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Pattern formation with pde2path -- a tutorial

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The pde2path tutorial establishes that the qswibra and cswibra routines can switch solution branches at symmetry-induced bifurcation points of multiplicity greater than one, reproducing amplitude-equation predictions in 1D, 2D, and 3D.

desk verdict A competent, candid software tutorial that deserves refereeing as a methods resource; judge it on reproducibility and usefulness, not on novelty. read the letter →

arxiv 1908.05211 v3 pith:S3GQ2PLM submitted 2019-08-02 nlin.PS

Hannes Uecker This is my paper · ORCID
classification nlin.PS MSC 35J4735B3237M20
keywords pde2pathnumericalcontinuationbifurcationpatternformationbranchswitchingSwift-Hohenbergequationreaction-diffusionsystemsfiniteelementmethod
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

pde2path is a numerical continuation package for pattern-forming PDEs, and this tutorial claims that its newer routines make a previously awkward step routine: switching from a trivial or homogeneous branch onto a patterned branch when the bifurcation point has multiplicity greater than one. The paper demonstrates qswibra and cswibra on the Swift–Hohenberg, Schnakenberg, Cahn–Hilliard, and related models in 1D, 2D, and 3D, obtaining branch directions that match the predictions of amplitude equations. A sympathetic reader should care because in many applications the interesting patterns—spots, hexagons, lamellas, tubes, localized states—bifurcate precisely at these multiple points, and older workarounds such as slightly distorting the domain made such branches easy to miss. If the routines work as claimed, numerical bifurcation analysis of pattern formation becomes more automatic and less dependent on the analyst knowing in advance which pattern to look for.

What carries the argument

The load-bearing object is the reduced bifurcation equation: at a bifurcation point $(u_0,\lambda_0)$ with $m$-dimensional kernel spanned by $\varphi_1,\dots,\varphi_m$ and adjoint kernel $\psi_1,\dots,\psi_m$, the tangent direction $u'(s_0)=\sum_{j=0}^m \alpha_j\varphi_j$ is found from the homogeneous quadratic system (25), or, for pitchfork-type branches, from the cubic system (27). The routines qswibra and cswibra set up these systems from second and third derivatives of $G(u,\lambda)$ projected onto the adjoint kernel, solve them by Newton loops, and store the resulting tangents; the user then selects a tangent and continues. This machinery carries the argument because it replaces the old trick of breaking the multiplicity by perturbing the domain, which hid branches such as spots, with a direct enumeration of all local branch directions. Continuous symmetries are factored out by choosing active kernel vectors (aux.ali), and phase conditions pin translations during continuation.

What would settle it

On the square-domain Swift–Hohenberg equation with $\nu=0$ and $\nu=0.7$, run cswibra at the first bifurcation point $\lambda=0$ and compare the returned tangents with the amplitude-equation solutions (12): stripes $(\alpha_1,\alpha_2)=(\pm\sqrt{\mu/c_1},0)$ and spots $(\alpha_1,\alpha_2)=(\pm\sqrt{\mu/(c_1+c_2)},\pm\sqrt{\mu/(c_1+c_2)})$. If either isotropy class is missing, or if an extra nonsymmetric tangent appears, the claim that the routine captures the true bifurcation structure fails; likewise, the claim fails if continuation of these branches swaps their predicted stability ordering from Lemma 2.2.

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Extended reading notes

Core claim

The paper's central claim is that steady bifurcation points of higher multiplicity—the typical situation in pattern-forming systems on symmetric domains—can be attacked directly instead of being perturbed away. It presents the branch-switching routines qswibra and cswibra, which solve the quadratic bifurcation equations (25) and cubic bifurcation equations (27) to obtain tangent vectors (or quadratic predictors) for every isolated bifurcating branch. Together with phase conditions for continuous symmetries, active kernel selection, and symmetry-respecting meshes, these routines reproduce, in 1D, 2D and 3D, the branch pictures predicted analytically from amplitude equations: stripes and spots on squares, hexagons and mixed modes on hexagonal domains, lamellas, tubes and rhombs on cubic lattices, and snaking branches of localized patterns. The same setup is shown to handle mass constraints, quasilinear cross-diffusion, global coupling, branch-point continuation, and pattern formation on spheres and tori.

