# Phase transition in dynamical systems

There are several occasions in the study of dynamical systems that are called phase transitions. For example consider a homeomorphism $f:X\to X$ and a potential function $\phi\in C(X,\mathbb{R})$. Consider the topological pressure $P(f,t\phi)$, where the parameter $t$ maybe viewed as a inverse-temperature of $(X,f,\phi)$. In particular

• a $k$-phase transition happens at the temperature $t$ if the pressure $P(f,t\phi)$ fails to be $C^k$ (for $k=1,2,\cdots,\infty,\omega$).

Do you have some more examples of phase transition in dynamical systems?

Thank you!

-
Do bifurcations count? –  Christopher A. Wong May 23 '13 at 5:36
This question is much too imprecise for an intelligent answer. For starters, what are P, f and $\phi$? I can sort of guess what $t$ is, but ... Voting to close as "not a real question." –  Michael Renardy May 23 '13 at 12:42
Phase transitions in dynamical systems is a subjective term. For example, in topological dynamical systems, one can have transitions in which the topology of the system undergoes changes as a critical parameter is changed. These are not bifurcations defined in the traditional sense i.e. no periodic points or fixed points are being created/destroyed, or going from stable->unstable, etc. You could look at : arxiv.org/abs/1206.2321 for more details. –  Piyush Grover May 23 '13 at 16:18
@nonlinearism Yes there are several different definitions of phase transition. Someone will say it is a phase transition for a family of systems $f_t:M\to M$ move from regular to chaotic, from integrable to non-integrable. I want to know some more similar phenomena :) –  Pengfei May 23 '13 at 19:53
@Christopher Technically bifurcation should count. As mentioned by nonlinearism, bifurcation always involves the status-change of periodic points and are easy to formulate/classify. –  Pengfei May 23 '13 at 20:00