Pair Of Complex Conjugate Eigenvalues Research Articles

Exactly solvable Stuart-Landau models in arbitrary dimensions.

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equationsin the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere S^{D-1} and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits, each of which has the geometry of a torus T^{N} on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalizations.

Medium induced mixing, spatial modulations, and critical modes in QCD

The mixing between the chiral condensate and the density in hot and dense quantum chromodynamics (QCD) matter is familiar. We show that the mixing relevant for the ground state is considerably more extensive, and in particular also involves gluonic degrees of freedom. As a result, the Hessian of the QCD effective action is non-Hermitian, but retains a symmetry under combined charge and complex conjugation. This can lead to complex-conjugate pairs of eigenvalues of this Hessian, signaling regimes with spatially modulated correlations. Furthermore, based on the analytic structure of the quark determinant at a chiral critical point, we demonstrate that the corresponding massless critical mode is composed of the chiral condensate, the density and the Polyakov loops. Due to an avoided crossing, the critical mode turns out to be disconnected from the chiral condensate in vacuum. We present general arguments for all these features and illustrate them through explicit model calculations. Published by the American Physical Society 2024

Asymptotics of Solutions of Autonomous Singularly Perturbed Equations when the Stability of the Equilibrium Position Changes at Several Points

Considers a system consisting of 2n equations with fast variables and one equation with a slow variable. The first approximation matrix of the system of fast variables has 2n pairwise complex conjugate eigenvalues. The stability of the equilibrium position, a system of fast variables, is influenced by all eigenvalues. Early studies considered cases where the stability of the equilibrium position is affected by only one pair of complex conjugate eigenvalues. The problem of delaying the solution of a system of fast variables near an unstable equilibrium position has been solved. The delay time for the decision is determined.

Dynamical transition and bifurcation of hydromagnetic convection in a rotating fluid layer

We study the stability and dynamic transition of a rotating electrically conducting fluid layer in the presence of an external magnetic field based on the Boussinesq approximation. By analyzing the spectrum of the linear part of the model and verifying the validity of the principle of exchange of stability, we take a hybrid approach combining theoretical analysis with numerical computation to study the transition from a simple real eigenvalue, a pair of complex conjugate eigenvalues and a real eigenvalue of multiplicity two, respectively. The center manifold reduction theory is applied to reduce the infinite dimensional system to the corresponding finite dimensional one together with several non-dimensional transition numbers that determine the dynamic transition types. Careful numerical computations are performed to determine these transition numbers as well as related flow patterns. Our results indicate that both continuous and jump transitions can occur at certain parameter region.

Which metrics are consistent with a given pseudo-hermitian matrix?

Given a diagonalizable N × N matrix H, whose non-degenerate spectrum consists of p pairs of complex conjugate eigenvalues and additional N − 2p real eigenvalues, we determine all metrics M, of all possible signatures, with respect to which H is pseudo-hermitian. In particular, we show that any compatible M must have p pairs of opposite eigenvalues in its spectrum so that p is the minimal number of both positive and negative eigenvalues of M. We provide explicit parameterization of the space of all admissible metrics and show that it is topologically a p-dimensional torus tensored with an appropriate power of the group Z2.

An update on coherent scattering from complex non-PT-symmetric Scarf II potential with new analytic forms

The versatile and exactly solvable Scarf II potential has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector, it has gone underutilised. Here, we present the most simple analytic forms for the scattering coefficients \((T(k),R(k),|\det S(k)|)\). On the one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand, they make new predictions – all simple and analytical. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of spectral singularity (SS) into a finite number of complex conjugate pairs of eigenvalues (CCPEs). NSDSS behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising coexistence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, so far, the complex Scarf II potential is not known to be pseudo-Hermitian (\(\eta ^{-1} H\eta =H^\dagger \)) under a metric of the type \(\eta (x)\).

