Math. Model. Nat. Phenom.
Volume 16, 2021
Control of instabilities and patterns in extended systems
Article Number 46
Number of page(s) 27
Published online 26 July 2021
  1. F.T. Arecchi, S. Boccaletti and P. Ramazza, Pattern formation and competition in nonlinear optics. Phys. Rep. 318 (1999) 1–83. [CrossRef] [Google Scholar]
  2. W. Barthel, C. John and F. Tröltzsch, Optimal boundary control of a system of reaction diffusion equations. Z. Angew. Math. und Mech. 90 (2010) 966–982. [CrossRef] [Google Scholar]
  3. I.V. Biktasheva, H. Dierckx and V.N. Biktashev, Drift of scroll waves in thin layers caused by thickness features: asymptotic theory and numerical simulations. Phys. Rev. Lett. 114 (2015) 068302. [CrossRef] [Google Scholar]
  4. R. Buchholz, H. Engel, E. Kammann and F. Tröltzsch, On the optimal control of the Schlögl model. Comput. Optim. Appl. 56 (2013) 153–185. [CrossRef] [Google Scholar]
  5. A.E. Bryson, Applied optimal control: optimization, estimation and control. CRC Press (1975). [Google Scholar]
  6. E. Casas, M. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity. Comp. Opt. Appl. 70 (2018) 239–266. [CrossRef] [Google Scholar]
  7. E. Casas, C. Ryll and F. Tröltzsch, Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems. Comp. Meth. Appl. Math. 13 (2013) 415–442. [CrossRef] [Google Scholar]
  8. E. Casas, C. Ryll and F. Tröltzsch, Optimal control of a class of reaction diffusion equations. Comput. Opt. Appl. 70 (2018) 677–707. [CrossRef] [Google Scholar]
  9. E. Casas, C. Ryll and F. Tröltzsch, Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation. SIAM J. Control Optim. 53 (2015) 2168–2202. [CrossRef] [MathSciNet] [Google Scholar]
  10. J.-X. Chen, H. Zhang and Y.-Q. Li, Synchronization of a spiral by a circularly polarized electric field in reaction–diffusion systems. J. Chem. Phys. 130 (2009) 124510. [CrossRef] [Google Scholar]
  11. S. Coombes, P. Beim Graben, R. Potthast and J. Wright, Neural Fields. Springer-Verlag Berlin Heidelberg (2014). [Google Scholar]
  12. S.M. Cox and P.C. Matthews, J. Comp. Phys. 176 (2002) 430–455. [CrossRef] [MathSciNet] [Google Scholar]
  13. A. Doelman, P. van Heijster and T.J. Kaper, Pulse dynamics in a three-component system: existence analysis. J. Dyn. Differ. Equ. 21 (2009) 73–115. [CrossRef] [Google Scholar]
  14. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445–466. [CrossRef] [PubMed] [Google Scholar]
  15. E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habitat creation. Phys. Rev. Lett. 93 (2004). [CrossRef] [Google Scholar]
  16. S.V. Gurevich and R. Friedrich, Instabilities of localized structures in dissipative systems with delayed feedback. Phys. Rev. Lett. 110 (2013). [Google Scholar]
  17. S.V. Gurevich, S. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three-component reaction–diffusion system. Phys. Rev. E 74 (2006) 066201. [CrossRef] [MathSciNet] [Google Scholar]
  18. S. Gurevich, H. Bödeker, A. Moskalenko, A. Liehr and H.G. Purwins, Physica D 199 (2004) 115–128. [CrossRef] [Google Scholar]
  19. G. Haas, M. Bär, I.G. Kevrekidis, P.B. Rasmussen, H.H. Rotermund and G. Ertl, Phys. Rev. Lett. 75 (1995) 3560. [CrossRef] [PubMed] [Google Scholar]
  20. K.-H. Hoffmann, I. Lasiecka, G. Leugering, J. Sprekels and F. Tröltzsch, eds., Optimal Control of Complex Structures. Vol. 139 of ISNM. Birkhäuser Verlag (2002). [Google Scholar]
  21. K.-H. Hoffmann, G. Leugering and F. Tröltzsch, Optimal Control of Partial Differential Equations. Vol. 133 of ISNM. Birkhäuser Verlag (1998). [Google Scholar]
