Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-22T00:05:49.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Isolas of multi-pulse solutions to lattice dynamical systems

Part of: General theory in ordinary differential equations Smooth dynamical systems: general theory Qualitative theory

Published online by Cambridge University Press:  20 July 2020

Jason J. Bramburger
Jason J. Bramburger*
Affiliation:
Department of Mathematics Statistics, University of Victoria, Victoria, BC, V8P 5C2, Canada and Division of Applied Mathematics, Brown University, Providence, RI02906, USA (jbramburger7@uvic.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.

Keywords

BifurcationLattice Dynamical SystemLocalized StructureSnakingIsola

MSC classification

Primary: 34C23: Bifurcation
Secondary: 34A33: Lattice differential equations 37C05: Smooth mappings and diffeomorphisms
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 916 - 952
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aougab, T., Beck, M., Carter, P., Desai, S., Sandstede, B., Stadt, M. and Wheeler, A.. Isolas versus snaking of localized rolls. J. Dyn. Differ. Eqns. 31 (2019), 1199–1222.CrossRefGoogle Scholar
Bramburger, J. J. and Sandstede, B.. Spatially localized structures in lattice dynamical systems. J. Nonlinear Sci. 30 (2020), 603–644.CrossRefGoogle Scholar
Bramburger, J. J. and Sandstede, B.. Localized patterns in planar bistable lattice systems. Nonlinearity 33 (2020), 3500–3525.CrossRefGoogle Scholar
Beck, M., Knobloch, J., Lloyd, D., Sandstede, B. and Wagenknecht, T.. Snakes, ladders, and isolas of localized patterns. SIAM J. Math. Anal. 41 (2009), 936–972.CrossRefGoogle Scholar
Berestycki, H., Wei, J. and Winter, M.. Existence of symmetric and asymmetric spikes for a crime hotspot model. SIAM J. Math. Anal. 46 (2014), 691–719.CrossRefGoogle Scholar
Bordeu, I., Clerc, M. G., Couteron, P., Lefever, R. and Tlidi, M.. Self-replicating of localized vegetation patches in scarce environments. Sci. Rep.-UK 6 (2016), 33703–00.00CrossRefGoogle Scholar
Brucal-Hallare, M. and Van Vleck, E.. Traveling wavefronts in an antidiffusion lattice Nagumo model. SIAM J. Appl. Dyn. Syst. 10 (2011), 921–959.CrossRefGoogle Scholar
Burke, J. and Knobloch, E.. Multi-pulse states in the Swift-Hohenberg equation. Discret. Contin. Dyn. Syst. 25 (2009), 109–117.Google Scholar
Chapman, S. J. and Kozyreff, G.. Exponential asymptotics of localised patterns and snaking bifurcation diagrams. Physica D 238 (2009), 319–354.CrossRefGoogle Scholar
Chong, C., Carretero-González, R., Malomed, B. A. and Kevrekidis, P. G.. Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices. Physica D 238 (2009), 126–136.CrossRefGoogle Scholar
Chong, C. and Pelinovsky, D. E.. Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices. Discrete Cont. Dyn.-S 4 (2011), 1019–1031.Google Scholar
Clerc, M. G., Elias, R. G. and Rojas, R. G.. Continuous description of lattice discreteness effects in front propagation. Phil. Trans. R. Soc. A 369 (2011), 412–424.CrossRefGoogle ScholarPubMed
Elmer, C. E.. Finding stationary fronts for a discrete Nagumo and wave equation; construction. Physica D 218 (2006), 11–23.CrossRefGoogle Scholar
Elmer, C. E. and Van Vleck, E. S.. Spatially discrete Fitzhugh-Nagumo equations. SIAM J. Appl. Math. 65 (2005), 1153–1174.CrossRefGoogle Scholar
Fáth, G.. Propagation failure of traveling waves in a discrete bistable medium. Physica D 116 (1998), 176–190.CrossRefGoogle Scholar
Groves, M., Lloyd, D. and Stylianou, A.. Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation. Physica D 350 (2017), 1–12.CrossRefGoogle Scholar
Guo, J.-S. and Wu, C.-H.. Wave propagation for a two-component lattice dynamical system arising in strong competition models. J. Differ. Equations 250 (2011), 3504–3533.CrossRefGoogle Scholar
van der Heijden, G. H. M., Champneys, A. R. and Thompson, J. M. T.. Spatially complex localisation in twisted elastic rods constrained to a cylinder. Int. J. Solids. Struct. 39 (2002), 1863–1883.CrossRefGoogle Scholar
