Physical systems with loss or gain feature resonant modes that are decaying or growing exponentially with time. Whenever two such modes coalesce both in their resonant frequency and their rate of decay or growth, a so-called "exceptional point" occurs, around which many fascinating phenomena have recently been reported to arise [1][2][3][4][5][6] . Particularly intriguing behavior is predicted to appear when encircling an exceptional point sufficiently slowly 7,8 , like a state-flip or the accumulation of a geometric phase 9,10 . Experiments dedicated to this issue could already successfully explore the topological structure of exceptional points [11][12][13] , but a full dynamical encircling and the breakdown of adiabaticity inevitably associated with it 14-21 remained out of reach of any measurement so far. Here we

In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This 'adiabatic flip' effect is an outcome of the appearance of a degeneracy known as an 'exceptional point' inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.

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...

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.