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B10
Geometric desingularization of non-hyperbolic iterated maps

A coherent theory for geometrically resolving singularities in time-discrete dynamical systems

Singularities are ubiquitous in dynamical systems. They often mark boundaries between different dynamical regimes and also serve as organizing centers for the geometry of phase space and parameter space. In this project, we aim to extend geometric desingularization methods developed in the context of continuous-time systems to various classes of discrete-time maps as well as to desingularization of space-discretizations of fast-slow partial differential equations.

Scientific Details+

The project aims to study geometric desingularization methods of non-hyperbolic fixed points for various classes of iterated maps and space discretizations of partial differential equations with a focus on multiple time scale problems. Iterated maps are discrete-time dynamical systems and non-hyperbolic fixed points occur if the linearization of the dynamics does not locally dominate higher-order nonlinear terms. For continuous-time dynamical systems, one available method to analyze nonhyperbolic equilibria is geometric desingularization, i.e., blowing up an equilibrium to a co-dimension one manifold usually taken as a sphere. We are going to investigate several classes of iterated maps, where developing a suitable blow-up in the discrete-time context is expected to be a highly effective tool in dynamical systems. The key class of motivating problems are multiscale maps with different time scales. In addition, we have recently broadened the scope of the project to also study reaction-diffusion PDEs and more general classes of semilinear PDEs with a multiple time scale structure and non-hyperbolic singular points.
 

