homoclinic orbits to the origin in the example which is both reversible and Hamiltonian. The normal form coefficients are calculated explicitly and a region of parameter space found where q 4>0 and −1≪ q 2<0. The normal form then shows a small-amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit. A genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies. This is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies. At each limit point the homoclinic orbit generates another ‘bump’ close to the periodic orbit. As q 2 is further decreased from zero, the snake widens until, at a critical value, a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium (a ‘kink’). For q 2 less than this value the kinks, heteroclinic tangencies and snake-like curves no longer occur.