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pertubations do not drastically alter the behaviour. For a hurricain to happen the state would need to pass through a mathematical catastrophy to reach a very diferent state charterised by a hurricain. Probably an example of a cusp catastrophy where the state changes from the upper sheat to the lower sheat. --
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Another good addition would be a definition of a mathematical catastrophe at the top of the article. Wiktionary defines "catastrophe (mathematics)" as "a type of bifurcation, where a system shifts between two stable states." This change would prevent anyone from being confused by the figure of a fold
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As far as I know, V = x³ + ax has two unstable fix points and one stable fix point at negative a, which gives a subcritical bifurcation. V = ax - x³ would have been more illustrative because it has two stable fix points and one unstable fix point at positive a which provides pitchfork bifurkation as
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What this article needs, simply, is someone who understands catastrophe theory and at the same time has the ability to convey abstractions in clear, strong, analogical, and metaphorical ways that any intelligent person can understand. It is an error to claim that a reader should have mathematical or
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As a non-mathematician, I was able to understand virtually nothing in this article. A simple google search reveals numerous webpages that are able to conceptually explain what the theory means and why it exists. While this is a common problem in almost all mathematics articles, catastrophe theory is
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Seems like a good idea to me. Perhaps the best way to do it in this article would be to say that both catastrophe theory and chaos theory deal with physical systems where small changes in the parameters cause large changes in their behavior, but that catastrophe theory is limited to systems whose
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I don't think you need to be a full-fledged mathematician to understand this article, but you should have studied mathematics as far as calculus and physics as far as potential energy. It is unreasonable to expect that you should be able to understand an advanced mathematical topic without having
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The transition that you are talking about is a Hopf bifurcation, in this case from laminar flow to circular flow in a simple model of the flow of a compressible fluid. It can be revealed as an example of a cusp catastrophe if you look at 2-dimensional fluid flow, and convert to polar coordinates.
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What's the name of the theorem which prove "If the potential function depends on two or fewer active variables, and four (resp. five) or fewer active parameters, then there are only seven (resp. eleven) generic structures for these bifurcation geometries"? It seems to be an important theorem but
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physical qualifications to a certain level to grasp catastrophe theory in a useful and meaningful way. It is
Knowledge's purpose to explain to and inform the general reader. If the Encyclopaedia Britannica can provide an approachable explanation of the theory, Knowledge can and must do the same.
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Years ago, I read an explanation of catastrophe theory for the layperson; I also saw a clearminded documentary about it. I was able to follow these explanations well enough, without any mathematics involved. Regrettably, in the intervening years, my grasp of the theory has grown hazy, but it was
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Without understanding phase space and nonlinear system it is impossible to understand. Theory of
Catastrophy is not a branch of bifurcation study. hopf bifurcation is also just one type of phase space shifting. therefore this article needs to be revised, may be rewrite. Theory of Catastrophy is
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Interestingly, there is some mathematical connection with catastrophy modelling. In the terms that the catastrophic events happen when a situation moves away from a stable state. Consider a weather system, it normally is in a stable state (say with wind speeds in a certain range) and slight
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I reject the obfuscation by Robert O'Rourke, above. Its claim that readers need technical qualifications to understand the theory is wrong, and conflicts with the purpose of an encyclopaedia, which is not written for experts, but for the general, interested
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The crucial distinction between chaos theory and catastrophe theory is that chaos theory deals with changes in initial conditions, while bifurcation theory, which includes catastrophe theory, deals with changes in parameters. The
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Brian Greene wrote a book a few years ago explaining string theory for the lay reader. The theory is highly abstract and mathematical in origin, yet it was explained in language the non-physicist and non-mathematician could
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Needs more clear mathematical explanations; and a separate section on philosophical implications. Right now the philosophy is mixed up with the mathematics, making it quite confusing, and less useful to mathematicians.
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one distinguish chaos from catastrophe theory. Whilst I'm not an expert in either, I feel that at a bare minimum this article, which is concerned with the less known of the two, ought to explain why the two are
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The svg images used in this page are pretty messed up; the legends have a lot of overlapping text. I'm viewing using: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.9.1.3) Gecko/20090824 Firefox/3.5.3 --
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A famous person, whose name escapes me now, said that if you truly understand something properly, you can explain it clearly to an intelligent audience. I have often seen it done, beginning at school.
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Nevertheless, I think it is quite proper to separate catastrophe modeling as it is understood in the insurance business from the mathematical treatment of nonlinear dynamical systems —
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totally different branch of nonlinear dynamics. germs of catastrophies and phase space shiftings are related to interactions between parameters and variables.
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Einstein wrote a book for laypeople explaining the special and general theories of relativity. The book was in plain
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Television documentaries have often been used to explain difficult material in common terms, to convey a sense of what is going on.
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also discussed in non-mathematics circles and needs to be generally accesible in addition to providing more specific information.
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Congrats, you just stepped all over historical geographic catastrophic searching
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https://archive.is/20121211093251/http://lagrange.physics.drexel.edu/flash/zcm/
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behavior is determined by a potential function in one of the forms given here.
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catastrophe, which the caption describes as a bifurcation.
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for additional information. I made the following changes:
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a prerequisite for cusp catastrophe: V = b + ax - x³.
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is used in both chaos theory and bifurcation theory.
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