Publications+

Papers
  • Samuel Jelbart and Christian Kuehn.
    A Formal Geometric Blow-up Method for Pattern Forming Systems.
    preprint, February 2023.
    arXiv:2302.06343.
  • Samuel Jelbart, Sara-Viola Kuntz, and Christian Kuehn.
    Geometric blow-up for folded limit cycle manifolds in three time-scale systems.
    preprint, August 2022.
    arXiv:2208.01361.
  • Felix Hummel, Samuel Jelbart, and Christian Kuehn.
    Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation.
    preprint, July 2022.
    arXiv:2207.03967.
  • Samuel Jelbart and Christian Kuehn.
    Discrete Geometric Singular Perturbation Theory.
    Discrete and Continuous Dynamical Systems, January 2022.
    arXiv:2201.06996.
  • Samuel Jelbart, Nathan Pages, Vivien Kirk, James Sneyd, and Martin Wechselberger.
    Process-oriented geometric singular perturbation theory and calcium dynamics.
    SIAM Journal on Applied Dynamical Systems, September 2021.
    arXiv:2104.07304.
  • Samuel Jelbart, Kristian Uldall Kristiansen, and Martin Wechselberger.
    Singularly perturbed boundary-equilibrium bifurcations.
    Nonlinearity, March 2021.
    arXiv:2103.09613.
  • M. Engel and H. Jardon-Kojakhmetov.
    Extended and symmetric loss of stability for canards in planar fast-slow maps.
    preprint, January 2020.
    arXiv:1912.10286.
  • M. Engel and C. Kuehn.
    A random dynamical systems perspective on isochronicity for stochastic oscillations.
    preprint, November 2019.
    arXiv:1911.08993.
  • L. Arcidiacono, M. Engel, and C. Kuehn.
    Discretized fast-slow systems near pitchfork singularities.
    Journal of Difference Equations and Applications, Vol. 25, No. 7, pp. 1024-1051, August 2019.
    doi:10.1080/10236198.2019.1647185.
  • M. Engel, C. Kuehn, M. Petrera, and Y. Suris.
    Discretized fast-slow systems with canard points in two dimensions.
    preprint, July 2019.
    arXiv:1907.06574.
  • M. Engel, J.S.W. Lamb, and M. Rasmussen.
    Conditioned Lyapunov exponents for random dynamical systems.
    Transactions of the American Mathematical Society 372(9), 2019, May 2019.
    doi:10.1090/tran/7803.
  • M. Engel and C. Kuehn.
    Discretized fast-slow systems near transcritical singularities.
    Nonlinearity, Vol. 32, No. 7, 2365-2391, May 2019.
    doi:10.1088/1361-6544/ab15c1.
  • Hildeberto Jardón-Kojakhmetov and Christian Kuehn.
    On fast-slow consensus networks with a dynamic weight.
    preprint, April 2019.
    arXiv:1904.02690.
  • C. Kuehn and C. Münch.
    Duck traps: two-dimensional critical manifolds in planar systems.
    Dynamical Systems: An International Journal, Vol. 34, No. 4, pp. 584-612,, February 2019.
    doi:10.1080/14689367.2019.1575337.
  • P. Degond, M.Engel, J.Liu, and R. Pego.
    A Markov jump process modelling animal group size statistics.
    Communications in Mathematical Sciences, January 2019.
  • H. Jardon-Kojakhmetov and C. Kuehn.
    A survey on the blow-up method for fast-slow systems.
    Preprint, 2019.
    arXiv:1901.01402.
  • M. Engel, J.S.W. Lamb, and M. Rasmussen.
    Bifurcation analysis of a stochastically driven limit cycle.
    Communications in Mathematical Physics 365(3), 2019, January 2019.
    arXiv:1606.01137, doi:10.1007/s00220-019-03298-7.
  • C. Kuehn.
    Multiscale dynamics of an adaptive catalytic network model.
    Math. Model. Nat. Pheno., 14(4):402, 2019.
    doi:10.1051/mmnp/2019015.
  • Matteo Petrera and Yuri B. Suris.
    New results on integrability of the Kahan-Hirota-Kimura discretizations.
    In Nonlinear systems and their remarkable mathematical structures. Vol. 1, pages 94–121. CRC Press, Boca Raton, FL, 2019.
    URL: https://www.crcpress.com/Nonlinear-Systems-and-Their-Remarkable-Mathematical-Structures-Volume-I/Euler/p/book/9781138601000.
  • Thai Son Doan, Maximilian Engel, Jeroen S.W. Lamb, and Martin Rasmussen.
    Hopf bifurcation with additive noise.
    Nonlinearity 31 (2018), no. 10, 4567–4601, August 2018.
    arXiv:1710.09649, doi:10.1088/1361-6544/aad208.
  • C. Kuehn and P. Szmolyan.
    Multiscale geometry of the Olsen model and non-classical relaxation oscillations.
    J. Nonlinear Sci., 25(3):583–629, 2015.
    doi:10.1007/s00332-015-9235-z.
  • Christian Kuehn.
    Normal hyperbolicity and unbounded critical manifolds.
    Nonlinearity, 27(6):1351–1366, may 2014.
    URL: https://doi.org/10.1088%2F0951-7715%2F27%2F6%2F1351, doi:10.1088/0951-7715/27/6/1351.
  • M. Petrera, A. Pfadler, and Yu. Suris.
    On integrability of Hirota-Kimura type discretizations.
    Regular and Chaotic Dynamics, 16(3):245–289, June 2011.
    doi:10.1134/S1560354711030051.
  • C. Kuehn.
    A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics.
    Physica D, 240(12):1020–1035, 2011.
    doi:10.1016/j.physd.2011.02.012.
  • J. Guckenheimer and C. Kuehn.
    Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system.
    SIAM J. Appl. Dyn. Syst., 9:138–153, 2010.

Books

Team+

Prof. Dr. Christian Kühn   +

Projects: B10
University: TU München, Department of Mathematics, 03.06.061
Address: Boltzmannstraße 3, 85748 Garching, GERMANY
Tel: +49 89 28918334
E-Mail: ckuehn[at]ma.tum.de
Website: https://www-m8.ma.tum.de/bin/view/Allgemeines/ChristianKuehn

Prof. Dr. Yuri Suris   +

Projects: B02, B10
University: TU Berlin, Institut für Mathematik, MA 827
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31425759
Fax: +49 30 31424413
E-Mail: suris[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~suris/

Luca Arcidiacono   +

Projects: B10
University: TU München, Department of Mathematics, 03.06.039
Address: Boltzmannstraße 3, 85748 Garching, GERMANY
Tel: +49 89 28918336
E-Mail: luca.arcidiacono[at]tum.de
Website: https://www-m8.ma.tum.de/bin/view/Allgemeines/LucaArcidiacono

Dr. Samuel Jelbart   +

Projects: B10
University: TU München, Department of Mathematics, 03.06.021
Address: Boltzmannstraße 3, 85748 Garching, GERMANY
Tel: +49 89 28917487
E-Mail: samuel.jelbart[at]ma.tum.de
Website: https://www-m8.ma.tum.de/bin/view/Allgemeines/SamuelJelbart