Critical Eigenvalues Research Articles

Resonance and Interior Layers in an Inhomogeneous Phase Transition Model

We consider the problem $\ve^2 \Delta u + (u-a(x))(1-u^2)=0$ in $\Omega, \frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$, where Ω is a smooth and bounded domain in $\R^2$, $ -1 <a(x) <1$. Assume that $ \Gamma = \{ x \in \Omega, a(x)=0 \}$ is a closed, smooth curve contained in Ω in such a way that $\Omega = \Omega_{+} \cup\Gamma\cup \Omega_{-}$ and $\frac{\partial a}{\partial n } >0$ on Γ, where n is the outer normal to $\Omega_+$. Fife and Greenlee [Russian Math. Surveys, 29 (1974), pp. 103–131] proved the existence of an interior transition layer solution $u_\ve$ which approaches ‐1 in $\Omega_{-}$ and +1 in $\Omega_{+}$, for all ε sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in $\Omega_{-}$ and ‐1 in $\Omega_{+}$ exists. In this paper, we answer this question affirmatively when $n=2$, provided that ε is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator.

The Criticality Eigenvalue Problem for the Transport Operator with General Boundary Conditions

The effective multiplication factor (keff) is the dominant eigenvalue for the linear particle transport operator. A proof for the existence of a simple dominant eigenvalue with a positive eigenfunction is established in the L1 setting in the presence of particle reflection at the domain boundary as characterized by a general integral albedo operator. By arguing that physics demands to work on an L1‐weighted space, a direct physical approach based on linearity and positivity allows us to circumvent some of the limitations encountered in the classical proofs.

Characterization of critical eigenvalues of axial flow engine compressor PDE model†

In this article, we study the spectrum of axial flow engine compressor PDE model and show that there are three types of critical eigenvalues which determine the stability of the steady solution of the system depending on the compressor geometry. All of these critical eigenvalues cross the imaginary axis with nonzero speeds as the throttle coefficient increases, leading to Hopf bifurcations of the system. †Dedicated to Professor Wei Lin on the occasion of his 70th birthday.

Application of a Novel Eigenvalue Trajectory Tracing Method to Identify Both Oscillatory Stability Margin and Damping Margin

This paper is proposing an integration-based approach to trace a specific eigenvalue or a set of eigenvalues of a power system network. Indices are developed to identify critical eigenvalue to be traced for further analysis. This index takes into account the rate of change as well as the direction of movement. The rate of change is with respect to any parameter variation of interest. It finally provides information related to oscillatory stability and margin requirements to maintain a specific damping ratio. Eigenvalue and eigenvector sensitivities (with respect to any explicit/implicit parameters) are by-products of this approach.

Calculation of Rightmost Eigenvalues in Power Systems Using the Jacobi–Davidson Method

A new methodology for the calculation of critical eigenvalues in the small signal stability analysis of large power systems is presented in this paper. The Jacobi–Davidson method, which is a very recent subspace iteration method, is suggested to compute the rightmost eigenvalues. The method is attractive as a new powerful technique for solving a variety of large sparse eigenproblems. The method combined Davidson's idea of taking a different subspace with Jacobi's idea of restricting the search of an update to the subspace orthogonal to current eigenvector approximation. An exact solution of the correction equation leads to quadratic convergence for the selected Ritz values. Based on the iterative construction of a partial Schur form and the effective restart, the algorithms are suitable for the efficient computation of a large number of eigenvalues and clustering eigenvalues. The proposed method is applied to a practical 46-machine system, and the results of the experiment are described.

Nilpotent Hopf Bifurcations in Coupled Cell Systems

Network architecture can lead to robust synchrony in coupled systems and, surprisingly, to codimen- sion one bifurcations from synchronous equilibria at which the associated Jacobian is nilpotent. We prove three theorems concerning nilpotent Hopf bifurcations from synchronous equilibria to periodic solutions, where the critical eigenvalues have algebraic multiplicity two and geometric multiplicity one, and discuss these results in the context of three different networks in which the bifurcations occur generically. Phenomena stemming from these bifurcations include multiple periodic solutions, solutions that grow at a rate faster than the standard λ 1 2 , and solutions that grow slower than the standard λ 1 2 . These different bifurcations depend on the network architecture and, in particular, on the flow-invariant subspaces that are forced to exist by the architecture.

Nonlinear time-response optimisation method for tuning power system stabilisers

A power system is an inherently highly nonlinear and complex system. Many tuning methods involving parameter optimisation used in the past, were often based on small-signal stability theory and linearised system equations. Such optimisation usually involves the optimising of frequency-domain characteristics such as damping and damping factors of the system critical eigenvalues. The use of frequency-domain information does not allow for the direct improvement of the time-domain characteristics of the system response, e.g. settling time. A simple method is proposed that directly optimises the nonlinear system by using system time response to large disturbances. The properties of the method and its application specifics are discussed in detail. The effectiveness of the method in the co-ordinated tuning of multiple power system stabilisers is illustrated on a multimachine test power system.

Calculation of eigenfunction fluxes in nuclear systems

A new Monte Carlo method is being developed to calculate eigenfunction fluxes in critical or near-critical nuclear systems. The correct estimation of fluxes is essential for radiation protection and shielding near these systems, in addition to isotope production, isotope depletion, nuclear criticality and other applications. The proposed method applies to Monte Carlo criticality eigenvalue calculations in which the fission sites in one generation are used as fission sources in subsequent generations. The usual Monte Carlo power iteration method for such problems often calculates fluxes (eigenfunctions) that are inaccurate and very different in symmetric parts of a problem geometry. The proposed method calculates flux distributions by estimating an approximate fission matrix. The way the fission matrix is estimated and used differs from other recent works. Preliminary results are promising.

