I am a physics undergrad interested in stuff like dynamical systems, chaos theory etc. Is there ongoing research in these fields? I am talking about pure research and not applications to things like weather etc? I hope this question is appropriate for Physics SE. I asked this question, because I browsed through the websites of the physics departments of a few renowned universities (MIT, Princeton, Caltech) etc, but nowhere it mentioned research in these areas.
Yes, there is investigation.
Some random names on the field (more on the physics side, NO specific order): Carl Dettmann, Tamás Tél, Ott, Ying-Cheng Lai, Adilson Motter, Celso Grebogi, Holger Kantz, Alessandro Moura, Eduardo G. Altmann, etc, etc, etc.
A quick search on some of these names should help you to find some recent papers on what is being done (not restricted to!).
Some specific topics on the subject with some activity:
Billiards, Transient chaos, Hamiltonian systems, quantum chaos, control theory.
Do not be biased by this information, use it has a shortcut to search more and more. It is not meant to represent anything in terms of importance, quantity or quality of the research.
Myself is working on this area and I will tell you why I found it fascinating. This area contain many interesting new mathematics, for example if you analyze dissipative chaotic system you will encounter fractals in phase-space. Fractals are beautiful mathematical objects which cannot be treated with standard differential geometry. You can use topology to study the phase-space trajectories. Lie groups and Lie algebra is used to explore the symmetry properties of the system/trajectory. You may think that the subject is simple, but once you try to track the nonlinear problems analytically this is more challenging. Often people relay on numerical computation. If you want do the research in classical chaos you should know about the chaotic maps and related things. Mainly mathematicians are working on this area and they call it "Dynamical system theory". Prof. James Yorke is one of the reputed person in this area. An another guy is Miguel A. F. Sanjuan who works in Classical Chaos. If you think about Quantum Mechanics there is an area called Quantum Chaos. Gutzwiller trace formula (Periodic orbit theory) and Predrag Civitanovic's Cycle expansion are the hottest topics.Cycle expansion can be utilized to study the fluid turbulence. Related to the periodic orbit theory people are trying to find the zeros of zeta functions using a quantum hamiltonian. If you can quantize H=xp properly then you can become a reputed guy in this area. In short, quantum chaos have applications in number theory. Jonathan Keating and Michel V. Berry are the reputed figures in this area. Then it comes the chaotic billiard, Sinai Billiard and the Bunimovich stadium are the few examples of it. Yakov Sinai, Leonid Bunimovich, and Marko Robnik are the reputed figures in this area. This topic has intimate connection to the statistical mechanics and the ergodic theory. Chaos theory also appear in the context of Renormalization group theory. Related to quantum billiard you can study the Quantum unique ergodicity, Quantum Scars etc. Terence Tao, Eric Heller etc. are working on this topic. Terence Tao works in the area of Random Matrix theory which is an area of pure Mathematics related to Quantum Chaos. By the way, Madan Lal Mehta and Dyson had done a great contribution to Random Matrix theory. Actually Eugene Wigner had started the idea of using Radom Matrix theory in Quantum Mechanics.
Now people are also studying chaos in connection with Quantum Entanglement (Entanglement and Chaos). In relation to quantum gravity, Gerard 't Hooft is studying the deterministic quantum mechanics using the ideas of cellular automata. (Cellular automata and iterating function system(IFS) appear in connection with fractals). There are other people like Laurent Nottale who study the non-differentiable space-time and the emergence of quantum mechanics from it. Some people use the geodesic deviation equation from general relativity to quantify the chaotic property. In short this is a vast and rich topic and you can work in in all areas of physics to apply the ideas of Chaos theory and the nonlinear dynamics.
There are many other topics and interesting people in this field and this information is my limited point of view!
You can look at University of Maryland, Georgia Tech, University of Bristol UK etc where they have good groups. If the university has a reputation, it doesn't mean that in all areas they are strong enough !!!
There is of course research going on. For example the field of billiards is still very active. People study e.g. billiards with non-trivial reflection rules.
In general this is done in Math departments rather than Physics, although there are a lot of physicists working there.