Hopf Bifurcation within Thermodynamic Representation

TL;DR

This paper demonstrates the occurrence of Hopf bifurcation within a thermodynamic Hamiltonian framework, revealing a revolving phase plane with limit cycles analogous to superfluid He$^4$, and simplifies the dynamics using complex order parameters.

Contribution

It introduces a novel thermodynamic Hamiltonian representation that captures Hopf bifurcation and revolving phase planes, linking phase dynamics to superfluid analogies.

Findings

Hopf bifurcation leads to the formation of a revolving phase plane with limit cycles.

The phase of a complex order parameter encapsulates fast angle variations.

Vector potential relates to the relative velocity of limit cycle movement.

Abstract

On base of Hamiltonian formalism, we show that Hopf bifurcation arrives, in the course of the system evolution, at creation of revolving region of the phase plane being bounded by limit cycle. A revolving phase plane with a set of limit cycles is presented in analogy with revolving vessel containing superfluid He$^4$. Within such a representation, fast varying angle is shown to be reduced to phase of complex order parameter whose module squared plays a role of action. Respectively, vector potential of conjugate field is reduced to relative velocity of movement of the limit cycle interior with respect to its exterior.