Load-bearing premise

The finite element discretization and the numerical solution of the bifurcation equations faithfully represent the continuous PDE, so the demo branches are not artifacts of mesh asymmetry, tolerances, or software bugs.

Editorial extensions

If this is right

  • Users can trust the primary bifurcation diagrams produced by pde2path on symmetric domains: on a square, the routines return both stripes and spots; on a hexagon-compatible rectangle, they return stripes, hexagons, and mixed modes, with the same stability ordering predicted by amplitude equations.
  • Branch switching can be followed by continuation of secondary bifurcations, so localized patterns, snaking branches, and fronts between patterns and the trivial state can be computed in 1D, 2D, and 3D without hand-built initial guesses.
  • For systems with continuous symmetries, combining active-kernel selection with phase conditions yields the same information on periodic domains, tori, and spheres, where the sphere case recovers the isotropy classes predicted by O(3)-equivariant theory.
  • Auxiliary tools—branch-point continuation, deflation, and time integration for initial guesses—let the same software approximate Eckhaus curves, find disconnected branches, and reach desired patterns that are far from a known branch.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the branch-switching machinery only needs the discretized equations and the derivative projections, it should transfer to other symmetry-induced multiple bifurcations—rhombic or oblique lattices, quasiperiodic domains, or systems with additional gauge symmetries—without new analysis.
  • The tutorial stops where the cubic amplitude equations degenerate ($c_1=0$ or $|c_1|=|c_2|$); a natural extension is to feed fifth-order bifurcation equations into the same Newton framework and check whether the predicted vertical branches appear, which would test the determinacy threshold directly.
  • On spheres, the computed branches suggest that numerical continuation can turn the curvature-dependent spot-versus-stripe observations, mostly obtained by time integration, into quantitative phase boundaries in $(R,\lambda)$ or $(R/\rho,\lambda)$ space.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This tutorial-style paper documents how the Matlab continuation package pde2path can be used for pattern-formation problems in one, two, and three space dimensions. The central technical focus is branch switching at steady bifurcation points of higher multiplicity, implemented in the routines qswibra and cswibra via quadratic and cubic bifurcation equations. The paper walks through a large set of demos: the Swift-Hohenberg equation in 1D/2D/3D, including snaking and localized patterns; the Schnakenberg system; Cahn-Hilliard and functionalized Cahn-Hilliard models with mass constraints; a chemotaxis system with quasilinear cross-diffusion; global coupling with rank-one Jacobian corrections; and pattern formation on spheres and tori. Throughout, the paper compares numerical results with analytical amplitude-equation predictions and gives practical guidance on meshes, mesh adaptation, phase conditions, deflation, and time integration as a fallback for computing isolated solutions.

Significance. If the software performs as documented, the paper is a substantial practical contribution: it makes higher-multiplicity branch switching accessible in a finite-element continuation environment and provides a large, reproducible set of demos. Its strengths are that the mathematical background is standard, the numerical demonstrations reproduce known analytical amplitude-equation results in several settings, and the software, including all demo directories, is publicly available. The paper is also honest about limitations: it explicitly states in Section 5.4 that the QBE/CBE solvers 'are not fail safe,' and Section 6.3 documents that on a sphere with a 13-dimensional kernel only a subset of branches is typically found after tuning aux.soltol and aux.isotol. This honesty is a point in favor of the paper, but it should be made visible in the abstract and introduction. The paper does not provide formal verification of the FEM discretizations or of the CBE coefficient computations; this is typical for a tutorial and does not by itself undermine the contribution.