  22. B.S. Kerner and V.V. Osipov, Vol. 61 of Autosolitons: a new approach to problems of self-organization and turbulence. Springer Science & Business Media (2013). [Google Scholar]
  23. M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A.S. Mikhailov, H.H. Rotermund and G. Ertl, Controlling chemical turbulence by global delayed feedback: Pattern formation in catalytic CO oxidation on Pt(110). Science 292 (2001) 1357. [CrossRef] [PubMed] [Google Scholar]
  24. C.R. Laing, W.C. Troy, B. Gutkin and G.B. Ermentrout, Multiple bumps in a neuronal model of working memory. SIAM J. Appl. Math. 63 (2002) 62–97. [CrossRef] [Google Scholar]
  25. T. Le Goff, B. Liebchen and D. Marenduzzo, Pattern Formation in polymerizing actin flocks: spirals, spots, and waves without nonlinear chemistry. Phys. Rev. Lett. 117 (2016) 238002. [CrossRef] [Google Scholar]
  26. J. Löber and H. Engel, Controlling the position of traveling waves in reaction–diffusion systems. Phys. Rev. Lett. 112 (2014) 148305. [CrossRef] [Google Scholar]
  27. J. Löber, R. Coles, J. Siebert, H. Engel and E. Schöll, Control of chemical wave propagation, in Engineering of Chemical Complexity II, edited by A. Mikhailov and G. Ertl. World Scientific Singapore (2015). [Google Scholar]
  28. J. Löber, Stability of position control of traveling waves in reaction–diffusion systems. Phys. Rev. E 89 (2014) 062904. [CrossRef] [Google Scholar]
  29. J. Löber, S. Martens and H. Engel, Shaping wave patterns in reaction–diffusion systems. Phys. Rev. E 90 (2014) 062911. [CrossRef] [Google Scholar]
  30. O. Lüthje, S. Wolff and G. Pfister, Control of chaotic Taylor-Couette flow with time-delayed feedback. Phys. Rev. Lett. 86 (2001) 1745–1748. [CrossRef] [PubMed] [Google Scholar]
  31. J. Löber, Control of reaction–diffusion systems. Springer International Publishing, Cham (2017) 195–220. [Google Scholar]
  32. B. Marwaha and D. Luss, Chem. Eng. Sci. 58 (2003) 733–738. [CrossRef] [Google Scholar]
  33. A. Mikhailov and K. Showalter, Control of waves, patterns and turbulence in chemical systems. Phys. Rep. 425 (2006) 79–194. [CrossRef] [Google Scholar]
  34. A. Mikhailov, L. Schimansky-Geier and W. Ebeling, Stochastic motion of the propagating front in bistable media. Phys. Lett. A 96 (1983) 453–456. [CrossRef] [Google Scholar]
  35. Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems. Chaos 15 (2005) 047509. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  36. Y. Nishiura, T. Teramoto and X. Yuan, Heterogeneity-induced spot dynamics for a three-component reaction–diffusion system. Comm. Pure Appl. Anal. 11 (2011) 307–338. [CrossRef] [Google Scholar]
  37. V. Odent, E. Louvergneaux, M.G. Clerc and I. Andrade-Silva, Optical wall dynamics induced by coexistence of monostable and bistable spatial regions. Phys. Rev. E 94 (2016) 052220. [CrossRef] [Google Scholar]
  38. T. Pierre, G. Bonhomme and A. Atipo, Controlling the chaotic regime of nonlinear ionization waves using the time-delay autosynchronization method. Phys. Rev. Lett. 76 (1996) 2290–2293. [CrossRef] [PubMed] [Google Scholar]
  39. H.-G. Purwins, H.U. Bödeker and A.W. Liehr, Dissipative solitons in reaction–diffusion systems. Dissipative solitons. Springer (2005) 267–308. [CrossRef] [Google Scholar]
  40. H.-G. Purwins, H. Bödeker and S. Amiranashvili, Dissipative solitons. Adv. Phys. 59 (2010) 485–701. [CrossRef] [Google Scholar]
  41. L. Qiao, X. Li, I.G. Kevrekidis, C. Punckt and H.H. Rotermund, Enhancement of surface activity in CO oxidation on Pt(110) through spatiotemporal laser actuation. Phys. Rev. E 77 (2008) 036214. [CrossRef] [Google Scholar]
