Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Hannes Uecker1
1Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany

Tóm tắt

AbstractNumerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

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Tài liệu tham khảo

Angenent, S.B., Mallet-Paret, J., Peletier, L.A.: Stable transition layers in a semilinear boundary value problem. J. Differ. Equ. 67(2), 212–242 (1987)

Avitabile, D., Lloyd, D.J.B., Burke, J., Knobloch, E., Sandstede, B.: To snake or not to snake in the planar Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst. 9(3), 704–733 (2010)

Barrett, J.W., Wood, P.A.: The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities. Numer. Math. 71(2), 135–157 (1995)

Bergeon, A., Burke, J., Knobloch, E., Mercader, I.: Eckhaus instability and homoclinic snaking. Phys. Rev. E 78, 046201 (2008)

Bollhöfer, M.: ILUPACK V2.4 (2011). www.icm.tu-bs.de/~bolle/ilupack/

Breden, M., Kuehn, C., Soresina, C.: On the influence of cross-diffusion in pattern formation. J. Comput. Dyn. 8(2), 213–240 (2021)

Burke, J., Knobloch, E.: Localized states in the generalized Swift-Hohenberg equation. Phys. Rev. E 73, 056211 (2006)

Burke, J., Knobloch, E.: Homoclinic snaking: structure and stability. Chaos 17(3), 037102 (2007)

Carter, P., Rademacher, J.D.M., Sandstede, B.: Pulse replication and accumulation of eigenvalues. SIAM J. Math. Anal. 53(3), 3520–3576 (2021)

Chapman, S.J., Kozyreff, G.: Exponential asymptotics of localised patterns and snaking bifurcation diagrams. Physica D 238, 319–354 (2009)

Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

Cross, M., Greenside, H.: Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press, Cambridge (2009)

Cross, M.C., Hohenberg, P.C.: Pattern formation outside equilibrium. Rev. Mod. Phys. 65, 854–1190 (1993)

Dankowicz, H., Schilder, F.: Recipes for Continuation. Comp. Sc. & Eng., vol. 11. SIAM, Philadelphia (2013)

De Wit, A., Lima, D., Dewel, G., Borckmans, P.: Spatiotemporal dynamics near codimension-two point. Phys. Rev. E 54(1), 261–271 (1996)

de Witt, H., Dohnal, T., Rademacher, J.D.M., Uecker, H., Wetzel, D.: pde2path - Quickstart guide and reference card (2020)

Decker, D., Keller, H.B.: Multiple limit point bifurcation. J. Math. Anal. Appl. 75(2), 417–430 (1980)

Delgado, M., Suárez, A.: On the existence of dead cores for degenerate Lotka-Volterra models. Proc. R. Soc. Edinb. A 130(4), 743–766 (2000)

Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003)

di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems. Applied Mathematical Sciences, vol. 163. Springer, London (2008)

di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Olivar Tost, G., Piiroinen, P.T.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50(4), 629–701 (2008)

Díaz, J.I., Hernández, J., Mancebo, F.J.: Branches of positive and free boundary solutions for some singular quasilinear elliptic problems. J. Math. Anal. Appl. 352(1), 449–474 (2009)

Dijkstra, H.A., Wubs, F.W., Cliffe, A.K., Doedel, E., Dragomirescu, I., Eckhardt, B., Gelfgat, A.Y., Hazel, A.L., Lucarini, V., Salinger, A.G., Phipps, E.T., Sanchez-Umbria, J., Schuttelaars, H., Tuckerman, L.S., Thiele, U.: Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys. 15, 1–45 (2014)

Doedel, E.J.: Lecture Notes on Numerical Analysis of Nonlinear Equations. In: Numerical Continuation Methods for Dynamical Systems, pp. 1–49. Springer, Dordrecht (2007)

Doedel, E., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang AUTO, X.: Continuation and bifurcation software for ordinary differential equations (with HomCont) (1997). http://indy.cs.concordia.ca/auto/

Ehstand, N., Kuehn, C., Soresina, C.: Numerical continuation for fractional PDEs: sharp teeth and bloated snakes (2020)

Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems. Software, Environments, and Tools, vol. 14. SIAM, Philadelphia (2002)

Fairgrieve, T.F., Jepson, A.D.: O. K. Floquet multipliers. SIAM J. Numer. Anal. 28(5), 1446–1462 (1991)

