Bifurcation diagrams, scalings

The first case to study is a system with fixed strength and periodicity of the heterogeneity, but varying ratio of the mobilities. The question is if the ODE system shows the same behavior as the droplets studied in the direct numerical simulations. The parameters to consider are again the velocity and the baselength as the system goes through the periodic motion.

Fig. 5.17(a) shows the obtained average velocity (i.e. the inverse of the periodicity) over driving force for the periodically moving solutions depending on the ratio of mobilities β.

The chosen equilibrium baselength is l0=0.1, i.e. the case where the droplet is not quite commensurable with the substrate, thus breaking the symmetry in baselength direction.

The system shows the same bistability where both moving and pinned solutions coexist.

As observed in the full simulations, the minimal driving force where a periodically moving droplet solution exists changes with the ratio of mobilities. A higher mobility in the transla-tion directransla-tion leads to a lower minimum driving force. The velocity also drops increasingly sharp at the critical point with increasingβ.

0.0 0.5 1.0 1.5 2.0

Fig. 5.17: (a) time averaged velocity over driving force for different mobility ra-tios showing a changing range of bistability with changing mobility ratioβ

(b) baselength over driving force for different mobility ratios plus the static solution branches (black) showing the transition from a homo-clinic bifurcation to a SNIPER bifurcation

(c) velocity over driving force rescaled to show logarithmic diver-gence, as predicted for a homoclinic bifurcation

(d) velocity over driving force rescaled to show square root behavior, as predicted for a SNIPER bifurcation

To confirm the similarity, Fig. 5.17(b) shows the average baselength of the solution branches over the driving force. The point where the periodically moving solution approaches the static branch shifts with increasing beta, with the slope of the approach becoming increas-ingly sharp.

To confirm the change in the bifurcation scenario, it is possible to check if the scaling of the velocity changes as expected. Fig. 5.17(c) shows the periodicity over the logarithm of the distance from the critical point, indicating that the results for β=10,15,20 display a homoclinic bifurcation. Fig. 5.17(d) shows the linear scaling when plotting over(µ−µcrit)^0.5 which corresponds to a SNIPER bifurcation forβ=1, while the other three cases deviate.

It turned out that a two-dimensional configuration space is sufficient to obtain the transition between sniper and homoclinic bifurcation. Introducing an additional degree of freedom to represent the configuration of the free interface was not necessary.

Around the saddle point(xe,le)that is approached by the periodically moving solution, the system can be linearized usingxc=xe+δxandl=le+δlin the form

Alternatively it can be written as

x⁰₁=λ₁x1

x⁰₂=λ₂x1

with λ1, λ2 as the eigenvalues of A plusx1 andx2 as the two coordinates in the base of the corresponding eigenvectors. For a saddle point,λ_1,2have opposite signs, as the system is stable in one direction and unstable in the other.

For a homoclinic bifurcation the periodically moving solution will approach the saddle point from the stable direction and eject along the unstable direction of the linearized system.

If the periodically moving solution is stable, the time scale of this process is dominated by the time required to leave the proximity of the saddle point, determined by the unstable eigen-value. The scaling of the period close to the critical point and the eigenvalues at the critical point of a homoclinic bifurcation can be related by the following equation, as explained in detail by Gaspard[79, 28]:

µ−µc=Ce^−λuT

Here, µ−µc is distance from the critical driving force, λu the unstable eigenvalue of the linearized system at the saddle point,T the periodicity of the obtained solution andC deter-mined by the dynamics away from the region close to the saddle point. The dynamics away from the saddle point is assumed to change only slowly withµ, even if the system is close to the critical point.

Figure 5.18(a) plots the eigenvalues of the unstable branch that is approached by the peri-odically moving solutions. It shows the unstable eigenvalue approaching zero as the branch approaches the turning point. The negative eigenvalue of the solution branch changes too, but relatively little, compared to the absolute value. As shown in Fig. 5.17(a), the point where the homoclinic bifurcation occurs moves to higher driving forces, as the mobility ratioβ de-creases. Therefore, even asλuincreases with decreasingβfor a fixed driving force,λuatµc

decreases and approaches zero.

The previously presented velocity data can now be rescaled with the eigenvalues at the branching points of the homoclinic bifurcation, as presented in Fig. 5.18(b). It shows a very good agreement with the predicted scaling, with the point where the scaling breaks down moving closer to the critical driving force withβapproaching the transition point.

0.0 0.1 0.2 0.3 0.4 0.5

Fig. 5.18: (a) Eigenvalues of the unstable branch that is approached by the peri-odically moving solutions for different mobility ratiosβ, showing the unstable eigenvalue approach 0 at the driving force where the last sta-ble solution annihilates.

(b) Velocity rescaled with the unstable eigenvalue showing the scaling predicted for homoclinic bifurcations, the black line represents a slope of−1

The critical driving force where the bifurcation occurs can be determined for different mobility ratios and periodicity mismatchesl0, as shown in Fig. 5.19(a). As the periodicity mismatch changes, the critical driving force where no more pinned solutions exist changes, as discussed previously. At a value ofβjust above 5 that varies slightly withl0, the critical driving force starts decreasing for all cases presented here. That means the critical point where no more moving solution exists is not at the turning point of the static branch anymore and the bifurcation scenario changed.

The transition region for the different curves has been replotted in 5.19(b). While the curves are qualitatively similar, no sensible scaling law to describe the transition could be derived from them. For the casel0=0.6 the range of bistability is decreasing most slowly, as the system is close to the state where the baselength of the pinned droplet equals the equilibrium baselength. This is also indicated by the fact that this state has the highest critical driving force.