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Analysis and Computation of Equilibria and Regions of Stability

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Analysis and Computation of Equilibria and Regions of Stability 1 INTERNATIONAL INSTITUTE FOR 1 IASAAPPL~EDSYSTEMS ANALYSIS CONFERENCE PROCEEDINGS ANALYSIS AND COMPUTATION OF EQUILIBRIA AND REGIONS OF STABILITY With Applications in Chemistry, Climatology, Ecology, and Economics RECORD OF A WORKSHOP July 21 -August 1,1975 H. R. Griimm, Editor SCHLOSS LAXENBURG A-2361 AUSTRIA ANALYSIS AND COMPUTATION OF EQUILIBRIA AND REGIONS OF STABILITY With Applications in Chemistry, Climatology, Ecology, and Economics RECORD OF A WORKSHOP July 21-August 1,1975 H. R. Griimm, Editor The views expressed are those of the contributors and not necessarily those of the Institute. The Institute assumes full responsibility for minor editorial changes made in grammar, syntax, or wording, and trusts that these modifications have not abused the sense of the writers' ideas. This record has been put together in a limited time for prompt distribution. It is not a proceedings volume. Rather it is a collection of all memoranda, diagrams, and literature references that were circulated before the workshop, used to support presentations during the workshop, or written down to preserve some ideas and some outcomes of computations that arose from the workshop. The only organizing principle is the temporal sequence in which the materials were presented or prepared. TABLE OF CONTENTS Preface iii List of Participants from outside IIASA ix from IIASA staff xi Workshop Organization xii Advance Description of Workshop xiii RECORD OF PLENARY OPENING SESSIONS, JULY 21 AND 22 Welcoming Remarks to the Participants R. Levien Opening Remarks on the Proposed Activities T.C. Koopmans Dynamic and Equilibrium Problems in Climatology J.G. Charney Computational Aspects of the Modeling of Atmospheric Dynamics and Estimates of the Influence of Different Factors V.V. Penenko ~ethodologicalProblems in the odel ling and Analysis of Ecological Systems C. Walters and W. Clark Fixed Point Methods H.E. Scarf Description of Fixed Point Algorithms T. Hansen An Outline of Structural Stability Theory P. Walters Extrapolation Methods for ~quilibriaCalculations M.L. Juncosa 5 7 Newton's Method for Systems of Nonlinear Equations S .M. Robinson 67 Self-organizing Chemical Systems P. Schuster FU2CORD OF WORKING SESSIONS AND IIASA SEMINARS, JULY 23-31 Bifurcations and the Appearance of Attracting Tori H.R. Griimm 83 A Geometrical View of Fixed Point Algorithms H.E. Scarf Analysis of a Compact Predator-Prey Model The Basic Equations and Behaviour D.D. Jones Volterra's System and the Equation of Michaelis-Menten A.D. Bazykin Stability Analysis of Predator-Prey Models Via the Liapunov Method M. Gatto and S. Rinaldi Zubov Procedures for Estimating the Domain of Attraction J.L. Casti Some Elements of the Response Function of the Enterprise in the Socialist Economy J. Beksiak Fixed Points, Periodic Orbits, etc., in Climatological Models J.G. Charney and K. Fraedrich Immunity - A Mathematical Model A. Molchanov Time Averages Near Strange Attractors K. Sigmund On Stochastic Stability and Resilience Yu. Rozanov Elementary Model of Eutrophication A.D. Bazykin Drought as a Biogeophysical Feedback Elechanism J.G. Charney MEMORANDA CIRCULATED DURING THE WORKSHOP Rigorousness May Be Dangerous A. Molchanov Rigorousness May Be Dangerous But Not Necessarily A. Molchanov Stability Versus Resilience A. Molchanov Equilibria in Local and Distributed Systems A.D. Bazykin RECORD OF PLENARY CLOSING SESSION, JULY 31 What Have We Learned? T.C. Koopmans The Role of Fixed Points and Closed Orbits in Dynamical Models of Climate J.G. Charney and K. Fraedrich An Approach for Solving Non-Stationary and Stationary Problems of Meteorology V.V. Penenko What Have We Learned? W. Clark and A.D. Bazykin Dynamic Characteristics of a System and the Choice of Techniques H.E. Scarf Fast computation of Planar Limit Cycles M.L. Juncosa Computing Closed Orbits and Periodic Points H.R. ~r%rtm Computational Experiments T. Hansen, assisted by H.R. ~rknm, D.D. Jones, and P. Schuster Retrospect and Prospect 279 Appendix: List of Papers Distributed During the Workshop 281 vii List of Participants (from outside IIASA) Dr. Alexander D. Bazykin Research Computing Center USSR Academy of Sciences Pushchino Moscow Region USSR Prof. N.J. Beksiak Central School of Planning and Statistics Warsaw POLAND Prof. Jule G. Charney Department of Meteorology Massachusetts Institute of Technology Cambridge, Plassachusetts USA Dr. William Clark* Institute of Animal Resource Ecology University of British Columbia Vancouver, B.C. CANADA Prof. Klaus Fraedrich Meteorology Institute Free University Berlin Berlin FEDERAL REPUBLIC OF GEWANY Prof. Terje Hansen* Institute of Econonics Norwegian School of Economics and Business Administration