Type of dynamic phase transition in bistable equations

We consider a class of bistable periodically perturbed ordinary differential equations of importance in mathematical physics and derive an asymptotic criterion for the existence of a tricritical point (TCP). Surprisingly, in the adiabatic limit the criterion is local and very simple. It also allows one to calculate the location of a TCP in parameter space, which we illustrate with three examples.

Figure 1

Two possible minimal scenarios of the emergence of a single periodic solution. In scenario (a) three solutions merge in a subcritical pitchfork bifurcation. In scenario (b) the central solution first undergoes a supercritical pitchfork bifurcation, emitting two unstable solutions. These unstable solutions disappear in fold bifurcations upon meeting the stable solutions.

Figure 2

If the bifurcation of the zero-mean solution is a supercritical pitchfork [scenario (b) of Fig. 1], one can effect a discontinuous change (a first-order phase transition) on the stable periodic cycle by adiabatically changing $h$. Here the fragment of the bifurcation diagram is drawn in thin lines and the evolution of the mean of the periodic cycle is indicated by the thick lines. When increasing the $h$, the transition from ferro (nonzero-mean) to para (zero-mean) cycle happens at $h=h_{fp}$ [part (a)]. However, if $h$ is decreased the para-ferro transition happens at a different value, $h=h_{pf}\ne h_{fp}$ [part (b)].

Figure 3

Dependence of the type of the periodic solutions on the parameters $\beta$ and $h$ in Eq. (2) with $ϵ=1/2$. F marks the region of existence of a nonzero-mean (ferromagnetic) stable periodic solution and P marks the region with a stable zero-mean solution (paramagnetic). For large values of $\beta$ there is a range of $h$ in which F and P solutions can co-exist. The point at which this range shrinks to zero is the tricritical point (TCP). Inset: location of the TCP as a function of $ϵ$. For small values of $ϵ$ the computation becomes unstable [5] due to the exponential narrowing of the overlap $\left(\text{F}+\text{P}\right)$ region [9].