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Chaos and social -spatial dynamics
BOOK REVIEWS Chaos and Social-Spatial Dynamics, Applied Mathematical Sciences, Volume 86 D. S. Dendrinos and M. Sonis Springer-Verlag, New York, 1990, $39.80 The modern theory of dynamical systems is finding a large number of applications in numerous fields such as classical and quantum mechanics, biology, epidemics, statistics, population dynamics, and social sciences. This book presents some key features of a discrete dynamical map that seems to accommodate a wide range of the dynamics found in spatially distributed social systems and population dynamics. Although the book contains many models, such as multistock/multilocation problems, many new phenomena that are reproduced by the map still wait to be uncovered. The most innovative features presented in the book are twodimensional discrete dynamical bifurcation, the onset of local and partial turbulence, and the existence of strange attractors and containers. The book consists of four parts and two appendices. The first part is a brief introduction to continuous and discrete dynamics. The second part considers a one-stock/two-location model and its equilibrium points, stability, and bifurcation behavior. It also considers a loglinear model and its higher iterates and fundamental bifurcations, including period doubling, Feigenbaum’s slope sequences, and Hopf’s equivalent bifurcations. The third part presents a onestock/mulltiple-location model with a log-linear model that exhibits local and partial turbulence, strange attractors, and containers when there are three locations. For the case of four locations it is difficult to obtain analytical results, and the results presented in the book illustrate the properties of such a model with computer-generated figures that show the first and second iterates of the map. The last part of the book deals multiple-stock/multiple-location with models with a log-linear specification. The discrete-time/discrete-space model presented in the book exhibits a rich dynamical behavior that describes the evolution and spatial distribution of populations and may have applications in the stock market, psychology, politics, and demography.
0 1992 Butterworth-Heinemann
The book is well written and will be of interest to those working on discretetime systems and maps, sociology, population dynamics, psychology, and the like. The authors are to be congratulated for their clear exposition and detailed analytical and numerical analysis of the one-stock models presented in the book, as well as for the rich behavior that they have found in the model. Dr. J. 1. Ramos
Numerical Modeling in Science and Engineering Myron B. Allen Ill, lsmael Herrera, and George F. Pinder John Wiley & Sons, New York, 1988 This book is an attempt to treat in a single volume three disciplines that have traditionally been considered as separate entities: continuum mechanics, differential equations, and numerical analysis. These disciplines together are the key elements of numerical modelling. An introduction to continuum mechanics is presented in the first chapter, which includes conservation (balance) equations in both integral and differential forms, constitutive equations, and the theory of mixtures. Chapter I also includes examples from fluid and solid mechanics and problems for students. Chapter 2 presents an introduction to numerical methods (polynomial interfinite elepolation/approximation, ments, finite differences, collocation, boundary elements) for the solution of partial differential equations. The second chapter also considers the stability and consistency of finite difference techniques and error bounds in finite element and collocation methods. Maximum and minimum principles, finite difference, finite element, and collocation methods for steady-state problems are treated in Chapter 3, which also considers the stability and consistency of, and iterative and direct solution methods for, finite difference techniques. Finite difference and finite element methods for the heat conduction equation and dissipative systems are presented in Chapter 4, while Chapter 5 considers finite difference, finite element, and orthogonal collocation techniques for hyperbolic equations and nondissipative systems. The last chap-
numerical ter, Chapter 6, contains methods for the biharmonic equation and nonlinear problems such as solid deformation and oil reservoir models. Most of the numerical techniques presented in the book are well known, and the contents of the first chapter can be found in numerous books on continuum mechanics. The book is not a numerical modelling text; it is a collection of numerical methods. Numerical modelling is an art based on the scientific method, whereby the problem to be solved must be formulated in terms of simple models whose validity must be verified both physically and mathematically. Numerical modelling is not a collection of numerical techniques and conservation principles of continuum mechanics, since the numerical modeller has to make assumptions in order to model the problem that he or she has to solve and the assumptions depend on the model and its application. The book may be used as an introductory text to finite difference and finite element methods. These topics are covered in greater detail in numerous books, including the one by Lapidus and Pinder (Numerical Solution oj Purtiul D&ferentiul Equcltions in Science, John Wiley
