Chaos in Nonlinear Dynamics and the Logistic Substitution Model

A rekindled appreciation of an old cliche has touched off a flurry of activity in the field of nonlinear dynamics lately. The truism that nonlinearities often lead to wild and exotic behavior has been known for a long time, but only recently has it been studied carefully, and the discoveries are startling and profound. The simplest equation illuminating these features is the logistic equation (in discrete form), which has a long history of application to growth phenomena in biology and population dynamics. This equation is also the basis for the logistic substitution model developed at IIASA by Cesare Marchetti and Nebojsa Nakicenovic. This model is a highly effective tool for modeling the dynamics of economic market substitution, and has been extensively applied to primary energy markets. This Working Paper begins with a brief review of the recent developments in nonlinear dynamics, followed by a study of the implications that these phenomena have for the logistic substitution model. The key finding is that only highly unrealistic parameter values can induce chaotic behavior in this model

A Critical Appraisal of the IIASA Energy Scenarios

This paper presents some disturbing findings about one aspect of a major scientific study of the world's energy system. The final report of the seven-year study was published in 1981, entitled "Energy in a "Finite World." Although the study claims to provide an objective, factual analysis for political decision making, some of the major conclusions are not scientifically justified. Principal results include detailed projections of the world's energy supply systems for the coming half-century. These were produced from an apparently sophisticated set of iterative computer models. However, the models are found to be largely trivial, because their final outputs are nearly identical to their inputs, which are arbitrary, unsubstantiated assumptions. Furthermore, despite claims of robustness, the energy supply projections are found to be highly sensitive to minor variations in data that are well known to be uncertain. The sizable contribution from the nuclear fast breeder reactor (FBR), is due to a 2% cost advantage that is introduced 25 years from now. Since future energy costs are highly uncertain, cost-minimization linear programming models are unsuitable for describing robust energy supply futures. In addition to these analytic findings, some aspects of the work are improperly presented in the published documentation. In one case, the important role of the FBR is traced to undocumented input data. Frequent statements that the computer models formed an iterative loop are contradicted elsewhere. Preliminary work that revealed serious difficulties with robustness is not cited, and standard sensitivity tests are not included. Nevertheless, several "robust" conclusions have been drawn from the projections and widely publicized. One of these implies that nuclear power plants must be built at the average rate of one plant every few days for the next 50 years. The overall conclusion in this paper is that the energy supply projections are opinion, rather than credible scientific analysis, and they therefore cannot be relied upon by policy makers seeking a genuine understanding of the energy choices for tomorrow

Cost overruns and financial risk in the construction of nuclear power reactors: a critical appraisal

Lovering and colleagues attempt to advance understanding of construction cost escalation risks inherent in building nuclear reactors and power plants, a laudable goal. Although we appreciate their focus on capital cost increases and overruns, we maintain in this critical appraisal that their study conceptualizes cost issues in a limiting way. Methodological choices in treating different cost categories by the authors mean that their conclusions are more narrowly applicable than they describe. We also argue that their study is factually incorrect in its criticism of the previous peer-reviewed literature. Earlier work, for instance, has compared historical construction costs for nuclear reactors with other energy sources, in many countries, and extending over several decades. Lastly, in failing to be transparent about the limitations of their own work, Lovering et al. have recourse to a selective choice of data, unbalanced analysis, and biased interpretation

MULTIRATE INTEGRATION OF TWO-TIME-SCALE DYNAMIC SYSTEMS

Simulation of large physical systems often leads to initial value problems in which some of the solution components contain high frequency oscillations and/or fast transients, while the remaining solution components are relatively slowly varying. Such a system is referred to as two-time-scale (TTS), which is a partial generalization of the concept of stiffness. When using conventional numerical techniques for integration of TTS systems, the rapidly varying components dictate the use of small stepsizes, with the result that the slowly varying components are integrated very inefficiently. This could mean that the computer time required for integration is excessive. To overcome this difficulty, the system is partitioned into "fast" and "slow" subsystems, containing the rapidly and slowly varying components of the solution respectively. Integration is then performed using small stepsizes for the fast subsystem and relatively large stepsizes for the slow subsystem. This is referred to as multirate integration, and it can lead to substantial savings in computer time required for integration of large systems having relatively few fast solution components. This study is devoted to multirate integration of TTS initial value problems which are partitioned into fast and slow subsystems. Techniques for partitioning are not considered here. Multirate integration algorithms based on explicit Runge-Kutta (RK) methods are developed. Such algorithms require a means for communication between the subsystems. Internally embedded RK methods are introduced to aid in computing interpolated values of the slow variables, which are supplied to the fast subsystem. The use of averaging in the fast subsystem is discussed in connection with communication from the fast to the slow subsystem. Theoretical support for this is presented in a special case. A proof of convergence is given for a multirate algorithm based on Euler's method. Absolute stability of this algorithm is also discussed. Four multirate integration routines are presented. Two of these are based on a fixed-step fourth order RK method, and one is based on the variable step Runge-Kutta-Merson scheme. The performance of these routines is compared to that of several other integration schemes, including Gear's method and Hindmarsh's EPISODE package. For this purpose, both linear and nonlinear examples are presented. It is found that multirate techniques show promise for linear systems having eigenvalues near the imaginary axis. Such systems are known to present difficulty for Gear's method and EPISODE. A nonlinear TTS model of an autopilot is presented. The variable step multirate routine is found to be substantially more efficient for this example than any other method tested. Preliminary results are also included for a pressurized water reactor model. Indications are that multirate techniques may prove fruitful for this model. Lastly, an investigation of the effects of the step-size ratio (between subsystems) is included. In addition, several suggestions for further work are given, including the possibility of using multistep methods for integration of the slow subsystem

Breathing New Life into Social Transformation: Holotropic Breathwork for social change leaders

Key experiences and insights are presented from the application of Holotropic Breathwork to innovative leadership for social, political, and environmental change. Inspiring anecdotes and breakthroughs are recounted from prominent social change leaders, showing how their breathwork experience impacted their leadership for social transformation