Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

Figure 1.  Schematic representation of simple laminates microstructures as periodic solutions with $X \in [0,1]$. (a) Microstructures in one space dimension: austenite (A) and martensite (M) alternate, while the transition area is shown in gray. (b) Structure in space of the variable $u_X$, whose values $\pm 1$ represent the two different phases of the material of width of order $\mathcal{O}(\varepsilon^\alpha)$, with $\alpha = 1/3$ for minimizers (as shown in [38]) and $\alpha = 0$ for other critical points. The width of the transition interval is of order $\mathcal{O}(\varepsilon)$.

Figure 2.  Critical manifold ${\mathcal C}_0$ in $(w,u,v)$-space. The magenta dashed lines are the fold lines ${\mathcal L}_\pm$. The blue solid curves correspond to $\mathcal{C}_{0,l}^\mu$ and $\mathcal{C}_{0,r}^\mu$, i.e., the intersection of $\mathcal{C}_{0,l}$ and $\mathcal{C}_{0,r}$ and the hypersurface $H(u,v,w,z) = \mu$ for $\mu = 0$.

Figure 3.  $\mathcal{C}_{0,l}^\mu$ and $\mathcal{C}_{0,r}^\mu$ in $(w,u)$-space with $\mu = 0$; cf. Figure 2.

Figure 4.  Fast flow in the $(w,z)$-space for 20. Equilibria are marked with blue dots and the stable and unstable manifold trajectories in green. The heteroclinic fast connections are indicated with double arrows.

Figure 5.  Singular periodic orbit $\gamma_0^\mu$ for a fixed value of $\mu$ ($\mu = 0$), obtained by composition of slow (blue) and fast (green) pieces. (a) Orbit in $(w,z,u)$-space. (b) Orbit in the $(w,u,v)$-space. The fast pieces are indicated via dashed lines to illustrate the fact we are here considering their projection in $(w,u,v)$, while they actually occur in the $(w,z)$-plane. Consequently, they do not intersect ${\mathcal C}_{0,m}$.

Figure 6.  Transversal intersection in the $(w,z,v)$ space between $W_u({\mathcal C}_{0,l})$ (in orange) and $W_s({\mathcal C}_{0,r})$ (in magenta). The blue line represents the critical manifold ${\mathcal C}_0$.

Figure 7.  Schematic representation of the SMST algorithm applied to $ \mathcal{C}_{0, l}^\mu $ (an analogous situation occurs for $ \mathcal{C}_{0, r}^\mu $). The critical manifold is indicated by a dotted blue line, while the red line represents the slow manifold for $ \varepsilon = 0.001 $. The orange point corresponds to $ (0, w_L, 0) $, which actually belongs to both manifolds

Figure 8.  Continuation in $\mu$: (a) bifurcation diagram in $(\mu, P)$-space, where two periodic solutions corresponding to $\mu = -0.124$ are marked by crosses; (b) corresponding solutions in $\left(w,z,u \right)$-space: the one on the lower branch (magenta) is almost purely fast, while the one on the upper branch (purple) contains long non-vanishing slow pieces.

Figure 9.  Continuation in $\mu$. (a) Zoom on the upper part of the bifurcation diagram in-$(\mu,P)$ space, where two periodic orbits corresponding to $\mu = 0.0025$ are marked by crosses. (a1)-(a2) The orbits are shown in $\left(w,z,u\right)$-space. The periodic orbit on the bottom part of the upper branch (purple) corresponds to analytical expectations with two fast and two slow segments. The periodic orbit on the top part of the upper branch (magenta) includes two new fast "homoclinic excursions". (b) Zoom on the lower part of the bifurcation diagram in $(\mu, P)$-space, where three solutions are marked. (b1) The solutions in phase space all correspond to periodic orbits around the center equilibrium $p_0$; note that the scale in the $u$-coordinate is extremely small so the three periodic orbits almost lie in the hyperplane $\{u = 0\}$.

Figure 10.  Continuation in $\varepsilon$: on the left side bifurcation diagrams in $(\varepsilon, P)$ are shown, on the right the corresponding solutions in $\left( w, z, u \right)$-space are displayed. (a) $\mu = \mu_l$, (b) $\mu = \mu_c$, (c) $\mu = \mu_r$.

Figure 11.  Illustration of two-parameter continuation. (a) Three different bifurcation diagrams have been computed, each starting from a solution at $\mu = 0$ for three different values of $\varepsilon = 0.1,0.01,10^{-5}$ (red, green, blue). It is already visible and confirmed by the computation that the sequence of leftmost fold points on each branch converges to $\mu = -1/8$ as $\varepsilon\rightarrow 0$. However, the period scaling law of the orbits precisely at these fold points, which is shown in (b) as three dots corresponding to the three folds in (a) and a suitable interpolation (black line), does not converge as $\mathcal{O}(\varepsilon^{1/3})$ (grey reference line with slope $\frac13$).

Figure 12.  Possible fits of the form $P \simeq \varepsilon^{\alpha}$ for the numerical data computed with $\mu = \mu_l$ (black line): $\alpha = 2$, blue; $\alpha = 1/3$, green; $\alpha = 1$, red.

Figure 13.  Parabola-shaped diagram obtained by fixing $\varepsilon = 0.001$ and numerically computing the value of the functional $\mathcal{I}^{\varepsilon}$ along the solutions computed via continuation in $\texttt{AUTO}$. The plot presents a minimum, and the value of $P$ corresponding to $\varepsilon$ where this minimum is realized is recorded in order to check the period law 3.

Figure 14.  Comparison between the values of $P$ minimizing $\mathcal{I}^{\varepsilon}$ for several discrete values in the range $I_{\varepsilon}$ (red circles) and the period law 3 (black line). (a) Zoom on the range $\left[ 10^{-7}, 10^{-2} \right]$, where it is expected that large values of $\varepsilon$ tend to deviate from the $\mathcal{O}(\varepsilon^{1/3})$ leading-order scaling, while for low values the scaling the scaling agrees. (b) The same plot as in (a) on a log-log scale.