Reaction–diffusion equations are ubiquitous in various scientific domains and their patterns represent a fascinating area of investigation. However, many of these patterns are unstable and, therefore, challenging to observe. To overcome this limitation, we present new noninvasive feedback controls based on symmetry groupoids. As a concrete example, we employ these controls to selectively stabilize unstable equilibria of the Chafee–Infante equation under Dirichlet boundary conditions on the interval. Unlike conventional reflection-based control schemes, our approach incorporates additional symmetries that enable us to design new convolution controls for stabilization. By demonstrating the efficacy of our method, we provide a new tool for investigating and controlling systems with unstable patterns, with potential implications for a wide range of scientific disciplines.
REFERENCES
1.
H.
Meinhardt
, A.-J.
Koch
, and G.
Bernasconi
, “Models of pattern formation applied to plant development
,” in Symmetry in Plants
, edited by R. V. Jean and D. Barabé (World Scientific, 1998), pp. 723
–758
.2.
J. D.
Murray
, “How the leopard gets its spots
,” Sci. Am.
258
, 80
–87
(1988
). 3.
H.
Meinhardt
, The Algorithmic Beauty of Sea Shells
(Springer-Verlag
, New York
, 1995
).4.
A. T.
Winfree
, “The prehistory of the Belousov–Zhabotinsky oscillator
,” J. Chem. Educ.
61
, 661
(1984
). 5.
A. M.
Zhabotinsky
, “A history of chemical oscillations and waves
,” Chaos
1
, 379
–386
(1991
). 6.
D.
Rand
, “Dynamics and symmetry. Predictions for modulated waves in rotating fluids
,” Arch. Ration. Mech. Anal.
79
, 1
–37
(1982
). 7.
J. D.
Crawford
and E.
Knobloch
, “Symmetry and symmetry-breaking bifurcations in fluid dynamics
,” Annu. Rev. Fluid Mech.
23
, 341
–387
(1991
). 8.
R.
Hoyle
, Pattern Formation: An Introduction to Methods
(Cambridge University Press
, 2006
).9.
W.
Lu
, D.
Yu
, and R. G.
Harrison
, “Control of patterns in spatiotemporal chaos in optics
,” Phys. Rev. Lett.
76
, 3316
(1996
). 10.
A. M.
Turing
, “The chemical basis of morphogenesis
,” Philos. Trans. R. Soc., B
237
, 37
–72
(1952
).11.
B.
Fiedler
and A.
Scheel
, “Spatio-temporal dynamics of reaction-diffusion patterns,” in Trends in Nonlinear Analysis (Springer, Berlin, 2003), pp. 23–152.12.
J. D.
Murray
, Mathematical Biology II: Spatial Models and Biomedical Applications
(Springer
, New York
, 2003
). Vol. 18
.13.
M.
Golubitsky
and I.
Stewart
, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics Vol. 200 (Birkhäuser, Basel, 2003)).14.
P.
Hövel
and E.
Schöll
, “Control of unstable steady states by time-delayed feedback methods
,” Phys. Rev. E
72
, 046203
(2005
).15.
S. J.
Jensen
, M.
Schwab
, and C.
Denz
, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering
,” Phys. Rev. Lett.
81
, 1614
(1998
). 16.
Y.
Kyrychko
, K.
Blyuss
, S.
Hogan
, and E.
Schöll
, “Control of spatiotemporal patterns in the Gray–Scott model
,” Chaos
19
, 043126
(2009
). 17.
J.
Lehnert
, P.
Hövel
, V.
Flunkert
, P. Y.
Guzenko
, A. L.
Fradkov
, and E.
Schöll
, “Adaptive tuning of feedback gain in time-delayed feedback control
,” Chaos
21
, 043111
(2011
). 18.
E.
Ott
, C.
Grebogi
, and J. A.
Yorke
, “Controlling chaos
,” Phys. Rev. Lett.
64
, 1196
(1990
). 19.
W.-J.
Rappel
, F.
Fenton
, and A.
Karma
, “Spatiotemporal control of wave instabilities in cardiac tissue
,” Phys. Rev. Lett.
83
, 456
(1999
). 20.
S.
Schikora
, P.
Hövel
, H.-J.
Wünsche
, E.
Schöll
, and F.
Henneberger
, “All-optical noninvasive control of unstable steady states in a semiconductor laser
,” Phys. Rev. Lett.
97
, 213902
(2006
). 21.
S.
Yanchuk
, M.
Wolfrum
, P.
Hövel
, and E.
Schöll
, “Control of unstable steady states by long delay feedback
,” Phys. Rev. E
74
, 026201
(2006
). 22.
K.
Pyragas
, “Continuous control of chaos by self-controlling feedback
,” Phys. Lett. A
170
, 421
–428
(1992
). 23.
I.
Schneider
, “An introduction to the control triple method for partial differential equations,” in Patterns of Dynamics (Springer, Berlin, 2016), pp. 269–285.24.
I.
Schneider
, B.
de Wolff
, and J.-Y.
Dai
, “Pattern-selective feedback stabilization of Ginzburg–Landau spiral waves
,” Arch. Ration. Mech. Anal.
246
, 631
–658
(2022
). 25.
N.
Chafee
and E. F.
Infante
, “A bifurcation problem for a nonlinear partial differential equation of parabolic type
,” Appl. Anal.