major comments (1)
  1. [Abstract, §1, §5.4, §6.3] The abstract and introduction present qswibra and cswibra as the main focus of the tutorial without the substantial caveat that these routines are heuristic and can require expert tuning. Sections 5.4 and 6.3 explicitly say the methods 'are not fail safe' and that for the sphere demo 'typically only a subset of the expected branches is found' after manual adjustment of auxiliary parameters. Since the claimed added value of the tutorial is branch switching at higher-multiplicity bifurcation points, this limitation should be stated in the abstract and in Section 1, not only deep in the technical sections. I recommend adding a sentence to the effect that the routines are intended as practical tools, that exhaustive branch enumeration is not guaranteed, and that gentau and manual selection of kernel vectors remain necessary fallbacks.
minor comments (4)
  1. [§3.8.3, Eq. (44)] The initial guess for the BCC-to-zero front contains an evident typo: the second condition '0 z≥0' should presumably be '0 for z<0', so that the guess is nonzero on the upper half and zero on the lower half of the domain.
  2. [§5.3] The word 'intuitely' should be 'intuitively' in the first paragraph of Section 5.3.
  3. [§3.5 and §4.2] The captions and text refer to 'ten bifurcation directions' and 'four bifurcation directions' from cswibra, but only a subset is plotted; it would help the reader if the captions stated explicitly how many directions were found and how many are shown, and whether the remainder are obtained by symmetry.
  4. [Table 3] The description of the default isotol value and its effect on classifying solutions as isolated is somewhat terse; a one-sentence explanation of why a too-small or too-large isotol can suppress or spurioulsy create isolated solutions would make the table more usable.

Circularity Check

0 steps flagged · score 2.0 of 10

Self-citations are pervasive but non-load-bearing; no prediction reduces to a fit or to the cited work.

full rationale

The tutorial's analytic predictions are derived in-line from the Swift-Hohenberg equation by explicit multiple-scale amplitude-equation computations (§2), and the pde2path computations then solve the full PDE with finite elements. No fitted parameter is renamed as a prediction, and no equation used as an output is identical to an input by construction. The branch-switching routines qswibra/cswibra are presented through the QBE/CBE equations (25) and (27), with the existence of distinct branches justified by standard implicit-function arguments, and the paper explicitly warns in §5.4 and §6.3 that the routines are not fail-safe and that the sphere demo finds only a subset of branches after manual tuning. Self-citations, such as [Uec19b] for algorithmic details and [UWR14] for earlier demos, are used as pointers and background, not as the sole evidence for the numerical results. The score of 2 reflects the tutorial's heavy reliance on the author's own software and prior papers, but no actual circular reduction was found.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted parameters. It relies on standard PDE theory, finite element methodology, and the correctness of the author's own software package.

assumptions (3)
  • domain assumption Finite element discretizations converge and preserve the qualitative bifurcation structure of the continuous problems.
    Used throughout the tutorial (e.g., §3.1); the paper relies on standard FEM theory rather than proving it.
  • standard math The quadratic and cubic bifurcation equations (QBE/CBE) characterize all relevant bifurcating branches at multiple bifurcation points.
    Invoked in §3.3, citing [KL72] and [Uec19b].
  • ad hoc to paper The pde2path software implements its documented algorithms correctly for the presented demos.
    The central claim depends on the code being bug-free or at least correct for the demo outputs; this is not formally verified in the manuscript.

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Pith. "Pith review of Pattern formation with pde2path -- a tutorial." pith.science (2026). https://pith.science/paper/S3GQ2PLM

@misc{pith2026190805211,
  author       = {Pith},
  title        = {Pith review of: Pattern formation with pde2path -- a tutorial},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S3GQ2PLM}},
  note         = {Machine review of arXiv:1908.05211}
}
read the original abstract

We explain some pde2path setups for pattern formation in 1D, 2D and 3D. A focus is on new pde2path functions for branch switching at steady bifurcation points of higher multiplicity, typically due to discrete symmetries, but we also review general concepts of pattern formation and their handling in pde2path, including localized patterns and homoclinic snaking, again in 1D, 2D and 3D, based on the demo sh (Swift-Hohenberg equation). Next, the demos schnakpat (a Schnakenberg reaction-diffusion system) and chemtax (a quasilinear RD system with cross-diffusion from chemotaxis) simplify and unify previous results in a simple and concise way, CH (Cahn-Hilliard) deals with mass constraints, hexex deals with (multiple) branch points of higher degeneracy in a scalar problem on a hexagonal domain, and shgc illustrates some global coupling. The demos acS, actor, schnakS and schnaktor (the Allen-Cahn and Schnakenberg models on spheres and tori) consider pattern formation on curved surfaces, cpol considers a problem of cell polarization described by bulk-surface coupling, and bruosc (Brusselator) explains how to augment autonomous systems by a time periodic forcing. Along the way we also comment on the choice of meshes and mesh adaptation, on time integration, and we give some examples of branch point continuation and Hopf point continuation to approximate stability boundaries.