Hupkes, H. J., Morelli, L., Schouten-Straatman, W. M. and Van Vleck, E. S.. (2020) Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations. In: Bohner M., Siegmund S., imon Hilscher R., Stehlk P. (eds) Difference Equations and Discrete Dynamical Systems with Applications. ICDEA 2018. Springer Proceedings in Mathematics & Statistics, vol 312. Springer, Cham.CrossRefGoogle Scholar
Keener, J. P.. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47 (1987), 556–572.CrossRefGoogle Scholar
Knobloch, J., Lloyd, D. J. B., Sandstede, B. and Wagenknecht, T.. Isolas of 2-pulse solutions in homoclinic snaking scenarios. J. Differ. Equ. 23 (2011), 93–114.Google Scholar
Kozyreff, G. and Chapman, S. J.. Asymptotics of large bound states of localised structures. Phys. Rev. Lett. 97 (2006), 044502.CrossRefGoogle Scholar
Kusdiantara, R. and Susanto, H.. Homoclinic snaking in the discrete Swift–Hohenberg equation. Phys. Rev. E 96 (2017), 062214.CrossRefGoogle ScholarPubMed
Lloyd, D. and O'Farrell, H.. On localised hotspots of an urban crime model. Physica D 253 (2013), 23–39.CrossRefGoogle Scholar
Makrides, E. and Sandstede, B.. Existence and stability of spatially localized patterns. J. Differ. Equ. 266 (2019), 1073–1120.CrossRefGoogle Scholar
McCullen, N. and Wagenknecht, T.. Pattern formation on networks: from localized activity to Turing patterns. Sci. Rep.-UK 6 (2016), 27397.CrossRefGoogle Scholar
Meron, E.. Pattern-formation approach to modelling spatially extended ecosystems. Ecol. Model. 234 (2012), 70–82.CrossRefGoogle Scholar
Michaels, T. C. T., Kusters, R., Dear, A. J., Storm, C., Weaver, J. C. and Mahadevan, L.. Geometric localization in supported elastic struts. P. Roy. Soc. A-Math. Phy. 475 (2019), 20190370.Google ScholarPubMed
Palis, J. and Melo, W.. Geometric theory of dynamical systems: an introduction (New York: Springer-Verlag, 1982).CrossRefGoogle Scholar
Papangelo, A., Grolet, A., Salles, L., Hoffman, N. and Ciavarella, M.. Snaking bifurcations of self-excited oscillator chain with cyclic symmetry. Commun. Nonlinear Sci. Numer. Simul. 44 (2006), 642–647.Google Scholar
Sandstede, B.. Stability of multiple-pulse solutions. T. Am. Math. Soc. 350 (1998), 429–472.CrossRefGoogle Scholar
Sheffer, E., Yizhaq, H., Shachak, M. and Meron, E.. Mechanisms of vegetation-ring formation in water-limited systems. J. Theor. Bio. 273 (2011), 138–146.CrossRefGoogle ScholarPubMed
Smale, S.. Differentiable dynamical systems. B. Am. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
Subramanian, P., Archer, A. J., Knobloch, E. and Rucklidge, A. M.. Spatially localized quasicrystalline structures. New J. Phys. 20 (2018), 122022.CrossRefGoogle Scholar
Susanto, H., Kusdiantara, R., Li, N., Kirikchi, O. B., Adzkiya, D., Putri, E. R. M. and Asfihani, T.. Snakes and ghosts in a parity-time-symmetric chain of dimers. Phys. Rev. E 97 (2018), 062204.CrossRefGoogle Scholar
Taranenko, V. B., Ganne, I., Kuszelewicz, R. J. and Weiss, C. O.. Patters and localized structures in bistable semiconductor resonators. Phys. Rev. A 61 (2000), 063818.CrossRefGoogle Scholar
Taylor, C. and Dawes, J. H. P.. Snaking and isolas of localised states in bistable discrete lattices. Phys. Rev. A 375 (2010), 14–22.Google Scholar
Tse, W. H. and Ward, M. J.. Hotspot formation and dynamics for a continuum model of urban crime. Eur. J. Appl. Math. 27 (2015), 583–624.CrossRefGoogle Scholar
Vanag, V. K., Zhabotinksky, A. M. and Epstein, I. R.. Pattern formation in the Belousov-Zhabotinksky reaction with photochemical global feedback. J. Phys. Chem. A 104 (2000), 11566–11577.CrossRefGoogle Scholar
Wadee, M. K., Coman, C. D. and Bassom, A. P.. Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation. Physica D 163 (2002), 26–48.CrossRefGoogle Scholar
Wiggins, S.. Introduction to applied nonlinear dynamical systems and chaos (New York: Springer-Verlag, 2003).Google Scholar
Yulin, A. V. and Champneys, A. R.. Discrete snaking: multiple cavity solitons in saturable media. SIAM J. Appl. Dyn. Syst. 9 (2010), 391–431.CrossRefGoogle Scholar
Yulin, A. V. and Champneys, A. R.. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discret. Contin. Dyn. S 4 (2011), 1341–1357.Google Scholar