Bergen NORWAY Dr. Dixon D. Jones* Institute of Animal Resource Ecology University of British Columbia Vancouver, B.C. CANADA Dr. Mario L. Juncosa The RAND Corporation Santa Monica, California USA Prof. Tjalling C. Koopmans* Cowles Foundation Department of Economics Yale University New Haven, Connecticut USA Prof. A. Molchanov Research Computing Center USSR Academy of Sciences Pushchino Moscow Region USSR Dr. Vladimir V. Penenko Computing Center of the Siberian Branch of the USSR Academy of Sciences Novosibirsk USSR Prof. Stephen PI. Robinson Mathematics Research Center University of Wisconsin Madison, Wisconsin USA Prof. Herbert E. Scarf Cowles Foundation Yale University New Haven, Connecticut USA Prof. Peter Schuster ~nstitutefor Theoretical Chemistry University of Vienna Vienna AUSTRIA Prof. Karl Sigmund Institute for Mathematics University of Vienna Vienna AUSTRIA Mr. Armand Taranco Institute of Advanced Scientific Studies Bures-Sur-Yvette FRANCE Prof. Peter Walters Department of Mathematics University of Warwick Coventry UNITED KINGDOM *former IIASA staff members List of Participants (from IIASA staff) Dr. John L. Casti Research Plathemat ician Methodology Project Dr. Hans R. GrUrnm Consultant Energy Project Dr. Yuri Rozanov Research Scholar Methodology and Water Projects Dr. Carl Walters Project Leader Ecology Project Workshop Organization Chairman of the Workshop T. C. Koopmans Assisting the Chairman in preparation of the Workshop J. L. Casti Advisory Committee A. D. Bazykin, J. L. Casti Wm. Clark, K. Fraedrich H. R. ~rknm,T. Hansen T. C. Koopmans, S. M. Robinson Editor of the Record Editorial Committee J. L. Casti, Wm. Clark H. R. Grh, T. C. Koopmans Secretarial and organizing Linda Berg, Brigitte Gromus assistance prior to, Eva Matt, of the staff of during and after the IIASA, and Lydia Zimmerman Workshop of the Cowles Foundation Yale University New Haven, Conn., U.S.A. xii Advance Description of Workshop In the last few years problems have come to the fore in c-, c-, in ecology, and in economics that have a common mathematical structure. In climatology and ecology, these problems concern systems described by a set of differential equations in which non- linearities are an important aspect of the problem. Mathematical treatment has therefore emphasized simplifying assumptions or complex simulations. The former destroys many subtle behavior characteristics while the latter can be expensive, and may lack the generality needed for transfer of findings to other situations Another handhold for analysis is present if the system has the property that as time proceeds the motion of the state variables approaches an asymptote (or rest point). Mathemat- ically, such a rest point also qualifies as a stationary solution of the differential equations. Moreover, such a solution can also be considered as a fixed point of a continuons mapping of the set of possible initial states into itself. In the economic theory the notion of a fixed point is the principal mathematical tool for the analysis of economic equilibria. However, unlike in the other two fields, the re- presentation of the path to equilibrium has not received com- parabel emphasis in the research in economics of the last decade. In all three fields there is a need for methods to find the fixed points or rest points of the system if such exist, and also for each such point the "basin" (or region of stability), that is, the set of initial conditions from which the solution ultimately approaches a given fixed point. The ecological concept of resilience is closely related to the notion of the basin. As to the computation of fixed points, the methods most in use by the climatolo~istsdepend on tracing a path from a suit- able initial state through time until it stabilizes. For an- alyzing the sensitivity of climate to specified present or possible future effects of man's activity such a calculation has been made for both the unperturbed ("base-line") equilibrium and fro the perturbed alternative. G.I. Marchuk has developed a procedure that replaces the second calculation by an approxi- mation utilizing the result of the first and exploiting the bilinearity of the equations. One should ask further whether, if knowledge of the equilibrium without the details of the path has value in itself, one could also dispense with the first calculation, using methods and algorithms to approximate the unperturbed and perturbed equilibria directly. Among the methods that should be explored and tried out on moderate-size examples are any one or possible combinations of (a) Direct solution of some finite-difference approximation to the differential equations defining a steady state, (b) Fixed point algorithms such as have been developed in economics in the last 6 years (Scarf, Hansen, Ruhn, and others), (c) Extrapolation procedures such as those developed by Aitken,
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