Stochastic asset flow equations: Interdependence of trend and volatility

The complex interaction between the volatility and the price trend is examined. We model stochastic asset prices using the asset flow model with randomness arising directly from supply and demand. We show that the volatility is smallest at the extrema of the expected logarithm of the price. Linearizing the stochastic differential equation (SDE) about equilibrium, we obtain an exact relation for the autocovariance function, and relate it to the (3 by 3) Jacobian of the linearized SDE. In particular, we find the conditions under which one has a pair of complex conjugate eigenvalues of the Jacobian resulting in oscillations. The frequency of the oscillations depends only on the imaginary part of the complex pair, while the decay rate depends only on the real eigenvalue and the real parts of the complex pair. For the deterministic system, oscillations typically decay rapidly. However, randomness induces oscillations to continue indefinitely with a frequency that depends on the parameters of the deterministic system. The computations and analytical results presented here demonstrate that volatility increases when traders place greater emphasis on trend, confirming a generally held belief among practitioners.

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

Stochastic Asset Flow Equations: Interdependence of Trend and Volatility

The interaction between the volatility and price dynamics is explored. We model stochastic asset prices using the asset flow model with randomness arising directly from supply and demand. We show that the volatility is smallest at the extrema of the price. Linearizing the stochastic differential equation (SDE) about equilibrium, we obtain an exact relation for the autocovariance function, and relate it to the (3 by 3) Jacobian of the linearized SDE. In particular, we find the conditions under which one has a pair of complex conjugate eigenvalues of the Jacobian resulting in oscillations. The frequency of the oscillations depends only on the imaginary part of the complex pair, while the decay rate depends only on the real eigenvalue and the real parts of the complex pair. For the deterministic system, oscillations typically decay rapidly. However, randomness induces oscillations to continue indefinitely with a frequency that depends on the parameters of the deterministic system. The computations and analytical results presented here demonstrate that volatility increases when traders place greater emphasis on trend, confirming a generally held belief among practitioners.

Global stability analysis of a 90°-bend pipe flow

The present work investigates the stability properties of the flow in a 90°-bend pipe with curvature δ=R/Rc=1/3, with R being the radius of the cross-section of the pipe and Rc the radius of curvature at the pipe centreline. Direct numerical simulations (DNS) for values of the bulk Reynolds number Reb=UbD/ν between 2000 and 3000 are performed. The bulk Reynolds number is based on the bulk velocity Ub, the pipe diameter D, and the kinematic viscosity ν. The flow is found to be steady for Reb⩽2500, with two main pairs of symmetric, counter-rotating vortices in the section of the pipe downstream of the bend. The presence of two recirculation regions is detected inside the bend: one on the outer wall and the other on the inner side. For Reb⩾2550, the flow exhibits a periodic behaviour, oscillating with a fundamental non-dimensional frequency St=fD/Ub=0.23. A global stability analysis is performed in order to determine the cause of the transition from the steady to the periodic regime. The spectrum of the linearised Navier-Stokes operator reveals a pair of complex conjugate eigenvalues with positive real part, hence the transition is ascribed to a Hopf bifurcation occurring at Reb,cr≈2531, a value much lower than the critical Reynolds number for the flow in a torus with the same curvature. The velocity components of the unstable direct and adjoint eigenmodes are investigated, and they display a large spatial separation, most likely due to the non-normality of the linearised Navier-Stokes operator. Thus, the core of the instability, also known in the literature as the wavemaker, is sought performing an analysis of the structural sensitivity of the unstable eigenmode to spatially localised feedbacks. The region located 15° downstream of the bend inlet, on the outer wall, is the most receptive to this kind of perturbations, and thus corresponds to where the instability originates. Since this region coincides with the outer-wall separation bubble, it is concluded that the instability is linked to the strong shear by the backflow phenomena. The present results are relevant for technical applications where bent pipes are frequently used, and their stability properties have hitherto not been studied.

Quasiperiodic dynamics and a Neimark-Sacker bifurcation in nonlinear random walks on complex networks

We study the dynamics of nonlinear random walks on complex networks. In particular, we investigate the role and effect of directed network topologies on long-term dynamics. While a period-doubling bifurcation to alternating patterns occurs at a critical bias parameter value, we find that some directed structures give rise to a different kind of bifurcation that gives rise to quasiperiodic dynamics. This does not occur for all directed network structure, but only when the network structure is sufficiently directed. We find that the onset of quasiperiodic dynamics is the result of a Neimark-Sacker bifurcation, where a pair of complex-conjugate eigenvalues of the system Jacobian pass through the unit circle, destabilizing the stationary distribution with high-dimensional rotations. We investigate the nature of these bifurcations, study the onset of quasiperiodic dynamics as network structure is tuned to be more directed, and present an analytically tractable case of a four-neighbor ring.

