This book presents an introduction to the geometric theory of infinite dimensional dynamical systems. Many of the fundamental results are presented for asymptotically smooth dynamical systems that have applications to functional differential equations as well as classes of dissipative partial differential equations. However, as in the earlier edition, the major emphasis is on retarded functional differential equations. This updated version also contains much material on neutral functional differential equations. The results in the earlier edition on Morse-Smale systems for maps are extended to a class of semiflows, which include retarded functional differential equations and parabolic partial differential equations.

The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.

In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.

This book showcases cutting-edge research papers from the XIIth international Milton Symposium hosted by the University of Strasbourg, 17-21 June 2019. Strasbourg is home to Martin Bucer, the Protestant reformer from whom Milton drew support for his theory of divorce, and to Gustave Dore, the famous French illustrator of Paradise Lost. The 26 essays gathered in the present volume are by international scholars, including ones from countries outside the Anglosphere, young or experienced. Opening with a tribute to all Milton symposia organized since 1981, the book falls into eight parts, covering all aspects of Milton studies. "Milton and Materiality" starts with an essay by James G. Turner on personal bodily reference in Milton. In "Milton's Style and Language", the polemicist's use of satire is scrutinized and his relation to enthusiasm is examined, while a new light is shed on his sonnets. In "Milton's Prose", in a rare essay on Observations upon the Articles of Peace (1649), David H. Sacks compares Milton's view of Ireland with what he thought of Russia, delving into the notions of "civilization" and "tyranny". Then the reader will find six essays on Paradise Lost, including one by Hiroko Sano, followed by three essays on his minor poems by promising scholars. The debate on the authorship of De Doctrina Christiana is reopened, with many stylometric tables and charts. A new track leads us to Silesia. In "Reception Studies", two Brazilian contributors study Milton through the lens of French philosophers, and the next essay by Christophe Tournu focuses on the first French verse translation of Paradise Lost. The concluding part, "Milton and his Audience", considers Milton's relationship to his readers, music in Haydn's Creation, while Beverley Sherry analyses portraits of Milton and his works in stained glass.

Debate about the authorship of the manuscript known to us as De Doctrina Christiana has bedevilled Milton studies over recent years. In this book four leading scholars give an account of the research project that demonstrated its Miltonic provenance beyond reasonable doubt. But the authors do much more besides, locating Milton's systematic theology in its broader European context, picking open the stages and processes of its composition, and analysing its Latinity.

Milton's poetry is one of the glories of the English language, and yet it owes everything to Milton's widespread knowledge of other languages: he knew ten, wrote in four, and translated from five. In Milton's Languages, John K. Hale first examines Milton's language-related arts in verse-composition, translations, annotations of Greek poets, Latin prose and political polemic, giving all relevant texts in the original and in translation. Hale then traces the impact of Milton's multilingualism on his major English poems. Many vexed questions of Milton studies are illuminated by this approach, including his sense of vocation, his attitude to print and publicity, the supposed blemish of Latinism in his poetry, and his response to his literary predecessors. Throughout this full-length study of Milton's use of languages, Hale argues convincingly that it is only by understanding Milton's choice among languages that we can grasp where Milton's own unique English originated.

The present book builds upon an earlier work of J. Hale, "Theory of Func tional Differential Equations" published in 1977. We have tried to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a complete new presentation of lin ear systems (Chapters 6 9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global at tractors was completely revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (see Chapters 1, 2, 3, 9, and 10). Chapter 12 is completely new and contains a guide to active topics of re search. In the sections on supplementary remarks, we have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive. Jack K. Hale Sjoerd M. Verduyn Lunel Contents Preface............................................................ v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1. Linear differential difference equations . . . . . . . . . . . . . . 11 . . . . . . 1.1 Differential and difference equations. . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . 1.2 Retarded differential difference equations. . . . . . . . . . . . . . . . 13 . . . . . . . 1.3 Exponential estimates of x( cents, f) . . . . . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . 1.4 The characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . 1.5 The fundamental solution. . . . . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . 1.6 The variation-of-constants formula............................. 23 1. 7 Neutral differential difference equations . . . . . . . . . . . . . . . . . 25 . . . . . . . 1.8 Supplementary remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 34 . . . . . . . . . . . . . 2. Functional differential equations: Basic theory . . . . . . . . 38 . . 2.1 Definition of a retarded equation. . . . . . . . . . . . . . . . . . . . . . 38 . . . . . . . . . 2.2 Existence, uniqueness, and continuous dependence . . . . . . . . . . 39 . . . 2.3 Continuation of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . ."

