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

Abstract

Numerical 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.

Stationary and oscillatory localized patterns in ratio-dependent predator–prey systems

“…We show that it is a consequence of the nonlinear interaction between a supercritical Hopf bifurcation of the trivial state and a nearby subcritical Turing instability, and in particular of the presence of tertiary Hopf bifurcations on both the Turing branches and the associated LS inherited from the primary Hopf bifurcations of the trivial state. As such this behavior appears characteristic of the interaction between the Turing and Hopf instabilities 17,27,[38][39][40][41] and we demonstrate its presence in two different RD systems.…”

“…We have revisited the Hopf-Turing interaction that arises in a number of two-species reaction-diffusion systems. 17,27,39,41 In particular, in Ref. 17 the authors considered the Brusselator model with supercritical Hopf and Turing branches in a regime with bistability between the two, finding a large multiplicity of stable Turing states embedded in an oscillating background obtained via DNS.…”

Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems

“…The software is specifically designed to provide bifurcation diagrams for PDEs by applying a modified (pseudo-)arclength parametrisation of solution branches to a spatial discretisation of the PDEs. Critically, the software has been optimised to deal with a large number of degrees of freedom, which will arise due to the PDE being turned into a large ODE system, and compounded further if higher spatial dimensions are considered (Uecker 2021b). It should be noted that, since we have turned to numerical continuation techniques, we can only follow bifurcation branches if we they are found to be connected to known branches.…”

Mesh selection strategies of the code TOM for Boundary Value Problems

“…Recently a comparison of some of the codes available in Matlab for the solution of optimal control problems has been made in [14]. General purpose codes for BVPs are used for the solution of many problems in applications see, for example, [15][16][17] for optimal control problems and path planning, [18][19][20] for numerical simulation, and [21,22] for analysis of bifurcation. General purpose codes are really useful for many real world problems.…”