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Welcome to Chaos General!

This thread is to discuss chaos theory, dynamical systems, many-body systems, nonlinear dynamics, fractals… You name it!

Untill recently, classical mechanics was thought to be the realm of determinism, opposed to quantum mechanics where the only certainty is uncertainty. Chaos theory shows us that we don't have to travel to the microscopic to find rich, unpredictable behaviour.

Prerequisites for studying chaos theory:
Since chaos theory is a branch of dynamical systems, a decent maths preparation is necessary. Calculus and also some abstract algebra are needed to undertake the study of chaotic dynamics.
A basic course on ODE and mathematical physics is also necessary. A classic would be Arnol'd's book.
Some knowledge in measure theory helps, there are heaps of different resources and approaches to it, for example Folland's book, although it is quite advanced.
Files and more resources in the next posts!

The world of dynamical systems is large, and chaos is a part of it. My personal interest in chaos theory is at a crossroad between chaos and what is not chaos, namely, integrability. This area is called KAM theory (Kolmogorov-Arnol'd-Moser). It's a fascinating branch of mathematical physics that shows what is lost and what is not lost after perturbations of integrable systems. A lot of natural systems are in a balance between chaos and order, so this theory gives us deep insight on such phenomena. Also, it has the best fractals!

There is an article on Lainzine 4 about an elementary chaotic system, and there will be followups in the next Lainzines. What do you think of it?


File: 1497008430938.pdf (18.87 MB, (Graduate Texts in Mathema….pdf)

Some books - I


File: 1497009300171-0.epub (5.74 MB, Gerald_B._Folland_Real_An….epub)

Some books - II


File: 1497017334945.pdf (2.25 MB, wolfram_statmechCA.pdf)

Statistical mechanics and cellular automata


File: 1497095742785.pdf (11.92 MB, Heisenberg-PhysicsPhilosop….pdf)

There is a good discussion to be had on point of philosophical implications. We know from history that all advancements in science have followed with philosophical systems that act as sort of meta-proof or build on methodology of that particular science - i.e. Hobbs builds hes political philosophy on newtonian mechanics which can be seen from his fundamental principle of inertia which he adopted and applied it to social science. Who would have thought that we would have political philosophy (that is considered basis for modern liberal society) founded on new laws discovered in physics?

In quantum physics we have physicist fight over philosophical systems that quantum theory implicates. Bohr and Heisenberg have idealist philosophical outlook rejecting positivist, deterministic explanation of nature, for them everything is phenomena going that far that they claim that object of observation doesn't exist until we observe it.

Most interesting in my opinion dialectical materialism. Marx builds it on Hegels dialectics which reject concepts of isolated causality in classical mechanics. Bohm expresses similar criticism: Thus, there is no real case known of a set of perfect one-to-one causal relationships that could in principle make possible predictions of unlimited precision, without the need to take into account qualitatively new sets of causal factors existing outside the system of interest or at other levels.

Problems of isolated cause and effect and determinism that challenged classical mechanics in its contradiction to quantum mechanics give legitimate consideration in Hegels philosophical system. Therefor there are plenty of marxist philosophers that make effort to merge chaos theory with dialectical materialism. I guess that could be said for classical hegelians.

i.e. these are some accounts on hegel and chaos theory

It is easy to identify cause and effect in isolated cases, as when one hits a ball with a bat. But in a wider sense, the notion of causality becomes far more complicated. Individual causes and effects become lost in a vast ocean of interaction, where cause becomes transformed into effect and vice versa. Just try tracing back even the simplest event to its "ultimate causes" and you will see that eternity will not be long enough to do it.

The impossibility of establishing a "final cause" has led some people to abandon the idea of cause altogether. Everything is considered to be random and accidental. In the 20th century this position has been adopted, at least in theory, by a large number of scientists on the basis of an incorrect interpretation of the results of quantum physics, particularly the philosophical positions of Heisenberg. Hegel answered these arguments in advance, when he explained the dialectical relation between accident and necessity.

Hegel explains that there is no such thing as causality in the sense of an isolated cause and effect. Every effect has a counter-effect, and every action has a counter-action. The idea of an isolated cause and effect is an abstraction taken from classical Newtonian physics, which Hegel was highly critical of, although it enjoyed tremendous prestige at that time. Here again, Hegel was in advance of his time. Instead of the action-reaction of mechanics, he advanced the notion of Reciprocity, of universal interaction. Everything influences everything else, and is in turn, influenced and determined by everything. Hegel thus re-introduced the concept of accident which had been rigorously derezzed from science by the mechanist philosophy of Newton and Laplace.

[quote] At first sight, we seem to be lost in a vast number of accidents. But this confusion is only apparent. The accidental phenomena which constantly flash in and out of existence, like the waves on the face of an ocean, express a deeper process, which is not accidental but necessary. At a decisive point, this necessity reveals itself through accident. This idea of the dialectical unity of necessity and accident may seem strange, but it is strikingly confirmed by a whole series of observations from the most varied fields of science and society. The mechanism of natural selection in the theory of evolution is the best-known example. But there are many others. In the last few years, there have been many discoveries in the field of chaos and complexity theory which precisely detail how "order arises out of chaos," which is exactly what Hegel worked out one and a half centuries earlier. [quote]

Sources on some articles for marxism and quantum mehanics:

Heisenberg physics and philosophy in attachment


Thank you for contributing, I thought this thread would die of neglect.

What you say is very interesting. Indeed, certain advances in maths, for example category theory, have had great rebound on philosophy, in lots of different areas (logic, but also ontology!). But I'd develop this on a more maths-specific thread.

I would like to stress on the fact that chaos is not only quantum. Actually, since quantum physics has a hard foundations problem, more mathematically-flavored accounts of chaos are realized in the realm of classical mechanics.

But the determinism hard coded in classical mechanics is nothing to be against. In fact, chaotic systems are, in a certain sense, ultradeterministic.

What I mean with this is that you may chose a property you want a solution to realize, and a chaotic system always has at least one such solution. So the dynamics in the chaotic realm is so rich it is akin to random processes: you may flip a coin and decide from the outcome that some solution should have some property, and the chaotic system will have such solution. This is a twisted version of determinism, where the possibilities, whose evolution along their "worldlines" is perfectly causal, are so many that the result is non-causal, entropic.

To expand, there always is a sort of "microdeterminism" in classical systems: the dynamics of one single point are totally determined by its equations of motion, well in contrast to the equations that govern quantum mechanics, which determine probability distributions.
So determinism is still there, but to really experience it, you must restrict yourself to an insignificant part of the system. Globally, or even locally (in the topology sense), determinism is not true: the evolution of a set of points is not guaranteed to mimic the evolution of a single point, and this is true also for non-chaotic systems (Gromov's non-squeezing theorem)!
For example we may take the single points of the set very close (small initial uncertainty), but the dynamics can separate them exponentially in time. As a result, the small initial uncertainty is propagated, deterninistically, to an enormous uncertainty at a further time.

All these considerations can be made without taking in account the more recent concepts of physics, especially (quantum) field theory, of which your quotations are surprisingly suggestive.
Even the emergence of order out of chaos can be observed in far simpler systems: just think of that astonishing piece of software that is the game of life!


I just found this thread. It is a very interesting topic.
I'm going to study a course on ODEs next semester, so this is very good motivation.
I want to understand what the 'edge of chaos' means to physics.
Thank you for this intro Opie.


In regards to this post in particular: as above so below.

This all looks incredibly fascinating. I'll be reading the books posted here in the coming months for sure. Hopefully I'll have something to add when I've delved a little deeper into the supporting math and theory.

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