Complex Conjugate Eigenvalues Research Articles

Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

We investigate a generalized Hopf bifurcation emerged from a corner located at the origin which is the intersection of n discontinuity boundaries in planar piecewise smooth dynamical systems with the Jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues. We obtain a novel result that the generalized Hopf bifurcation can occur even when the Jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues. According to the eigenvalues of the Jacobian matrices and the number of smooth subsystems, we provide a general method and prove some generalized Hopf bifurcation theorems by studying the associated Poincaré map.

A Novel Arnoldi Method to Calculate the Critical Eigenvalues in Power System

Hopf bifurcation is regarded as one of the three kinds of bifurcations which can cause voltage instability. The Hopf bifurcation was obtained if a pair of complex conjugate eigenvalues of Jacobian matrix crossing the imaginary axis. To calculate the critical eigenvalues of bulk power system, a restarted-refined Arnoldi method is introduced. The refined Ritz vectors are used as the approximations based on the refined projection theory in order to enrich the information of the eigenvectors in the projective subspace. Two examples indicate that the method can work out a set of conjugate eigenvalues which will show whether the Hopf bifurcation appears and provide the datum for the dynamic voltage stability analysis ulteriorly.

Dynamic properties of structures with dampers modelled using fractional order derivatives

The focus of this paper is on determination of the dynamic parameters of structural systems with viscoelastic (VE) dampers described by Maxwell rheological models. Such parameters could be obtained after solving the appropriately defined nonlinear eigenvalue problem for frames with VE dampers. The solution to the nonlinear eigenvalue problem is obtained by equating to zero the determinant of the considered system of equations. Apart from complex conjugate eigenvalues, the real ones occurred when dampers that are described by the classic Maxwell model, are also determined.

Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type

In this paper, we deal with the classical question of the existence of polynomials in momenta integrals for geodesic flows on the 2-torus. For the quasilinear system on the coefficients of the polynomial integral, we investigate the region (so-called elliptic region) where two of the eigenvalues are complex conjugate. We show that for quartic integrals the other two eigenvalues are real and necessarily genuinely nonlinear. This observation, together with the property of the system to be rich (semi-Hamiltonian), enables us to classify elliptic regions completely. We prove that on these regions the integral is always reducible. The case of complex-conjugate eigenvalues for the system corresponding to the integral of degree 3 is done similarly. These results show that if new integrable examples exist, they can be found only within the region of hyperbolicity of the quasilinear system.

Generalized Hopf bifurcation in a class of planar switched systems

This article presents an analysis on a generalized Hopf bifurcation in a class of planar switched systems with the phase portraits of the subsystems being locally those of similarly oriented foci at the origin, where the appearance or disappearance of a periodic orbit is not due to the crossing of complex conjugate eigenvalues of the linearization of smooth subsystems through the imaginary axis, but due to the switching law between these smooth subsystems. The mechanism of the generalized Hopf bifurcation dealt with in this article is of significance from switched systems point of view.

The rank of a 2 × 2 × 2 tensor

As computing power increases, many more problems in engineering and data analysis involve computation with tensors, or multi-way data arrays. Most applications involve computing a decomposition of a tensor into a linear combination of rank-1 tensors. Ideally, the decomposition involves a minimal number of terms, i.e. computation of the rank of the tensor. Tensor rank is not a straight-forward extension of matrix rank. A constructive proof based on an eigenvalue criterion is provided that shows when a 2 × 2 × 2 tensor over ℝ is rank-3 and when it is rank-2. The results are extended to show that n × n × 2 tensors over ℝ have maximum possible rank n + k where k is the number of complex conjugate eigenvalue pairs of the matrices forming the two faces of the tensor cube.

Complex conjugate eigenvalues in the spectrum of an operator for resonant activation

We consider the exit problem for an overdamped Brownian particle in a potential undergoing dichotomic fluctuations. The system exhibits resonant activation. We compute the corresponding exit times distribution and show that the resonance is associated with the presence of a finite number of complex conjugate eigenvalue pairs in the spectrum of the evolution equation. The properties of these eigenvalues and their influence on the exit times distribution and on the possible dynamics of the system are discussed in detail.

