ff-bifbox: A scalable, open-source toolbox for bifurcation analysis of nonlinear PDEs

Abstract

Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution requirements. This work introduces ff-bifbox: a new open-source toolbox for performing numerical branch tracing, stability/bifurcation analysis, resolvent analysis, and time integration of large, time-dependent nonlinear PDEs discretized on adaptively refined meshes in two and three spatial dimensions. Spatial discretization is handled using finite elements in FreeFEM, with the discretized operators manipulated in a distributed framework via PETSc. Following a summary of the underlying theory and numerics, results from three examples are presented to validate the implementation and demonstrate its capabilities. The considered examples, which are provided with runnable ff-bifbox code, include: a 3-D Brusselator system, a 3-D plate buckling system, and a 2-D compressible Navier--Stokes system. In addition to reproducing results from prior studies, novel results are presented for each system.

Figures (5)

• Figure 1: Bifurcation diagram showing the oscillation period $T=2\pi/\omega$ of selected time-periodic solutions to the 3-D Brusselator system with varying $L$ for $A=2$, $B=5.45$, $D_X=0.008$, and $D_Y=0.004$. Respectively, pitchfork and Neimark--Sacker (NS) bifurcations of periodic orbits are denoted using two-tone diamonds and circles; red, blue, and green lines and markers denote 1-D, 2-D, and 3-D branches; and thick and thin lines denote asymptotically stable and unstable solution branches. Hollow and solid markers along $T\approx2.94$ represent the exact and numerical Hopf points, respectively. Hopf, pitchfork, and NS bifurcations of the 1-D system as reported by Lust & Roose lust_roose_2000 are overlaid in black.
• Figure 2: Schematic of the benchmark problem for the hinged cylindrical section under point loading parameterized by $P$ with lines of symmetry indicated. Note that the pinned boundary conditions are enforced on a line only along the neutral axis of the body leahu-aluas_abed-meriam_2011.
• Figure 3: Bifurcation diagram showing the $z$-deflection of the center of the plate under parametric variations in $P$ and isometric visualizations of the $z$-deflection for the three stable solutions at $P=170\,N$. Linearly stable and unstable solutions are indicated by thick and thin lines, respectively. Saddle--node and symmetry breaking bifurcations are indicated on the diagram by circle and square symbols, respectively, with colors indicating the symmetry or symmetries of the bifurcating solutions. The $S$ and $H_x$ reference data from the thin shell model of Leahu-Aluas & Abed-Meriam leahu-aluas_abed-meriam_2011 are shown.
• Figure 4: Illustration (not to scale) of the flow configuration over the circular cylinder. The blue interior region corresponds to the physical domain of interest and the orange outer region denotes the absorbing layer.
• Figure 5: Bifurcation diagrams of $\bar{c}_d$ and $St$ versus $M$ and $Re$, $M$--$Re$ stability map, and flow visualization summarizing the dynamics of the 2-D compressible flow past a circular cylinder. The projections of the Hopf bifurcation curve associated with time-periodic vortex shedding are plotted in black, with the gray shaded portion of the stability map indicating linear instability of the steady state. The two-tone diamonds indicate codimension-2 Bautin bifurcation points. The projections of the LPC curves are drawn in yellow, with the yellow shaded area in the stability map indicating the bistable region. In the bifurcation diagrams, Hopf and LPC points are indicated with square and round markers, respectively, with asymptotically stable and unstable solutions indicated by thick and thin lines, respectively. Bifurcation diagrams are shown for $Re=[40,50,70,90]$ over varying $M$ and $M=[0,0.4,0.6,0.8]$ over varying $Re$. Reference data from direct numerical simulations of Canuto & Taira canuto_taira_2015 are shown by color-matched diamond markers. Snapshots of the stable steady state, saddle periodic solution, and stable periodic solution at $(M,Re)=(0.8,80)$ are visualized via instantaneous pressure contours and velocity streamlines over $(x,y)\in[-3,-3]\times[21,3]$.