A study of fixed points and hopf bifurcation of hindmarsh-rose model

In this article, a class of Hindmarsh-Rose model is studied. First, all necessary conditions for the parameters of system are found in order to have one stable fixed point which presents the resting state for this famous model. After that, using the Hopf’s theorem proofs analytically the existence of a Hopf bifurcation, which is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. Moreover, with the suitable assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point

Homoclinic saddle to saddle-focus transitions in 4D systems

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

A new bifurcation in non-smooth continuous system inducing semi-limit cycle

In this paper, we presented a study on a non-smooth continuous system with emphasis on a special bifurcation. As the parameter varies, a series of concentric closed orbits appear near the equilibrium point. Moreover, the outermost closed orbit attracts all the trajectories outside. It is called as a semi-limit cycle as the trajectories at only one side of this orbit are attracted. By using the theory of generalized Jacobian matrix, it is revealed that this bifurcation can be featured by a pair of complex conjugate eigenvalues reaching exactly but not crossing the imaginary axis. The bifurcation can somewhat be considered to be a degenerate case of the Hopf bifurcation, in which the eigenvalues cross the imaginary axis totally. This study enriches the knowledge of bifurcation analysis for non-smooth dynamical systems.

On the stability of an equilibrium and the small motions of a rigid body containing a liquid, suspended in a uniform flow

In this paper, we consider a planar motion of a rigid body partially filled with an inviscid liquid and suspended in a uniform horizontal flow. At first, we write the equations of the problem, prove the existence of an equilibrium under a suitable condition and, using a first integral, we give a sufficient condition of stability of this one. Afterwards, we give the equations of the small oscillations of the system about its equilibrium position. Writing these equations in an operatorial form, we prove the existence of a denumerable infinity of complex conjugate pairs of eigenvalues having the infinity as a point of accumulation and obtain the characteristic equation permitting the calculation of the eigenvalues.

Eigen-Decomposition of Quaternions

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.

Traveling Waves Bifurcating from Plane Poiseuille Flow of the Compressible Navier–Stokes Equation

Plane Poiseuille flow in viscous compressible fluid is known to be asymptotically stable if Reynolds number R and Mach number M are sufficiently small. On the other hand, for R and M being not necessarily small, an instability criterion for plane Poiseuille flow is known, and the criterion says that, when R increases, a pair of complex conjugate eigenvalues of the linearized operator cross the imaginary axis. In this paper it is proved that a spatially periodic traveling wave bifurcates from plane Poiseuille flow when the critical eigenvalues cross the imaginary axis.

On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents

This study examines the instability and dynamical transitions of the two-layer western boundary currents represented by the Munk profile in the upper layer and a motionless bottom layer in a closed rectangular domain. First, a bound on the intensity of the Munk profile below which the western boundary currents are locally nonlinearly stable is provided. Second, by reducing the infinite dimensional system to a finite dimensional one via the center manifold reduction, non-dimensional transition numbers are derived, which determine the types of dynamical transitions both from a pair of simple complex eigenvalues as well as from a double pair of complex conjugate eigenvalues as the Reynolds number crosses a critical threshold. We show by careful numerical estimations of the transition numbers that the transitions in both cases are continuous at the critical Reynolds number. After the transition from a pair of simple complex eigenvalue, the western boundary layer currents turn into a periodic circulation, whereas a quasi-periodic or possibly a chaotic circulation emerges after the transition from a pair of double complex eigenvalues. Finally, a comparison between the transitions exhibited in one-layer and two-layer models is provided, which demonstrates the fundamental differences between the two models.

Conformally flat Lorentzian hypersurfaces inR14with a pair of complex conjugate principal curvatures

Abstract A three dimensional Lorentzian hypersurface x : M 1 3 → R 1 4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of R 1 4 . Using the projective light-cone model, for those ones whose shape operators have a pair of complex conjugate eigenvalues, we study the integrability condition by constructing a scalar conformal invariant and a canonical moving frame in this paper. It follows that these hypersurfaces can be determined by the solutions to a system of three-order partial differential equations.