  42. C. Ryll, J. Löber, S. Martens, H. Engel and F. Tröltzsch, Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction–diffusion systems, in Control of Self-Organizing Nonlinear Systems, edited by E. Schöll, S.H.L. Klapp and P. Hövel. Springer (2016) 189–210. [Google Scholar]
  43. T. Sakurai, E. Mihaliuk, F. Chirila and K. Showalter, Design and control of wave propagation patterns in excitable media. Science 296 (2002) 2009–2012. [CrossRef] [PubMed] [Google Scholar]
  44. J. Schlesner, V. Zykov and H. Engel, Feedback-mediated control of hypermeandering spiral waves, in Handbook of Chaos Control. Wiley-VCH Verlag (2008) 591–607. [Google Scholar]
  45. A. Schrader, M. Braune and H. Engel, Dynamics of spiral waves in excitable media subjected to external periodic forcing. Phys. Rev. E 52 (1995) 98. [CrossRef] [Google Scholar]
  46. O. Steinbock, V.S. Zykov and S.C. Müller, Control of spiral-wave dynamics in active media by periodic modulation of excitability. Nature 366 (1993) 322–324. [CrossRef] [Google Scholar]
  47. J.S. Taube and J.P. Bassett, Persistent neural activity in head direction cells. Cereb. Cortex 13 (2003) 1162–1172. [CrossRef] [Google Scholar]
  48. S. Totz, J. Löber, J.F. Totz and H. Engel, Control of transversal instabilities in reaction–diffusion systems. NJP 20 (2018) 053034. [CrossRef] [Google Scholar]
  49. P. van Heijster, A. Doelman and T.J. Kaper, Pulse dynamics in a three-component system: stability and bifurcations. Physica D 237 (2008) 3335–3368. [CrossRef] [Google Scholar]
  50. V.K. Vanag and I.R. Epstein, Design and control of patterns in reaction–diffusion systems. Chaos 18 (2008) 026107. [CrossRef] [Google Scholar]
  51. V.K. Vanag and I.R. Epstein, Localized patterns in reaction–diffusion systems. Chaos 17 (2007) 037110. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  52. G.A. Viswanathan, M. Sheintuch and D. Luss, Transversal hot zones formation in catalytic packed-bed reactors. Ind. Eng. Chem. Res. 47 (2008) 7509–7523. [CrossRef] [Google Scholar]
  53. J. Wolff, A.G. Papathanasiou, I.G. Kevrekidis, H.H. Rotermund and G. Ertl, Spatiotemporal addressing of surface activity. Science 294 (2001) 134–137. [CrossRef] [PubMed] [Google Scholar]
  54. J. Wolff, A.G. Papathanasiou, H.H. Rotermund, G. Ertl, X. Li and I.G. Kevrekidis, Gentle dragging of reaction waves. Phys. Rev. Lett. 90 (2003) 018302. [CrossRef] [PubMed] [Google Scholar]
  55. L. Yang, A.M. Zhabotinsky and I.R. Epstein, Jumping solitary waves in an autonomous reaction–diffusion system with subcritical wave instability. Phys. Chem. Chem. Phys. 8 (2006) 4647–4651. [CrossRef] [Google Scholar]
  56. A. Ziepke, S. Martens and H. Engel, Control of nonlinear wave solutions to neural field equations. arXiv:1806.10938 (2018). [Google Scholar]
  57. A. Ziepke, S. Martens and H. Engel, Wave propagation in spatially modulated tubes. J. Chem. Phys. 145 (2016) 094108. [CrossRef] [Google Scholar]
  58. V.S. Zykov, G. Bordiougov, H. Brandtstädter, I. Gerdes and H. Engel, Periodic forcing and feedback control of nonlinear lumped oscillators and meandering spiral waves. Phys. Rev. E 68 (2003) 016214. [CrossRef] [Google Scholar]
  59. V.S. Zykov, G. Bordiougov, H. Brandtstädter, I. Gerdes and H. Engel, Global control of spiral wave dynamics in an excitable domain of circular and elliptical shape. Phys. Rev. Lett. 92 (2004) 018304. [CrossRef] [Google Scholar]

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