Analysis of a corner layer problem in anisotropic interfaces

We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered $Cu_3Au$ state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number $0(\frac{1}{\epsilon})$, of critical eigenvalues, where $\frac{1}{\epsilon}$ » $1$ is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues $\lambda_n = \e^{2/3}\mu_n$ with $\mu_n$ ~ $Cn^{2/3}$, as $n \to + \infty$. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.

Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems

This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called ‘‘time-2τ2 map’’ of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T1 (Hopf invariant circles), tori 2T1 and tori 2T2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms’ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.

Optimal Placement of SVC for Static and Dynamic Voltage Security Enhancement

This paper presents a sensitivity based approach to determine optimal location of Static VAR Compensator (SVC) for voltage security enhancement. The proposed approach of SVC placement computes sensitivity of system loading factor with respect to reactive power generation, derived from reactive power balance equation. This sensitivity factor has been termed as Bus Static Participation Factor (BSPF). The sensitivity of the most critical eigen value with respect to bus voltage magnitude has been considered as Bus Dynamic Participation Factor (BDPF). BSPF and BDPF have been combined to obtain Bus Hybrid Participation Factor (BHPF), which have been computed for the system intact case and critical contingency cases and used for selecting the optimal site for the SVC installation. The effectiveness of the proposed approach of SVC placement has been validated on a practical 75-bus Indian system with respect to enhancement of the static as well as dynamic voltage stability margins.

AN OPTIMALITY CONDITION FOR THE ASSEMBLY DISTRIBUTION IN A NUCLEAR REACTOR

This paper presents an optimality condition for the optimization problem of the assembly distribution in a nuclear reactor, by using the homogenization method. In this paper the reactivity of the reactor core is measured by the critical eigenvalue for both continuous and multigroup neutron transport models. In particular, we extend the spectral theory of the critical eigenvalue and prove the differentiability of the latter with respect to the design parameter, the configuration of the fuels.

Numerical Normal Forms for Codim 2 Bifurcations of Fixed Points with at Most Two Critical Eigenvalues

In this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues. These include all strong resonances, as well as the degenerate flip and Neimark--Sacker bifurcations. Applying our results to n-dimensional maps, one avoids the computation of the center manifold, but one can directly evaluate the critical normal form coefficients in the original basis. The formulas remain valid also for n=2 and allow one to avoid the transformation of the linear part of the map into Jordan form. The developed techniques are tested on two examples: (1) a three-dimensional map appearing in adaptive control; (2) a periodically forced epidemic model. We compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models. To compute the required derivatives of the Poincaré map for the epidemic model, the automatic differentiation package ADOL-C is used.

Saddle-node index as bounding value for Hopf bifurcations detection

Some aspects related to modal analysis in voltage stability studies are discussed. The idea is to determine a set of dominant eigenvalues during the voltage collapse path. This set contains the most critical eigenvalues, which are not, necessarily, the least eigenvalues commonly focused in modal analysis. This tool is then used to detect Hopf and saddle-node bifurcations, since the least real part of the eigenvalues are monitored. The results are obtained and discussed with the help of two sample systems with two and nine buses each.

Calculation of electromechanical oscillation modes in large power systems using Jacobi-Davidson method

An efficient method for the calculation of electromechanical oscillation modes in the small-signal stability analysis of large power systems is presented. The method requires calculating only the critical eigenvalues with increasing damping ratios rather than all rightmost eigenvalues. The Jacobi-Davidson method is a very recent subspace iteration method. It is attractive as a new powerful technique for solving a variety of large sparse eigenproblems and is successfully applied to the calculation of critical eigenvalues. The computational efficiency of the Jacobi-Davidson method is improved by adopting Krylov subspace as the initial search subspace. The proposed method has been tested on 46 and 113 machine systems. The results show that it is capable of effectively finding all electromechanical modes of oscillation with unsatisfactory damping ratios, and suitable for the electromechanical oscillation analysis of large power systems.

New results on general observers for nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence ofgeneral observers (asymptotic or exponential) for nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential), for nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.

New results on general observers for discrete-time nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.

Multiscale analysis of defective multiple-Hopf bifurcations

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order- m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation ( m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.

Stability of spot and ring solutions of the diblock copolymer equation

The Γ-convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is stable if the sample size is small and unstable if the sample size is large. The stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems. In one study no two-interface ring solutions are found by the Γ-convergence method if the sample is small. A stable two-interface ring solution exists if the sample size is increased. It becomes unstable if the sample size is increased further.

Effects of the SVC and the STATCOM on Damping Power Systems Low Frequency Electromechanical Oscillations

This work investigates the effects of two FACTS (Flexible AC Transmission Systems) devices, the SVC (Static Var Compensator) and the STATCOM (Static Synchronous Compensator), in power systems small-signal angle stability. This investigation is carried out to a single generator connected to an infinite bus via a loss-less transmission line. The study is based in the investigation of the critical eigenvalues of the power system linearized model in the framework of the bifurcation theory. The presented simulation results allow a comparative analysis of the effects of these two controllers in damping power systems low frequency electromechanical oscillations.