"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)
"Geometry gives us a way of turning numbers into pictures." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"General relativity, one of the most famous theories, is formulated in terms of a nonlinear equation. This makes us wonder if some of the phenomena described by general relativity, namely black holes, objects orbiting black holes, and even the universe itself, can become chaotic under certain circumstances. [...] The problem is the equation itself, namely the equation of general relativity; it is so complex that the most general solution has never been obtained. It has, of course, been solved for many simple systems; if the system has considerable symmetry (e.g., it is spherical) the equation reduces to a number of ordinary equations that can be solved, but chaos does not occur in these cases. In more realistic cases—situations that actually occur in nature - chaos may occur, but the equations are either unsolvable or very difficult to solve. This presents a dilemma. If we try to model the system using many simplifications it won't exhibit chaos, but if we model it realistically we can't solve it." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"Most of us use the word 'chaos' rather loosely to represent anything that occurs randomly, so it is natural to think that the motion described by the erratic pendulum above is completely random. Not so. The scientific definition of chaos is different from the one you may be used to in that it has an element of determinism in it. This might seem strange, as determinism and chaos are opposites of one another, but oddly enough they are also compatible." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"Noise is a problem in most signals. [...] It's easy to see that noise is random; it fluctuates erratically with no pattern." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"Nonlinear equations have many properties and difficulties that linear equations do not have. If a nonlinear equation, for example, describes a collection of objects and we want to find the collective effect of these objects, we cannot merely add their individual effects. Because source and effect are independent of one another, their sum does not give the overall effect. With nonlinear phenomena there are strong interactions between the bodies and the contribution from each is modified by the others. Mathematically this means if we change a variable on one side of the equation it doesn't cause a proportional change in the variable on the other side." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"Nonlinearity is important because it can lead to chaos. That's not to say that we get chaos all the time with nonlinear equations; in practice it only occurs under certain conditions." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"One of the major problems with general relativity is that it is not a theory in the usual sense. In the case of most theories you have a stable background, or frame of reference, and you look for solutions within it. In general relativity the solution is the background - the space-time - and it is not necessarily stable," (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"One of the reasons we deal with the pendulum is that it is easy to plot its motion in phase space. If the amplitude is small, it's a two-dimensional problem, so all we need to specify it completely is its position and its velocity. We can make a two-dimensional plot with one axis (the horizontal), position, and the other (the vertical), velocity." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"The problems associated with the initial singularity of the universe bring us to what is called the theory of everything. It is an all-encompassing theory that would completely explain me origin of the universe and everything in it. It would bring together general relativity and quantum mechanics, and explain everything there is to know about the elementary particles of the universe, and the four basic forces of nature (gravitational, electromagnetic, weak, and strong nuclear forces). Furthermore, it would explain the basic laws of nature and the fundamental constants of nature such as the speed of light and Planck's constant." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"The reason general relativity cannot be unified with electromagnetic theory seems to be related to its nonlinearity. To unify the two fields properly we have to construct a "quantized" version of relativity; in other words, we have to quantize it, and so far no one has." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"What chaos implies is not catastrophes, but rather our inability to make long-range predictions [...]" (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"What is chaos? Everyone has an impression of what the word means, but scientifically chaos is more than random behavior, lack of control, or complete disorder. [...] Scientifically, chaos is defined as extreme sensitivity to initial conditions. If a system is chaotic, when you change the initial state of the system by a tiny amount you change its future significantly." (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)
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