"[…] a pink (or white, or brown) noise is the very paradigm of a statistically self-similar process. Phenomena whose power spectra are homogeneous power functions lack inherent time and frequency scales; they are scale-free. There is no characteristic time or frequency -whatever happens in one time or frequency range happens on all time or frequency scales. If such noises are recorded on magnetic tape and played back at various speeds, they sound the same […]" (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"All physical objects that are 'self-similar' have limited self-similarity - just as there are no perfectly periodic functions, in the mathematical sense, in the real world: most oscillations have a beginning and an end (with the possible exception of our universe, if it is closed and begins a new life cycle after every 'big crunch' […]. Nevertheless, self-similarity is a useful abstraction, just as periodicity is one of the most useful concepts in the sciences, any finite extent notwithstanding." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"Percolation is a widespread paradigm. Percolation theory can therefore illuminate a great many seemingly diverse situations. Because of its basically geometric character, it facilitates the analysis of intricate patterns and textures without needless physical complications. And the self-similarity that prevails at critical points permits profitably mining the connection with scaling and fractals." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"[…] power laws, with integer or fractional exponents, are one of the most fertile fields and abundant sources of self-similarity." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"The only prerequisite for a self-similar law to prevail in a given size range is the absence of an inherent size scale." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale or size, is an attribute of many laws of nature and innumerable phenomena in the world around us. Self-similarity is, in fact, one of the decisive symmetries that shape our universe and our efforts to comprehend it." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"[…] the world is not complete chaos. Strange attractors often do have structure: like the Sierpinski gasket, they are self-similar or approximately so. And they have fractal dimensions that hold important clues for our attempts to understand chaotic systems such as the weather." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic" (that is fixed) rules" (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order" (a pattern) within disorder" (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)
"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"What is renormalization? First of all, if scaling is present we can go to smaller scales and get exactly the same result. In a sense we are looking at the system with a microscope of increasing power. If you take the limit of such a process you get a stability that is not otherwise present. In short, in the renormalized system, the self-similarity is exact, not approximate as it usually is. So renormalization gives stability and exactness." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"The self-similarity of fractal structures implies that there is some redundancy because of the repetition of details at all scales. Even though some of these structures may appear to teeter on the edge of randomness, they actually represent complex systems at the interface of order and disorder. " (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)
"The self-similarity on different scales arises because growth often involves iteration of simple, discrete processes (e.g. branching). These repetitive processes can often be summarized as sets of simple rules." (David G Green, 2000)
"Fractals are self-similar objects. However, not every self-similar object is a fractal, with a scale-free form distribution. If we put identical cubes on top of each other, we get a self-similar object. However, this object will not have scale-free statistics: since it has only one measure of rectangular forms, it is single-scaled. We need a growing number of smaller and smaller self-similar objects to satisfy the scale-free distribution." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"The concept of bifurcation, present in the context of non-linear dynamic systems and theory of chaos, refers to the transition between two dynamic modalities qualitatively distinct; both of them are exhibited by the same dynamic system, and the transition (bifurcation) is promoted by the change in value of a relevant numeric parameter of such system. Such parameter is named 'bifurcation parameter', and in highly non-linear dynamic systems, its change can produce a large number of bifurcations between distinct dynamic modalities, with self-similarity and fractal structure. In many of these systems, we have a cascade of numberless bifurcations, culminating with the production of chaotic dynamics." (Emilio Del-Moral-Hernandez, "Chaotic Neural Networks", Encyclopedia of Artificial Intelligence, 2009)
"In the telephone system a century ago, messages dispersed across the network in a pattern that mathematicians associate with randomness. But in the last decade, the flow of bits has become statistically more similar to the patterns found in self-organized systems. For one thing, the global network exhibits self-similarity, also known as a fractal pattern. We see this kind of fractal pattern in the way the jagged outline of tree branches look similar no matter whether we look at them up close or far away. Today messages disperse through the global telecommunications system in the fractal pattern of self-organization." (Kevin Kelly, "What Technology Wants", 2010)
"Cyberneticists argue that positive feedback may be useful, but it is inherently unstable, capable of causing loss of control and runaway. A higher level of control must therefore be imposed upon any positive feedback mechanism: self-stabilising properties of a negative feedback loop constrain the explosive tendencies of positive feedback. This is the starting point of our journey to explore the role of cybernetics in the control of biological growth. That is the assumption that the evolution of self-limitation has been an absolute necessity for life forms with exponential growth." (Tony Stebbing, "A Cybernetic View of Biological Growth: The Maia Hypothesis", 2011)
"Laws of complexity hold universally across hierarchical scales (scalar, self-similarity) and are not influenced by the detailed behavior of constituent parts." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)
"Fractals are different from chaos. Fractals are self-similar geometric objects, while chaos is a type of deterministic yet unpredictable dynamical behavior. Nevertheless, the two ideas or areas of study have several interesting and important links. Fractal objects at first blush seem intricate and complex. However, they are often the product of very simple dynamical systems. So the two areas of study - chaos and fractals - are naturally paired, even though they are distinct concepts." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"The study of chaos shows that simple systems can exhibit complex and unpredictable behavior. This realization both suggests limits on our ability to predict certain phenomena and that complex behavior may have a simple explanation. Fractals give scientists a simple and concise way to qualitatively and quantitatively understand self-similar objects or phenomena. More generally, the study of chaos and fractals hold many fun surprises; it challenges one’s intuition about simplicity and complexity, order and disorder." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012
"Chaos theory is a branch of mathematics focusing on the study of chaos - dynamical systems whose random states of disorder and irregularities are governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of complex, chaotic systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is a sensitive dependence on initial conditions)." (Nima Norouzi, "Criminal Policy, Security, and Justice in the Time of COVID-19", 2022)
No comments:
Post a Comment