# 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!

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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

Hermann Haken in his theory of Synergetics discussed over years the huge variety of phase transitions and particularly the non-linear phase transition phenomena. His series with Springer publishing on synergetics (http://www.springer.com/series/712) brings extensive literature on this matter. He approached phase transition phenomena in very different disciplines (physics (from laser theory to thermodynamics...), chemistry, biology, economy, information theory, artificial intelligence, mathematics, politics and social phenomena...) and with very different methods, however systematised in one theory that combines stochastic (Fokker Planck and master equations) and deterministic apporaches. although emeritus his institute of theoretical physics and synergetics has a huge catalogue of works and the researcher over there are a wealth of knowledge on this matter.

The central work is concise in one monograph: http://www.amazon.com/Synergetics-Introduction-Advanced-Topics-Springer/dp/3642074057/ref=sr_1_1?ie=UTF8&qid=1376158698&sr=8-1&keywords=synergetics+and+introduction

You will find in this spectrum all you would need on phase transition examples and much exciting stuff more. It made me at least the last 30 years since my lectures with Haken again and again excited.

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