On complex matrices with simple spectrum that are unitarily similar to real matrices

Suppose that one should verify whether a given complex n × n matrix can be converted into a real matrix by a unitary similarity transformation. Sufficient conditions for this property to hold were found in an earlier publication of this author. These conditions are relaxed in the following way: as before, the spectrum is required to be simple, but pairs of complex conjugate eigenvalues \( \lambda ,\bar \lambda \) are now allowed. However, the eigenvectors corresponding to such eigenvalues must not be orthogonal.

Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics

Two metrics g and ḡ are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the (1, 1)-tensor G := g gkj has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalise the Topalov-Sinjukov (hierarchy) Theorem for pseudo-Riemannian metrics. © 2011 American Mathematical Society.

Conjugacy classes in Möbius groups

Let $${\mathbb H^{n+1}}$$ denote the n + 1-dimensional (real) hyperbolic space. Let $${\mathbb {S}^{n}}$$ denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of $${\mathbb {S}^{n}}$$ is denoted by M(n). Let M o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M o (n) depends on the conjugacy of T and T −1 in M o (n). For an element T in M(n), T and T −1 are conjugate in M(n), but they may not be conjugate in M o (n). In the literature, T is called real if T is conjugate in M o (n) to T −1. In this paper we classify real elements in M o (n). Let T be an element in M o (n). Corresponding to T there is an associated element T o in SO(n + 1). If the complex conjugate eigenvalues of T o are given by $${\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}$$ , then {θ1, . . . , θ k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M o (n) we have parametrized the conjugacy classes of regular elements in M o (n). In the parametrization, when T is not conjugate to T −1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as the expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart–Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble, we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.

Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms**In memoriam: Floris Takens.

Dynamical phenomena are studied near a Hopf-saddle-node bifurcation of fixed points of 3D-diffeomorphisms. The interest lies in the neighbourhood of weak resonances of the complex conjugate eigenvalues. The 1 : 5 case is chosen here because it has the lowest order among the weak resonances, and therefore it is likely to have a most visible influence on the bifurcation diagram. A model map is obtained by a natural construction, through perturbation of the flow of a Poincaré–Takens normal form vector field. Global bifurcations arise in connection with a pair of saddle-focus fixed points: homoclinic tangencies occur near a sphere-like heteroclinic structure formed by the 2D stable and unstable manifolds of the saddle points. Strange attractors occur for nearby parameter values and three routes are described. One route involves a sequence of quasi-periodic period doublings of an invariant circle where loss of reducibility also takes place during the process. A second route involves intermittency due to a quasi-periodic saddle-node bifurcation of an invariant circle. Finally a route involving heteroclinic phenomena is discussed. Multistability occurs in several parameter subdomains: we analyse the structure of the basins for a case of coexistence of a strange and a quasi-periodic attractor and for coexistence of two strange attractors. By construction, the phenomenology of the model map is expected in generic families of 3D diffeomorphisms.

About the ellipticity of the Chebyshev–Gauss–Radau discrete Laplacian with Neumann condition

The Chebyshev–Gauss–Radau discrete version of the polar-diffusion operator, 1 r ∂ ∂ r r ∂ ∂ r - k 2 r 2 , k being the azimuthal wave number, presents complex conjugate eigenvalues, with negative real parts, when it is associated with a Neumann boundary condition imposed at r = 1. It is shown that this ellipticity marginal violation of the original continuous problem is genuine and not due to some round-off error amplification. This situation, which does not lead per se to any particular computational difficulty, is taken here as an opportunity to numerically check the sensitivity of the quoted ellipticity to slight changes in the mesh. A particular mapping is chosen for that purpose. The impact of this option on the ellipticity and on the numerical accuracy of a computed flow is evaluated.

The chiral Gaussian two-matrix ensemble of real asymmetric matrices

We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter μ. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its eigenvalues are either real, purely imaginary or come in complex conjugate eigenvalue pairs. The eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for complex non-Hermitian matrices, generalizing the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed ν at strong and weak non-Hermiticity.

Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems

We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and parameterized nonlinear wave equations. The aim of the paper is to present and analyze a continuation method for invariant subspaces that belong to a group of eigenvalues, the number of which is much smaller than the dimension of the system. The continuation method is of predictor-corrector type, similar to the approach for the linear eigenvalue problem in [Beyn, Kles, and Thummler, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001], but we avoid linearizing the problem, which will double the dimension and change the sparsity pattern. The matrix equations that occur in the predictor and the corrector step are solved by a bordered version of the Bartels-Stewart algorithm. Furthermore, we set up an update procedure that handles the transition from real to complex conjugate eigenvalues, which occurs when eigenvalues from inside the continued cluster collide with eigenvalues from outside. The method is demonstrated on several numerical examples: a homotopy between random matrices, a fluid conveying pipe problem, and a traveling wave of a damped wave equation.

On the effect of AVR gain on bifurcations of subsynchronous resonance in power systems

This paper presents the effect of the automatic voltage regulator (AVR) gain on the bifurcations of subsynchronous resonance (SSR) in power systems. The first system of the IEEE second benchmark model of SSR is chosen for numerical investigations. The dynamics of both axes damper windings of the generator and that of the power system stabilizer (PSS) are included. The bifurcation parameter is the compensation factor. Hopf bifurcation, where a pair of complex conjugate eigenvalues of the linearized model around the operating point transversally crosses from left- to right-half of the complex plane, is detected in all AVR gains. It is shown that the Hopf bifurcation is of subcritical type. The results also show that the location of the Hopf bifurcation point i.e. the stable operating point regions are affected by the value of the AVR gain. The variation of the location of the Hopf bifurcation point as function of the AVR gain for two operating conditions is obtained. Time domain simulation results based on the nonlinear dynamical mathematical model carried out at different compensation factors and AVR gains agree with that of the bifurcation analysis.

Implications of complex eigenvalues in homogeneous flow: A three-dimensional kinematic analysis

We present an investigation of the kinematic properties of 3D homogeneous flow defined by complex eigenvalues. We demonstrate, using simple algebraic analysis, that the clear threshold between pulsating and non-pulsating fields, fixed for W n > 1 and valid for planar flow, is not easily defined in a 3D flow system. In 3D flows, one of the three eigenvalues is always real and gives rise to an exponential flow, coexisting with a pulsating pattern defined by the other two complex conjugate eigenvalues. Due to this mathematical property, the existence of a stable or pulsating pattern depends strongly on the relative dominance of the real eigenvector with respect to the complex ones. As a consequence, the pattern of behaviour is not simply imposed by the kinematic vorticity numbers, but is also determined by both the amount of strain accumulation and the extrusion component. It is also shown that complex flow can occur locally within shear zones and can sustain some predictable hyperbolic strain paths. These results are applied to the kinematic analysis of some non-dilational and dilational monoclinic and triclinic flows. Some geological implications of this investigation, and the limit of applying these algebraic and kinematic results to real rocks fabric analysis, are briefly discussed.

On the reduction of the normal Hankel problem to two particular cases

The normal Hankel problem (NHP) is to describe complex matrices that are normal and Hankel at the same time. The available results related to the NHP can be combined into two groups. On the one hand, there are several known classes of normal Hankel matrices. On the other hand, the matrix classes that may contain normal Hankel matrices not belonging to the known classes were shown to admit a parametrization by real 2 × 2 matrices with determinant 1. We solve the NHP for the cases where the characteristic matrix W of the given class has: (a) complex conjugate eigenvalues; (b) distinct real eigenvalues. To obtain a complete solution of the NHP, it remains to analyze two situations: (1) W is the Jordan block of order two for the eigenvalue 1; (2) W is the Jordan block of order two for −1.

Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems

This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings (eight wings) of these novel attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.

GENERATION OF A FOUR-WING CHAOTIC ATTRACTOR BY TWO WEAKLY-COUPLED LORENZ SYSTEMS

This paper presents a novel chaotic four-wing attractor generated by coupling two identical Lorenz systems. An analysis of the proposed system shows that its equilibria have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings of this new attractor are located around four equilibria with four unstable complex-conjugate eigenvalues. A generalization of the proposed system to realize an eight-wing attractor is also described.