Definitions of Resilience

During the past year, several research efforts at IIASA have tried to develop a precise mathematical definition of Holling's very general and rich resilience concept. This paper develops a mathematical language for resilience, using the terms and concepts of differential topology. Central to this treatment is the division of the state space of a system- into basins, each containing an attractor. The translation of Holling's concept into this language reads roughly as follows: a system is resilient if, after perturbation, it will still tend to the same attractor as before (or to an "only slightly changed" attractor). The reason for treating changes of state variables and changes of parameters separately is explained. All resilience measures conceived up to now, as defined within this language, are listed as well. The various definitions of resilience are then compared to the well-known concepts of structural stability and of Thorn's catastrophe theory. Finally, the author indicates some -- in his opinion -- important directions for further research into the general resilience concept

Stable Manifolds and Separatrices

In the last weeks, many people at IIASA have been concerned with the concept of resilience, after initial work in this direction by C.S. Holling. He talked about resilience as a "measure of the ability of systems to absorb change of state variables, driving variables and parameters and still persist". In my opinion, resilience, thus defined, is directly related to a) the basins of attraction of the system and b) their changes under variations of external parameters. i.e. variations in the time-evolution laws of the system. If one changes the state variables (= the point in phase space describing the system at a given time) but still remains within the same basin, the asymptotic behavior of the system will not change. (This can be made rigorous by a recent theorem of Ruelle and Bowen.) If the dynamics of the system are changed a little, the boundaries of the basins (the separatrices) might move only a little and the structure of the attractors within them might remain the same, such that a point would still trace out a trajectory of the same nature as before the change under the new dynamic laws. On the contrary crossing a separatrix will lead to drastic and catastrophic changes in the long-time behavior of the system, as illustrated in the recent model of Haefele. Haefele proposed therefore that the distance from the next separatrix should be put into a measure of resilience. It is the purpose of this paper to discuss the nature of these separatrices and illustrate their properties in a simple model

Economic Evolutions and Their Resilience: A Model

The model treated in this paper is the latest "societal model" of IIASA's Energy Systems Program. After a brief resume of historical developments, the structure of the model is described. Its structural properties are investigated, and the dynamics inside and outside the slack-free region are determined. Then two parameter sets -- one for a typical developed country and one for a typical less developed country -- are chosen, and scenario variables are introduced. Using this approach, the authors study the interaction between a developed country and a less developed one under the influence of different oil price levels

Analysis and Computation of Equilibria and Regions of Stability, With Applications in Chemistry, Climatology, Ecology, Economics

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

Thermal Radiation and Entropy

In line with the "negentropy approach" currently pursued at IIASA, the thermodynamic properties of electromagnetic radiation are described. We use the photon description of the electromagnetic field and clarify the question of the entropy content of radiation. Entropy is found not in the spectral distributions but in the incoherence of individual modes. As an illustration, we calculate the efficiency of a highly idealized photocell

New Societal Equations

In a recent paper, W. Haefele established a number of phenomenological equations describing the behavior of a model society. The state variables of this model society were gross national product, population, energy consumption and risk acceptance. In this paper, the state of the discussion within the IIASA energy project at the time being shall be fixed. Several new sets of equations will be established which extend the set given by Haefele and Manne in the following sense: capital will be included as another state variable; a finite asymptotic population will be assumed; there are several primary energy sources (fossil and nuclear). We will outline three different approaches, namely: (1) an approach where a complete system of equations, including one primary energy source, is established and where the topological features (separatrices, fix points, etc.) can be studied in detail; (2) a "control theoretical approach," including two primary energy sources, where we limit the number of state variables in such a way that there remains only one "control variable" subject to optimization with respect to an appropriate objective function; and (3) a "linear programming approach" where we introduce the same number of energy supply variables as in the work of Haefele and Manne, and where we optimize the (more than one) free state variables according to different objective functions. The total energy demand is either taken from a model of the first kind or is assumed to be an independent control variable subject to optimization