Friedman, A., Phillips, D.: The free boundary of a semilinear elliptic equation. Trans. Am. Math. Soc. 282(1), 153–182 (1984)

García-Melián, J., Rossi, J., Sabina de Lis, J.: A bifurcation problem governed by the boundary condition. II. Proc. Lond. Math. Soc. (3) 94(1), 1–25 (2007)

Golubitsky, M., Stewart, I.: The Symmetry Perspective. Birkhäuser, Basel (2002)

Govaerts, W.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000)

Hazel, A., Heil, M.: oomph-lib (2017). http://oomph-lib.maths.man.ac.uk/doc/html

Hoyle, R.B.: Pattern Formation. Cambridge University Press, Cambridge (2006)

Jensen, K.E.: A MATLAB script for solving 2D/3D miminum compliance problems using anisotropic mesh adaptation. In: 26th International Meshing Roundtable, vol. 203, pp. 102–114 (2017)

Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Application of bifurcation theory, Proc. adv. Semin., Madison/Wis., 1976 pp. 359–384 (1977)

Keller, H.B.: Lectures on Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 79. Springer, Berlin (1987)

Keller, H.B., Langford, W.F.: Iterations, perturbations and multiplicities for nonlinear bifurcation problems. Arch. Ration. Mech. Anal. 48, 83–108 (1972)

Knobloch, E.: Spatially localized structures in dissipative systems: open problems. Nonlinearity 21, T45–T60 (2008)

Knobloch, E.: Spatial localization in dissipative systems. Annu. Rev. Condens. Matter Phys. 6, 325–359 (2015)

Kolokolnikov, Th., Paquin-Lefebvre, F., Ward, M.J.: Competition instabilities of spike patterns for the 1D Gierer-Meinhardt and Schnakenberg models are subcritical. Nonlinearity 34(1), 273–312 (2021)

Kressner, D.: An efficient and reliable implementation of the periodic QZ algorithm. In: IFAC Workshop on Periodic Control Systems (2001)

Kuehn, C.: Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes. J. Phys. A 42(4), 045101 (2009)

Kuehn, C., Soresina, C.: Numerical continuation for a fast–reaction system and its cross-diffusion limit. SN Partial Differ. Equ. Appl. 1, 7 (2020)

Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Applied Mathematical Sciences, vol. 112. Springer, New York (2004)

Le Berre, M., Petrescu, A.S., Ressayre, E., Tallet, A.: Daisy patterns in the passive ring cavity with diffusion effects. Opt. Commun. 123, 810–824 (1996)

Lee, S.-Y., Wang, S.-H., Ye, C.-P.: Explicit necessary and sufficient conditions for the existence of a dead core solution of a $p$-Laplacian steady-state reaction-diffusion problem. Discrete Contin. Dyn. Syst. suppl, 587–596 (2005)

Leine, R.I.: Bifurcations of equilibria in non-smooth continuous systems. Phys. D 223(1), 121–137 (2006)

Lust, K.: Improved numerical Floquet multipliers. Int. J. Bifurc. Chaos 11(9), 2389–2410 (2001)

Makarenkov, O., Lamb, J.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241(22), 1826–1844 (2012)

Mazzia, F., Trigiante, D.: A hybrid mesh selection strategy based on conditioning for boundary value ODE problems. Numer. Algorithms 36(2), 169–187 (2004)

Meijer, H., Dercole, F., Oldeman, B.: Numerical bifurcation analysis. In: Mathematics of Complexity and Dynamical Systems. Vols. 1–3, pp. 1172–1194. Springer, New York (2012)

Meixner, M., De Wit, A., Bose, S., Schöll, E.: Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations. Phys. Rev. E 55(6, Part A), 6690–6697 (1997)

Nochetto, R.H.: Sharp $L^{\infty }$-error estimates for semilinear elliptic problems with free boundaries. Numer. Math. 54(3), 243–255 (1988)

Ophaus, L., Knobloch, E., Gurevich, S.V., Thiele, U.: Two-dimensional localized states in an active phase-field-crystal model. Phys. Rev. E 103(3), 032601 (2021)

Pismen, L.M.: Patterns and Interfaces in Dissipative Dynamics. Springer, Berlin (2006)

Pomeau, Y.: Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23, 3–11 (1986)

