This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems "of sensitivity analysis and optimum structural design with respect to multiple eigenvaiues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality conditioli (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.

A wave function picks up, in addition to the dynamic phase, the geometric ͑Berry͒ phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for ͑double͒ cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.

Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceedings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control.The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Selected Volumes in Series B ISBN 981-238-406-5Printed in Singapore by World Scientific Printers (S) Pte Ltd PrefaceStability theory is one of the most interesting and important fields of applied mathematics having numerous applications in natural sciences as well as in aerospace, naval, mechanical, civil and electrical engineering. Stability theory was always important for astronomy and celestial mechanics, and during last decades it is applied to stability study of processes in chemistry, biology, economics, and social sciences. Every physical system contains parameters, and the main goal of the present book is to study how a stable equilibrium state or steady motion becomes unstable or vice versa with a change of problem parameters. Thus, the parameter space is divided into stability and instability domains. It turns out that the boundary between these domains consists of smooth surfaces, but can have different kind of singularities. Qualitatively, typical singularities for systems of ordinary differential equations were classified and listed in [Arnold (1983a); Arnold (1992)]. One of the motivations and challenges of the present book was to bring some qualitative results of bifurcation and catastrophe theory to the space of problem parameters making the theory also quantitative, i.e., applicable and practical. It is shown in the book how the stability boundary and its singularities can be described using information on the system.Behavior of the eigenvalues near the stability boundary with a change of parameters determines stability or instability of the system. Fig. 0.1 reproduced from [ Thompson (1982)] shows interaction of eigenvalues for a specific mechanical system, a pipe conveying fluid, depending on a single parameter p. As we can see, the eigenvalues approach each other, collide and diverge with exciting loops and pirouettes making the system stable or unstable. Looking at this and similar figures several questions appear: What are the rules for movements of eigenvalues on the complex plane de- pending on problem parameters? What kind of collisions are possible and which of them are typical? Are there some special properties for behavior of eigenvalues of mechanical sy...

The paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceedings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control.The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Selected Volumes in Series B ISBN 981-238-406-5Printed in Singapore by World Scientific Printers (S) Pte Ltd PrefaceStability theory is one of the most interesting and important fields of applied mathematics having numerous applications in natural sciences as well as in aerospace, naval, mechanical, civil and electrical engineering. Stability theory was always important for astronomy and celestial mechanics, and during last decades it is applied to stability study of processes in chemistry, biology, economics, and social sciences. Every physical system contains parameters, and the main goal of the present book is to study how a stable equilibrium state or steady motion becomes unstable or vice versa with a change of problem parameters. Thus, the parameter space is divided into stability and instability domains. It turns out that the boundary between these domains consists of smooth surfaces, but can have different kind of singularities. Qualitatively, typical singularities for systems of ordinary differential equations were classified and listed in [Arnold (1983a); Arnold (1992)]. One of the motivations and challenges of the present book was to bring some qualitative results of bifurcation and catastrophe theory to the space of problem parameters making the theory also quantitative, i.e., applicable and practical. It is shown in the book how the stability boundary and its singularities can be described using information on the system.Behavior of the eigenvalues near the stability boundary with a change of parameters determines stability or instability of the system. Fig. 0.1 reproduced from [ Thompson (1982)] shows interaction of eigenvalues for a specific mechanical system, a pipe conveying fluid, depending on a single parameter p. As we can see, the eigenvalues approach each other, collide and diverge with exciting loops and pirouettes making the system stable or unstable. Looking at this and similar figures several questions appear: What are the rules for movements of eigenvalues on the complex plane de- pending on problem parameters? What kind of collisions are possible and which of them are typical? Are there some special properties for behavior of eigenvalues of mechanical sy...