Figures

Figures reproduced from arXiv: 1908.05211 by the authors.

Figure 1
Figure 1. (a) Dispersion relation with admissible k for Ω = (0, π) (*) and Ω = (0, 4π) (◦). (b) First 3 modes for Ω = (0, 8π). amplitude equation on the center manifold we make the ansatz λ = µε2 , where µ = 1 or µ = −1, and u(t, x) = εA1(T)e1 + ε 2  1 2 A0(T) + A2(T)e2  + c.c. + h.o.t, ej = eijx , (8) with complex coefficients Aj = Aj (T), which depend on the slow time scale T = ε 2 t. Furthermore, h.o.t denotes higher ord… view at source ↗
Figure 2
Figure 2. Spectral situation for the linearization of the SH equation around u ≡ 0 over square domains. (a) Admissible wave vectors k for Ω = (−π/2, π/2)2 (*) and Ω = (−2π, 2π) 2 (◦), respectively, with Neumann BCs. The amplitudes A1,2 and B1,2 are used in the amplitude equations below. (b,c) Kernels at the first two bifurcations. In (b), the left column shows the ’natural’ planforms of stripes, corresponding to A1 and A2, an… view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (41 more)
Figure 4
Figure 4. Figure 4: Admissible wave vectors for the domain Ω = (−π, π) × (−π/√ 3, π/√ 3), and associated planforms at bifurcation, for clarity plotted over Ω = (−4π, 4π) × (−4π/√ 3, 4π/√ 3). The ansatz u(x, t) = A1(t)e1 + A2(t)e2(t) + A3(t)e3 + h.o.t (14) and the pertinent symmetry consid…
Figure 5
Figure 5. Figure 5: (a1) Subcritical bifurcation of primary periodic patterns (black branch) in the SH equation with ν = 2, Ω = (−10π, 10π) and secondary bifurcations of snaking branches of a front (blue, 1Ds1) and a localized pattern (red 1Ds2). (a2) blow-up from (a1) showing how the sna…
Figure 6
Figure 6. Figure 6: The SH equation on a distorted square with D4 symmetry. (a) Pure modes at the first two BPs on the square Ω = (0, 2π) 2 . (b) Associated mixed modes. (c,d) BD and sample solutions over Ω0.35 with ν = 0.5, from cmds2dpsq. Concerning the implementation we refer to the sc…
Figure 7
Figure 7. Figure 7: Example results from cmds2dhex.m. Bifurcation diagrams and example plots SH over a small rectangular domain permitting hex solutions. For ν = 0 in (a), hex (unstable), str and pq (stable) bifurcate in supercritical pitchforks. The (up and down) hex become stable at λ =…
Figure 8
Figure 8. Figure 8: Results from cmds2dhexfro for the SH over a long rectangular domain, ν = 1.3. Primary branch of (up) hexagons (blue), and snaking branches of fronts (red, hf) between hexagons and u = 0 and of localized hex (magenta, hloc). %% init and zero - branch lx =8* pi; nx = rou…
Figure 9
Figure 9. Figure 9: Results from cmds2dhexfroada. We coarsen hf/pt40 (with np = 4650) from [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Selected results from demo sh/cmds3dcube. Primary bifurcations at λ = 0 in (3) over the cube Ω = (−π, π) 3 with homogeneous Neumann BCs, pseudo criss-cross mesh of np = 6006 points and nt = 33000 tetrahedral elements, see §3.7 for comments on the meshing. (a) Isosurfa…
Figure 11
Figure 11. Figure 11: (3) on a “BCC lattice cube” Ω = (− √ 2π, √ 2π) 3 , ν = 1. (a) kernel vectors (distorted square prisms) and bifurcation direction for a BCC obtained from qswibra. (b) BD, ’hot’ and ’cold’ BCCs and square prism example solutions. np = 10351, nt = 57024 tetrahedra, see c…
Figure 12