& Sons, New York, 1982). Dr. J. I. Ramos
Introduction to Finite Elements in Engineering Tirupathi R. Chandrupatla and Ashok D. Belengudu Prentice Hail, Englewood Cliffs, New Jersey, 1991 This is a textbook on finite element methods for undergraduate students that emphasizes applications to solid mechanics. The book consists of 12 chapters and contains a diskette with computer problems. Chapter 1 presents the constitutive equations for elastic solids, thermally induced stresses, and the methods of Rayleigh-Ritz and Galerkin. Chapter 2 is a brief introduction to matrix algebra and Gaussian elimination, which is the method used in most of the computer programs presented in the book. Chapter 3 considers one-dimensional problems and includes a detailed account of the division into and the numbering of finite elements, linear and quadratic shape functions, assembly and
Appl. Math. Modelling,
1992, Vol. 16, February
107
0 1992 Butterworth-Heinemann
The book is well written and will be of interest to those working on discretetime systems and maps, sociology, population dynamics, psychology, and the like. The authors are to be congratulated for their clear exposition and detailed analytical and numerical analysis of the one-stock models presented in the book, as well as for the rich behavior that they have found in the model. Dr. J. 1. Ramos
Numerical Modeling in Science and Engineering Myron B. Allen Ill, lsmael Herrera, and George F. Pinder John Wiley & Sons, New York, 1988 This book is an attempt to treat in a single volume three disciplines that have traditionally been considered as separate entities: continuum mechanics, differential equations, and numerical analysis. These disciplines together are the key elements of numerical modelling. An introduction to continuum mechanics is presented in the first chapter, which includes conservation (balance) equations in both integral and differential forms, constitutive equations, and the theory of mixtures. Chapter I also includes examples from fluid and solid mechanics and problems for students. Chapter 2 presents an introduction to numerical methods (polynomial interfinite elepolation/approximation, ments, finite differences, collocation, boundary elements) for the solution of partial differential equations. The second chapter also considers the stability and consistency of finite difference techniques and error bounds in finite element and collocation methods. Maximum and minimum principles, finite difference, finite element, and collocation methods for steady-state problems are treated in Chapter 3, which also considers the stability and consistency of, and iterative and direct solution methods for, finite difference techniques. Finite difference and finite element methods for the heat conduction equation and dissipative systems are presented in Chapter 4, while Chapter 5 considers finite difference, finite element, and orthogonal collocation techniques for hyperbolic equations and nondissipative systems. The last chap-
numerical ter, Chapter 6, contains methods for the biharmonic equation and nonlinear problems such as solid deformation and oil reservoir models. Most of the numerical techniques presented in the book are well known, and the contents of the first chapter can be found in numerous books on continuum mechanics. The book is not a numerical modelling text; it is a collection of numerical methods. Numerical modelling is an art based on the scientific method, whereby the problem to be solved must be formulated in terms of simple models whose validity must be verified both physically and mathematically. Numerical modelling is not a collection of numerical techniques and conservation principles of continuum mechanics, since the numerical modeller has to make assumptions in order to model the problem that he or she has to solve and the assumptions depend on the model and its application. The book may be used as an introductory text to finite difference and finite element methods. These topics are covered in greater detail in numerous books, including the one by Lapidus and Pinder (Numerical Solution oj Purtiul D&ferentiul Equcltions in Science, John Wiley
& Sons, New York, 1982). Dr. J. I. Ramos
Introduction to Finite Elements in Engineering Tirupathi R. Chandrupatla and Ashok D. Belengudu Prentice Hail, Englewood Cliffs, New Jersey, 1991 This is a textbook on finite element methods for undergraduate students that emphasizes applications to solid mechanics. The book consists of 12 chapters and contains a diskette with computer problems. Chapter 1 presents the constitutive equations for elastic solids, thermally induced stresses, and the methods of Rayleigh-Ritz and Galerkin. Chapter 2 is a brief introduction to matrix algebra and Gaussian elimination, which is the method used in most of the computer programs presented in the book. Chapter 3 considers one-dimensional problems and includes a detailed account of the division into and the numbering of finite elements, linear and quadratic shape functions, assembly and
Appl. Math. Modelling,
1992, Vol. 16, February
107