On the complexity of "simple" solids:
"The most fundamental question that one might be expected to answer is "why are there solids?"; That is, if we were given a large number of atoms of copper, why should they form themselves into the regular array that we know as a crystal of metallic copper?
We are ill-equipped to answer these questions in any other than a qualitative way, for they demand the solution of the many-body problem in one of its most difficult forms. We should have to consider the interactions between large numbers of identical copper nuclei-identical, that is, if we were fortunate enough to have an isotopically pure specimen- and even larger numbers of electrons. We should be able to omit neither the spins of the electrons nor the electric quadrupole moments of the nuclei. Provided we treated the problem with the methods of relativistic quantum mechanics, we could hope that the solution we obtained would be a good picture of the physical reality, and that we should then be able to predict all the properties of copper.
But, of course, such a task is impossible. Methods have not yet been developed that can find even the lowest-lying exact energy level of such a complex system. The best that we can do at present is to guess at the form the states will take, and then to try and calculate their energy."
Taylor, P. L., & Heinonen, O. (2002).
A quantum approach to condensed matter physics. Cambridge University Press.
On the complexity of brains & biology:
"In biological and cognitive systems, finally, myriad closed causal loops and networks with positive feedback obscure the relationships between cause and effect, leading both to the emergence of new dynamic entities with unanticipated properties and to chaotic interactions among them"
Scott, A. (2006). Physicalism, Reductionism, and Chaos. In J. A. Tuszynski (Ed.)
The Emerging Physics of Consciousness (pp. 171-191). Springer.
On the chaotic behavior of "simple" 1-dimensional systems like the pendulum:
"No analytical method is known for constructing the exact solution of this problem."
Greenspan, D. (2004).
N-Body Problems and Models. World Scientific
(see also
The Chaotic Pendulum)
For chaos in chemistry, see e.g.
Argoul, F., Arneodo, A., Richetti, P., Roux, J. C., & Swinney, H. L. (1987).
Chemical chaos: from hints to confirmation.
Accounts of Chemical Research,
20(12), 436-442.
On chaos in astrophysics:
"The three-body problem arises in many different contexts in nature. This book deals with the classical three-body problem,
the problem of motion of three celestial bodies under their mutual gravitational attraction...
At the suggestion of leading scientists, the King of Sweden Oscar II established a prize for the solution of the general three-body problem...Nobody was able to claim the prize for many years and finally it was awarded in 1889 to Poincar´e who was thought to have made the most progress...
Now that the orbits can be calculated quickly by computer,
it is quite obvious why this line of research could not lead to a real solution of the three-body problem: the orbits are good examples of chaos in nature, and deterministic series expansions are utterly unsuitable for their description."
(emphasis added)
Valtonen, M., & Karttunen, H. (2005).
The Three-Body Problem. Cambridge University Press.
Chaos Everywhere:
"One of the major concepts central to the deeper understanding of contemporary Physics is the concept of chaos.
It would not be an exaggeration to say that chaos is everywhere in Physics." (emphasis added)
Bolotin, Y., Tur, A., & Yanovsky, V. (2009).
Chaos: Concepts, Control and Constructive use (
Understanding Complex Systems). Springer.