Figure 12. Figure 12: The SH equation (3) over Ω = (−2π, 2π) 2 with pBCs in x and y, ν = 0. The kernels at the 1st, 2nd and 3rd BPs u ≡ 0 and λ = 0, 1/16, 1/4 are 4,8, and 4–dimensional, respectively. However, modulo the translational symmetries Γ they are only 2,4 and 2–dimensional, respe…
Figure 13
Figure 13. Figure 13: (a-c) 2D meshes on Ω = (−lx, lx) × (−ly, ly), lx = 2π, ly = 2π/√ 3, starting from a meshgrid with nx = 3, ny = 2. (a) Standard meshing (np = 6) in 2D destroys reflection and rotation symmetry. (b) criss￾cross via “refine-longest”, np = 18, and with one additional defa…
Figure 14
Figure 14. Figure 14: (3) on Ω=(−2lx, 2lx)×(−ly, ly), lx=2π, ly=2π/√ 3, ν = 1.3, example results from cmds2dhexb.m. See text and cmds2dhexb.m for details. the bifurcation points become ’sufficiently dense’ on a given branch, we can more or less • call q(c)swibra at any point, including reg…
Figure 15
Figure 15. Figure 15: Deflation for the SH equation (3) at λ = 1 over Ω = (−2π, 2π) 2 with pBCs in x and y, ν = 0. Using u ≡ 0 as primary known solution, ten searches from initial guesses with random (|k| ≈ 1) Fourier coefficients, yield the four solutions from (a). (b,c) Continuation of 2…
Figure 16
Figure 16. Figure 16: Obtaining solutions from guesses, if necessary including time-integration. (a) A ’reasonable’ initial guess for a hex-to-stripes front, yielding the desired solution directly from a Newton loop, ν = 1.3, λ = 0.2. (b) A ’bad’ initial guess for a hex-to-stripes front; h…
Figure 17
Figure 17. Figure 17: Results from sh/cmdsBCClong.m for (3) on Ω = (−lx, lx) 2×(−lz, lz), lx = π/√ 2, lz = 8lx, ν = 1.5. (a) BD of BCCs (red), tubes (blue), and b2z front branch (dark brown) and b2t isola (light brown). (b) BCC and tube plotted over small domain; guess for b2z front, and s…
Figure 18
Figure 18. Figure 18: Results from sh/cmdsBCClongref.m for (3) on Ω = (−lx, lx) 2 ×(−lz, lz), lx = π/√ 2, lz = 8lx, ν = 1.5. (a) comparison of the same branch on different meshes. Brown (b2z): uniform mesh, np = 11360; blue: original uniform mesh, np = 17375, on top of brown branch; magent…
Figure 19
Figure 19. Figure 19: Preparatory step (not strictly necessary) for (46): plotting the dispersion for different λ. Listing 8 shows the start of schnakpat/cmds1d.m. After finding kc in C1, in C2 we restart the computations on a domain tuned to the critical mode cos(kcx), i.e, of length 10π/…
Figure 20
Figure 20. Figure 20: (46), (σ, d) = (−0.6, 60), lx = 10π/kc ≈ 48.8132. Turing branches T1 (blue), T2 (dark blue), T3 (brown) and T6 (magenta), and a snaking branch of a front bifurcating from T1 but reconnecting to T6. In the solution plot of 1D1/pt30 we use the setting plotsol(’1D1’,’pt3…
Figure 21
Figure 21. Figure 21: Continuation of folds on T1,T3 and T6 in σ, and behavior of branches at σ = −0.4. over a domain Ω = (−lx, lx) × (−ly, ly) with lx = π/kc and ly = lx/ √ 3, we find a double branch point at λ = λc = √ 60p 3 − √ 8 ≈ 3.21. We then use qswibra to switch to the hexagon bran…
Figure 22
Figure 22. Figure 22: Results from schnakpat/cmds2da.m for (46) on a small rectangular domain Ω = (−lx, lx) × (−ly, ly), lx = π/kc, ly = lx/ √ 3, corresponding to a hexagonal dual lattice. (a) kernel at the first BP, and ’hex’ bifurcation direction obtained from qswibra. (b) Bifurcation di…
