"As with subtle bifurcations, catastrophes also involve a control parameter. When the value of that parameter is below a bifurcation point, the system is dominated by one attractor. When the value of that parameter is above the bifurcation point, another attractor dominates. Thus the fundamental characteristic of a catastrophe is the sudden disappearance of one attractor and its basin, combined with the dominant emergence of another attractor. Any type of attractor static, periodic, or chaotic can be involved in this. Elementary catastrophe theory involves static attractors, such as points. Because multidimensional surfaces can also attract (together with attracting points on these surfaces), we refer to them more generally as attracting hypersurfaces, limit sets, or simply attractors." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Chaos and catastrophe theories are among the most interesting recent developments in nonlinear modeling, and both have captured the interests of scientists in many disciplines. It is only natural that social scientists should be concerned with these theories. Linear statistical models have proven very useful in a great deal of social scientific empirical analyses, as is evidenced by how widely these models have been used for a number of decades. However, there is no apparent reason, intuitive or otherwise, as to why human behavior should be more linear than the behavior of other things, living and nonliving. Thus an intellectual movement toward nonlinear models is an appropriate evolutionary movement in social scientific thinking, if for no other reason than to expand our paradigmatic boundaries by encouraging greater flexibility in our algebraic specifications of all aspects of human life." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"[...] chaos and catastrophe theories per se address behavioral phenomena that are consequences of two general types of nonlinear dynamic behavior. In the most elementary of behavioral terms, chaotic phenomena are a class of deterministic processes that seem to mimic random or stochastic dynamics. Catastrophe phenomena, on the other hand, are a class of dynamic processes that exhibit a sudden and large scale change in at least one variable in correspondence with relatively small changes in other variables or, in some cases, parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Chaos and catastrophe theories directly address the social scientists' need to understand classes of nonlinear complexities that are certain to appear in social phenomena. The probabilistic properties of many chaos and catastrophe models are simply not known, and there is little likelihood that general procedures will be developed soon to alleviate the difficulties inherent with probabilistic approaches in such complicated settings." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Chaos has three fundamental characteristics. They are (a) irregular periodicity, (b) sensitivity to initial conditions, and (c) a lack of predictability. These characteristics interact within any one chaotic setting to produce highly complex nonlinear variable trajectories." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Fundamental to catastrophe theory is the idea of a bifurcation. A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed. This occurs when an attractor in the system changes in response to change in the value of a parameter. A catastrophe is one type of bifurcation. The broader framework within which catastrophes are located is called dynamical bifurcation theory." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"In addition to dimensionality requirements, chaos can occur only in nonlinear situations. In multidimensional settings, this means that at least one term in one equation must be nonlinear while also involving several of the variables. With all linear models, solutions can be expressed as combinations of regular and linear periodic processes, but nonlinearities in a model allow for instabilities in such periodic solutions within certain value ranges for some of the parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"In classical catastrophe theory, the various attracting static hypersurfaces are actually connected. However, there are portions of the overall surface that are unstable, and thus repelling. Thus nearby trajectories tend to 'fly' quickly past these unstable regions as they move from one stable area to another. It is this relatively rapid snapping movement that is typical of nearly all catastrophe phenomena." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"In its essence, chaos is an irregular oscillatory process. Because chaos is a subset of the more general classification of oscillatory dynamics, it is useful before venturing into chaos to review briefly the extent to which regular oscillatory processes influence human behavior." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Small changes in the initial conditions in a chaotic system produce dramatically different evolutionary histories. It is because of this sensitivity to initial conditions that chaotic systems are inherently unpredictable. To predict a future state of a system, one has to be able to rely on numerical calculations and initial measurements of the state variables. Yet slight errors in measurement combined with extremely small computational errors (from roundoff or truncation) make prediction impossible from a practical perspective. Moreover, small initial errors in prediction grow exponentially in chaotic systems as the trajectories evolve. Thus, theoretically, prediction may be possible with some chaotic processes if one is interested only in the movement between two relatively close points on a trajectory. When longer time intervals are involved, the situation becomes hopeless."(Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"The dimensionality and nonlinearity requirements of chaos do not guarantee its appearance. At best, these conditions allow it to occur, and even then under limited conditions relating to particular parameter values. But this does not imply that chaos is rare in the real world. Indeed, discoveries are being made constantly of either the clearly identifiable or arguably persuasive appearance of chaos. Most of these discoveries are being made with regard to physical systems, but the lack of similar discoveries involving human behavior is almost certainly due to the still developing nature of nonlinear analyses in the social sciences rather than the absence of chaos in the human setting." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"Thus my advice to model builders in the social sciences is to think in terms of social processes that might require algebraic structures that could yield catastrophe potential. Build models from an intimate knowledge of these processes while remaining aware of the algebraic requirements for catastrophes. The art of nonlinear model building is a delicate dance with two partners, algebraic forms that produce known effects and a substantive understanding of the complexities of social phenomena. Coordinating the two by mixing structure to match complexity is the job of the theorist, and it is the single greatest creative challenge of any researcher." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
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