Prüfert, U.: OOPDE (2021). https://tu-freiberg.de/fakult1/nmo/pruefert

Salinger, A.: LOCA (2016). www.cs.sandia.gov/LOCA/

Sánchez, J., Net, M.: Numerical continuation methods for large-scale dissipative dynamical systems. Eur. Phys. J. Spec. Top. 225, 2465–2486 (2016)

Schneider, G., Uecker, H.: Nonlinear PDE – A Dynamical Systems Approach. Graduate Studies Mathematics, vol. 182. Am. Math. Soc., Providence (2017)

Siero, E.: Resolving soil and surface water flux as drivers of pattern formation in Turing models of dryland vegetation: a unified approach. Phys. D 414, 132695 (2020)

Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15(1), 319–328 (1977)

Teixeira, E.V.: Regularity for the fully nonlinear dead-core problem. Math. Ann. 364(3–4), 1121–1134 (2016)

Tuckerman, L.S.: Computational challenges of nonlinear systems. In: Emerging Frontiers in Nonlinear Science, pp. 249–277. Springer, Berlin (2020)

Tzou, J.C., Xie, S., Kolokolnikov, T., Ward, M.J.: The stability and slow dynamics of localized spot patterns for the 3-D Schnakenberg reaction-diffusion model. SIAM J. Appl. Dyn. Syst. 16(1), 294–336 (2017)

Uecker, H.: Hopf bifurcation and time periodic orbits with pde2path – algorithms and applications. Commun. Comput. Phys. 25(3), 812–852 (2019)

Uecker, H.: Numerical Continuation and Bifurcation in Nonlinear PDEs. SIAM, Philadelphia (2021)

Uecker, H.: Optimal spatial patterns in feeding, fishing and pollution. DCDS-S (2021)

Uecker, H.: pde2path with higher order finite elements (2021). Available at [73]

Uecker, H.: pde2path without finite elements (2021). Available at [73]

Uecker, H.: Supplementary information for this paper (2021). Available at [73]

Uecker, H.: (2021). www.staff.uni-oldenburg.de/hannes.uecker/pde2path

Uecker, H., Wetzel, D.: Numerical results for snaking of patterns over patterns in some 2D Selkov-Schnakenberg reaction-diffusion systems. SIAM J. Appl. Dyn. Syst. 13(1), 94–128 (2014)

Uecker, H., Wetzel, D.: Snaking branches of planar BCC fronts in the 3D Brusselator. Physica D 406, 132383 (2020)

Uecker, H., Wetzel, D., Rademacher, J.D.M.: pde2path – a Matlab package for continuation and bifurcation in 2D elliptic systems. Numer. Math., Theory Methods Appl. 7, 58–106 (2014)

Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia (2011)

Upmann, T., Uecker, H., Hammann, L., Blasius, B.: Optimal stock enhancement activities for a spatially distributed renewable resource. J. Econ. Dyn. Control 123, 104060 (2021)

Veltz, R.: BifurcationKit.jl (2020). https://hal.archives-ouvertes.fr/hal-02902346

Verschueren, N.: Pattern formation on a finite disk using the SH35 equation (2021). https://nverschueren.bitbucket.io/sh35p2p.html. Online tutorial

Verschueren, N., Knobloch, E., Uecker, H.: Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk. Phys. Rev. E 104, 014208, (2021)

Wong, T., Ward, M.: Weakly nonlinear analysis of peanut-shaped deformations for localized spots of singularly perturbed reaction-diffusion systems. SIAM J. Appl. Dyn. Syst. 19(3), 2030–2058 (2020)

Woolley, T.E., Krause, A.L., Gaffney, E.A.: Bespoke Turing systems. Bull. Math. Biol. 83(5), 41 (2021)

Yang, L., Dolnik, M., Zhabotinsky, A.M., Epstein, I.R.: Pattern formation arising from interactions between Turing and wave instabilities. J. Chem. Phys. 117(15), 7259–7265 (2002)

Zeidler, E.: Nonlinear Functionalanalysis I. Springer, Berlin (1989)

Zhao, L.-X., Zhang, K., Siteur, K., Li, X.-Z., Liu, Q.-X., van de Koppel, J.: Fairy circles reveal the resilience of self-organized salt marshes. Sci. Adv. 7, 100 (2021)

Zhen, M.: Numerical Bifurcation Analysis for Reaction-Diffusion Equations. Springer, Berlin (2000)