Figure 23
Figure 23. Figure 23: (46) on a long rectangular domain Ω=(−lx, lx)×(−ly, ly), lx = 8π/kc, ly = π/( √ 3kc). On the bean branch (red) between stripes (blue) and spots (magenta), there are bifurcation points leading to snaking branches of localized patterns. Right column: Fourier spectrum of…
Figure 24
Figure 24. Figure 24: (a) First 3 bifurcating branches on Ω = (0, 20π), with wave numbers k = 1, 19/20 and 21/20, and secondary bifurcations. Inset zoom near λ = 0.025. (b) tangent plots at the BPs on u ≡ 0. (c) tangent plots at the ’Eckhaus points’ b2/bpt1 and b3/bpt2 (first two panels), …
Figure 25
Figure 25. Figure 25: The Cahn–Hilliard problem (52) with W(u) = − 1 2 u 2 + 1 4 u 4 on the unit square, ε = 1/100. (a) BD E over m. The spinodal region (where the alloy can lower its energy by decomposing) extends from m = −m0 to m = m0, where −m0 ≈ −0.55 is the first BP on the homogeneou…
Figure 26
Figure 26. Figure 26: Branch point continuation (and subsequent return to normal continuation) from CHb/cmds1D.m. Cahn–Hilliard problem (52) with W(u) = − 1 2 u 2 + 1 4 u 4 on (−1/2, 1/2). (a) The primary nontrivial branch(es) (blue) for different ε. (b) example plots at m = −0.3 for incre…
Figure 27
Figure 27. Figure 27: Example results from cmds1, Ω = (−2, 2) × (−3, 3), ε = 0.25, η1 = 1, η2 = 2, continuation in m starting with initial in the form of straight chanel with ≈= −0.853, which is unstable. Continuation to smaller m first yields meandering BPs (green branches), then a stable…
Figure 28
Figure 28. Figure 28: (a) Continuation of m1/pt10 in η1, giving pearling, with sample solutions and tangent in (b). Snapshots from the evolution starting from near s1 at m = −0.8. At t = 200 the solution is quasi steady, and a Newton loop for the steady problem yields a (stable) steady sol…
Figure 29
Figure 29. Figure 29: Results from the demo hexex for bifurcations at the second BP for (66). with homogeneous Neumann BCs for u1, u2. The trivial branch for (67) is (u1, u2) = (1, 1/2). We take the chemotaxis coefficient λ ∈ R as bifurcation parameter, fix d = 1/4 and r = 1.52, and now im…
Figure 30
Figure 30. Figure 30: Basic BD and example solutions for the chemotaxis model (67) over Ω = (−0.5, 0.5) × (−2, 2). Stripe branches q* in shades of blue, and spots q* in shades of red, with * increasing left to right. 5.6 Global coupling, and customized linear system solvers: Demo shgc Inte…
Figure 31
Figure 31. Figure 31: Slanted snaking for the globally coupled 2-3 SH equation (74). (a) Ω = (−10π, 10π), ν = 2, γ = 2, compare to [PITH_FULL_IMAGE:figures/full_fig_p058_31.png]
Figure 32
Figure 32. Figure 32: The Allen–Cahn equation (86) on tori, nt=3600 triangular elements. (a) Kernel vectors at BPs 2 to 7 (only one shown at the double BPs 2,3,6 and 7). (b-d) Continuation in λ and example solutions. (e) Continuation in R of two solutions from (b). 3. The translational inv…
Figure 33
Figure 33. Figure 33: The Allen–Cahn equation (86) on a sphere of radius 3 (with cutoff δ = 10−3 ). (a) Idea of the meshing, with coarser meshes for |y| → π/2 − δ. (b) Basic bifurcation diagram of branches from the first 4 BPs. (c,d) example solutions, where 2-1, 3-1a, 3-1b, 4-1, 4-2, 4-3 …
Figure 34