4
, 17
–37
(1974
). 26.
D.
Henry
, Geometric Theory of Semilinear Parabolic Equations
(Springer
, Berlin
, 1981
).27.
B.
Fiedler
and C.
Rocha
, “Heteroclinic orbits of semilinear parabolic equations
,” J. Differ. Equ.
125
, 239
–281
(1996
). 28.
K.
Pyragas
, “Delayed feedback control of chaos
,” Philosophical Trans. R. Soc. London A: Math., Phys. Eng. Sci.
364
, 2309
–2334
(2006
).29.
K.
Yamasue
, K.
Kobayashi
, H.
Yamada
, K.
Matsushige
, and T.
Hikihara
, “Controlling chaos in dynamic-mode atomic force microscope
,” Phys. Lett. A
373
, 3140
–3144
(2009
). 30.
H.
Nakajima
and Y.
Ueda
, “Half-period delayed feedback control for dynamical systems with symmetries
,” Phys. Rev. E
58
, 1757
–1763
(1998
). 31.
J. E.
Socolar
, D. W.
Sukow
, and D. J.
Gauthier
, “Stabilizing unstable periodic orbits in fast dynamical systems
,” Phys. Rev. E
50
, 3245
(1994
). 32.
A.
Kittel
, J.
Parisi
, and K.
Pyragas
, “Delayed feedback control of chaos by self-adapted delay time
,” Phys. Lett. A
198
, 433
–436
(1995
). 33.
I.
Schneider
and M.
Bosewitz
, “Eliminating restrictions of time-delayed feedback control using equivariance
,” Discrete Contin. Dyn. Syst. A
36
, 451
–467
(2016
). 34.
P.
Chossat
and R.
Lauterbach
, Methods in Equivariant Bifurcations and Dynamical Systems, Advanced Series in Nonlinear Dynamics Vol. 15 (World Scientific, Singapore, 2000).35.
M.
Golubitsky
, D. G.
Schaeffer
, and I.
Stewart
, Singularities and Groups in Bifurcation Theory, Applied Mathematical Sciences Vol. 69 (Springer-Verlag, New York, 1988), Vol. 2.36.
I.
Schneider
, Symmetry Groupoids in Dynamical Systems: Spatio-temporal Patterns and a Generalized Equivariant Bifurcation Theory (Freie Universität Berlin, in press).37.
R.
Brown
, “From groups to groupoids: A brief survey
,” Bull. London Math. Soc.
19
, 113
–134
(1987
). 38.
A.
Ibort
and M.
Rodriguez
, An Introduction to Groups, Groupoids and Their Representations
(CRC Press
, 2019
).39.
J. C.
Baez
and J.
Dolan
, “From finite sets to feynman diagrams,” in Mathematics Unlimited–2001 and Beyond (Springer, Berlin, 2001), pp. 29–50.40.
F. M.
Ciaglia
, A.
Ibort
, and G.
Marmo
, “A gentle introduction to Schwinger’s formulation of quantum mechanics: The groupoid picture
,” Mod. Phys. Lett. A
33
, 1850122
(2018
). 41.
C.
Sämann
and R. J.
Szabo
, “Groupoids, loop spaces and quantization of 2-plectic manifolds
,” Rev. Math. Phys.
25
, 1330005
(2013
).42.
D. S.
Freed
and G. W.
Moore
, “Twisted equivariant matter,” in Annales Henri Poincaré (Springer, 2013), Vol. 14, pp. 1927–2023.43.
C.
Blohmann
, M. C. B.
Fernandes
, and A.
Weinstein
, “Groupoid symmetry and constraints in general relativity
,” Commun. Contemp. Math.
15
, 1250061
(2013
). 44.
J.-Y.
Dai
, “Ginzburg–Landau spiral waves in circular and spherical geometries
,” SIAM J. Math. Anal.
53
, 1004
–1028
(2021
). 45.
J.-Y.
Dai
and P.
Lappicy
, “Ginzburg–Landau patterns in circular and spherical geometries: Vortices, spirals, and attractors
,” SIAM J. Appl. Dyn. Syst.
20
, 1959
–1984
(2021
). 46.
M. G.
Crandall
and P. H.
Rabinowitz
, “Bifurcation from simple eigenvalues
,” J. Funct. Anal.
8
, 321
–340
(1971
). 47.
S.
Angenent
, “The zero set of a solution of a parabolic equation
,” J. Reine Angew. Math.
390
, 79
–96
(1988
).48.
T.
Kato
, Perturbation Theory for Linear Operators
(Springer
, Berlin
, 1995
).49.
B.
Bamieh
, F.
Paganini
, and M. A.
Dahleh
, “Distributed control of spatially invariant systems
,” IEEE Trans. Autom. Control
47
, 1091
–1107
(2002
). 50.
R.
Martin
, A.
Scroggie
, G.-L.
Oppo
, and W.
Firth
, “Stabilization, selection, and tracking of unstable patterns by fourier space techniques
,” Phys. Rev. Lett.
77
, 4007
(1996
). 51.
P.
Olver
, “The symmetry groupoid and weighted signature of a geometric object
,” J. Lie Theory
26
, 235
–267
(2015
).52.
A.
Weinstein
, “Groupoids: Unifying internal and external symmetry. A tour through some examples
,” Notices AMS
43
, 744
–752
(1996
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.