A loosely defined concept, a Complex System presents a behavior nontrivially determined by the interactions between its parts. Complex systems often exhibit emergence phenomena, such as swarming and pattern formation. In such systems, nonlinear interactions can lead to memory and feedback mechanisms, self-organized criticality, and chaotic behavior. Network theory, systems biology, and adaptive/evolutionary systems also fall under this umbrella concept.
Questions tagged [complex-systems]
418 questions
45
votes
5 answers
Why are we sure that integrals of motion don't exist in a chaotic system?
The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$.
Why are we sure that no other, independent on $E$,…
Alexey Sokolik
- 2,453
37
votes
3 answers
Why can't many models be solved exactly?
I have been told that few models in statistical mechanics can be solved exactly. In general, is this because the solutions are too difficult to obtain, or is our mathematics not sufficiently advanced and we don't know how to solve many of those…
Daphne
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25
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9 answers
Are the physical structures in our sun of comparable complexity to those in the human brain?
The writings of Rupert Sheldrake tend to provoke strong emotions, be they ridicule, curiosity, outrage, sympathy, disgust, or otherwise. While Physics SE is not an appropriate forum in which either to debunk or to promote his general worldview, it…
Tom Hosker
- 361
21
votes
3 answers
What do physicists mean by an "integrable system"?
The notion of "integrability" is everywhere in physics these days. It's a hot topic in high energy theory, atomic physics, and condensed matter. I hear the word at least once a week, and every time, I ask the speaker what precisely they mean by it.…
knzhou
- 107,105
20
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5 answers
What are some of the best books on complex systems and emergence?
I'm rather interested in getting my feet wet at the interface of complex systems and emergence. Can anybody give me references to some good books on these topics? I'm looking for very introductory technical books.
Joebevo
- 2,291
18
votes
3 answers
Why did these algae grow like this in the pool? Are these curves the gravitational equivalents of the bell curve?
My friend sent me these pictures of a pool that has been abandoned for a long time, and we are curious about the reason behind the peculiar growth of algae in this pattern. The needle-like towers of algae seem to resemble the mathematical equation…
Tripasect
- 318
17
votes
2 answers
Are there any known models with limit cycles in their RG flow?
The text-book presentation of the renormalization group (RG) leaves one with the impression that all systems will eventually flow to a fixed point. This is somewhat enforced by the phenomenological scaling hypothesis in the sense that we empirically…
S.G.
- 587
17
votes
1 answer
Chaos and integrability in classical mechanics
An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any examples of non-integrable systems that are not…
AngusTheMan
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17
votes
1 answer
The natural metric of a phase space and the Lyapunov exponent
For me, it seems that there is no apparent metric on a phase space of a dynamical system. Of course one can naively define an Euclidean metric on it, but it seems that this metric has not much to do with the peculiar features of a phase space.
But…
Jiang-min Zhang
- 4,262
17
votes
4 answers
What creates the chaotic motion on a double pendulum?
As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random?
I'm just a high school kid. So, try to make answers…
Vaishnavi
- 1,107
16
votes
2 answers
What is the definition of a quantum integrable model?
What is the definition of a quantum integrable model?
To be specific: given a quantum Hamiltonian, what makes it integrable?
lagoa
- 321
15
votes
2 answers
Hamiltonian or not?
Is there a way to know if a system described by a known equation of motion admits a Hamiltonian function? Take for example
$$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$
where $\omega_i$ are constants.
How can I know if…
Martino
- 3,350
15
votes
2 answers
Which areas in physics overlap with those of social network theory for the analysis of the graphs?
I am studying social networks in terms of graph theory and linear algebra. I know that physicists have published and worked a lot in this field. This causes me to assume that there are sub-fields in physics which overlap in the essence of their…
Vass
- 782
15
votes
1 answer
Poincaré maps and interpretation
What are Poincaré maps and how to understand them?
Wikipedia says:
In mathematics, particularly in dynamical systems, a first recurrence
map or Poincaré map, named after Henri Poincaré, is the intersection
of a periodic orbit in the state…
user929304
- 4,910
14
votes
1 answer
Are all aperiodic systems chaotic?
So I understand that a chaotic system is a deterministic system, which produces aperiodic long-term behaviour and is hyper-sensitive to initial conditions.
So are all aperiodic systems chaotic? Are there counter-examples?
Julian Kurz
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