Figure 34. Figure 34: Pattern formation on spheres. (a) Silica skeletons of two radiolaria, scale about 1mm, from [VAB99]. (b)-(d) (92) on a sphere of radius R = 10, nt=10000 triangular elements. (b) Some branches bifurcating at the first BP (l = 6, hence transcritical) and second BP (l = …
Figure 35
Figure 35. Figure 35: Continuation in R of a pentagonal (blue branch) and a stripe (orange branch) solution from [PITH_FULL_IMAGE:figures/full_fig_p068_35.png]
Figure 36
Figure 36. Figure 36: (93) on a rather small torus, where however already many primary bifurcation occur close together near λ ≈ 3.2. Branches b1 (dark blue), b2 (light blue), b3 (red), b4 (blue again), and 2 secondary branches c3-4 (orange) and c3-5 (brown). This BD is only a small select…
Figure 37
Figure 37. Figure 37: (93) on a ’half torus’, with Neumann BCs in x direction. (a) BD of three primary bifurcations from the trivial branch, with two example solutions from the unstable branches d1 (blue) and d10 (black) in (b). (c) shows example solutions from the red branch d7 with for i…
Figure 38
Figure 38. Figure 38: Results for (94), (c, γ, ly) = (0.1, 1, π) throughout, discretization with 4800 (cylinder Ω1) + 500 (top lid Ω2) triangles. (a) shows the mesh of the top disk, and its boundary, plotted via [x2,i2]=p.p2.grid.bdseg(1). (b) shows a basic bifurcation diagram, with five p…
Figure 39
Figure 39. Figure 39: Bistability for f with k0 = 0.05 and γ = 1, for which w1 ≈ 1.8 and w2 ≈ 2.3. Remark 6.2. Problems similar to (97) are also studied in, e.g., [RR14, GKRR16, MC16, NRV19]. In [RR14], (fast) linear diffusion for w in the bulk Ω is coupled to a RD system for species u, v …
Figure 40
Figure 40. Figure 40: Results for (97); (R, k0, γ, ε) = (1, 0.1, 1, 0.1). (a)–(c) Bifurcation diagrams, homogeneous (black), primary (blue) and secondary (magenta) patterns; (c) shows the Lagrange multiplier λm for mass conservation. (d) Sample solutions, where the cut-away plots (pstyle=4…
Figure 41
Figure 41. Figure 41: Phase diagram for fixed a = 3 and Dv = 10, and sketches of dispersion relations µ(k; par) with parameters par = (a, b, Du, Dv) as indicated and wave number k horizontally. The dispersion relation for the linearization of (101) around (102) can be studied analytically.…
Figure 42
Figure 42. Figure 42: Period doubling bifurcations to oscillating Turing patterns in (101) in 1D, (a, Du, Dv) = (3, 6, 10), Ω = (−lx, lx), lx = 2π/0.7, with Neumann BCs. (a) BD of u ∗ = (a, b/a) (steady state, black branch), primary Hopf branch (blue), first period doubling (red), secondar…
Figure 43
Figure 43. Figure 43: Period doubling to stable squares and other osc. patterns for (101) over Ω = (−lx, lx) 2 , lx = √ 2π/0.7. (a) BD of u ∗ = (a, b/a) (steady state, black branch), primary Hopf branch (blue), squares (red), and two branches at the 2nd PD from the blue branch. The stripes…
Figure 44
Figure 44. Figure 44: BD under periodic forcing with amplitude α = √ δ, β = 0.42 fixed, (a, b, Du, Dv) = (3, 9.5, 6, 10). Subcritical oscillating Turing patterns under time-periodic forcing via period doubling bifurcations from the ’natural’ forced branch. References [AGH+05] M. Alber, T. …

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Pith tools

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