The 1986 NATO Advanced Study Insti tute on Dynamics of Infini te Dimensional Systems was held at the Instituto Superior Tecnico. Lisbon. Portugal. In recent years. there have been several research workers who have been considering partial differential equations and functional differential equations as dynamical systems on function spaces. Such approaches have led to the formulation of more theoretical problems that need to be investigated. In the applications. the theoretical ideas have contributed significantly to a better understanding of phenomena that have been experimentally and computationally observed. The investigators of this development come wi th several different backgrounds - some from classical partial differential equations. some from classical ordinary differential equations and some interested in specific applications. Each group has special ideas and often these ideas have not been transmitted from one group to another. The purpose of this NATO Workshop was to bring together research workers from these various areas. It provided asoundboard for the impact of the ideas of each respective discipline. We believe that goal was accomplished. but time will be a better judge. We have included the list of participants at the workshop. with most of these giving a presentation. Although the proceedings do not include all of the presentations. it is a good representative sampie. We wish to express our gratitude to NATO. and.to Dr. M. di Lullo of NATO. who unfortunately did not live to see the completion of this project.

In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.

Since the publication of my lecture notes, Functional Differential Equations in the Applied Mathematical Sciences series, many new developments have occurred. As a consequence, it was decided not to make a few corrections and additions for a second edition of those notes, but to present a more compre hensive theory. The present work attempts to consolidate those elements of the theory which have stabilized and also to include recent directions of research. The following chapters were not discussed in my original notes. Chapter 1 is an elementary presentation of linear differential difference equations with constant coefficients of retarded and neutral type. Chapter 4 develops the recent theory of dissipative systems. Chapter 9 is a new chapter on perturbed systems. Chapter 11 is a new presentation incorporating recent results on the existence of periodic solutions of autonomous equations. Chapter 12 is devoted entirely to neutral equations. Chapter 13 gives an introduction to the global and generic theory. There is also an appendix on the location of the zeros of characteristic polynomials. The remainder of the material has been completely revised and updated with the most significant changes occurring in Chapter 3 on the properties of solutions, Chapter 5 on stability, and Chapter lOon behavior near a periodic orbit."

State-of-the-art in qualitative theory of functional differential equations; Most of the new material has never appeared in book form and some not even in papers; Second edition updated with new topics and results; Methods discussed will apply to other equations and applications

Milton's poetry is one of the glories of the English language, and yet it owes everything to Milton's widespread knowledge of other languages: he knew ten, wrote in four, and translated from five. In Milton's Languages, John K. Hale first examines Milton's language-related arts in verse-composition, translations, annotations of Greek poets, Latin prose and political polemic, giving all relevant texts in the original and in translation. Hale then traces the impact of Milton's multilingualism on his major English poems. Many vexed questions of Milton studies are illuminated by this approach, including his sense of vocation, his attitude to print and publicity, the supposed blemish of Latinism in his poetry, and his response to his literary predecessors. Throughout this full-length study of Milton's use of languages, Hale argues convincingly that it is only by understanding Milton's choice among languages that we can grasp where Milton's own unique English originated.

The second of eleven volumes of Milton's Complete Works to be published contains his systematic theology, De Doctrina Christiana. It is his longest work and was, Milton said, his dearest possession. In it, he works out his religious beliefs from Scripture; what Scripture does not mention, such as the Trinity, he energetically refutes. The work exists in manuscript and was written in Latin for European as well as home consumption. Its chapters are conceived and arranged according to the binarizing logic devised by the Protestant martyr Ramus.
De Doctrina Christiana first appeared in print nearly two hundred years ago but the previous editions are now overdue for replacement. For this ground-breaking edition, the manuscript has been freshly transcribed, with fuller textual apparatus and commentary than in any of its few predecessors. The edition aims above all at accuracy, clarity, and completeness, presenting Latin and English on facing pages, amplifying the Biblical citations where necessary, and adding extensive annotations not only on the text and its transcription but also on the content and context of Milton's ideas. The provenance and history of the work are expertly narrated, enabling readers to get closer than ever before to its composition. Milton's Latin is examined in unprecedented detail, and the translation aims to reproduce the nuances and changes of register which characterize his Latin in all its individuality - from the high-flown rhetoric of his arguments in favour of divorce and polygamy, and against tithing, to the plainer style of those sections where he states his main points more dispassionately but bolsters them with strong and wide-ranging Biblical support. The structure of this massive edifice is clarified by the addition of charts which show the Ramist scheme he followed, whereby the primary division between faith (Book One) and worship (Book Two) is mirrored by smaller and smaller subdivisions whose relationship to the whole can be seen at a glance.