Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications

Uecker, Hannes

doi:10.4208/cicp.oa-2017-0181

Cited by 31 publications

(57 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [UW14] we numerically study planar fronts between stripes and hexagons in the 2D Schnakenberg model, using the package pde2path [UWR14,Uec19c]. Such fronts (or heteroclinic connections) can be naturally extended to localized patterns (or homoclinic cycles) by gluing together fronts and backs (i.e., considering heteroclinic cycles).…”

Section: Introductionmentioning

confidence: 99%

Snaking branches of planar BCC fronts in the 3D Brusselator

Uecker

Wetzel

2020

Physica D: Nonlinear Phenomena

Self Cite

View full text Add to dashboard Cite

We present results of the application of the numerical continuation and bifurcation package pde2path to the 3D Brusselator model, focusing on snaking branches of planar fronts between body centered cubes (BCCs) and the spatial homogeneous solution, and on planar fronts between BCCs and tubes (also called square prisms). These solutions also yield approximations of localized BCCs, and of BCCs embedded in a background of tubes (or vice versa). Additionally, we compute some moving fronts between lamellas and tubes. To give some theoretical background, and to aid the numerics for the full system, we use the Maxwell points for the cubic amplitude system over the BCC lattice.

show abstract

Section: Introductionmentioning

confidence: 99%

Snaking branches of planar BCC fronts in the 3D Brusselator

Uecker

Wetzel

2020

Physica D: Nonlinear Phenomena

Self Cite

View full text Add to dashboard Cite

show abstract

“…See, for example, Grass et al () for various ODE applications, and Uecker () and Grass and Uecker () for some PDE examples. Additionally, there may be optimal limit cycles (Uecker, , section 3.4; Wirl, ), or (slow, i.e., such that still holds) divergence of optimal solutions to infinity, see the AK example in Section .…”

Section: The Cs Formalismmentioning

confidence: 99%

“…See Grass and Uecker () for further comments on the significance of the SPP on the discrete level, and its (mesh‐independent) meaning for the CS as a PDE. Furthermore, see Uecker () for algorithmic details how to implement in pde2path, and how to find CPs connecting some

v_{0}

\overset{u}{true}

by a continuation process in the initial states.…”

Section: The Cs Formalismmentioning

confidence: 99%

Optimal fishery with coastal catch

Grass

Uecker²,

Upmann

2019

Natural Resource Modeling

Self Cite

View full text Add to dashboard Cite

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs). These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states. The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts. Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

show abstract

“…To study the bifurcation structure, we use and extend the continuation software for PDEs pde2path [20,46,48], based on a FEM discretization of the stationary problem. Since the general class of problems which can be numerically analyzed by the software does not include the cross-diffusion term appearing in (1.5), it requires an additional setup to be able to compare the fast-reaction PDE results with the singular limit cross-diffusion systems.…”

Section: Introductionmentioning

confidence: 99%

Numerical continuation for a fast-reaction system and its cross-diffusion limit

Kuehn

Soresina

2020

SN Partial Differ. Equ. Appl.

View full text Add to dashboard Cite

In this paper we investigate the bifurcation structure of the triangular SKT model in the weak competition regime and of the corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be adapted to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the Turing instability analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limits onto the cross-diffusion singular limit. Furthermore, we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability, and improve our understanding of the shape of patterns in 2D for the SKT model.

show abstract

Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications

Cited by 31 publications

References 52 publications

Snaking branches of planar BCC fronts in the 3D Brusselator

Snaking branches of planar BCC fronts in the 3D Brusselator

Optimal fishery with coastal catch

Numerical continuation for a fast-reaction system and its cross-diffusion limit

Contact Info

Product

Resources

About