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.

Scientific Details+

The project aims to study geometric desingularization methods of non-hyperbolic fixed points for various classes of iterated maps with a focus in 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.

Publications+

Papers
Extended and symmetric loss of stability for canards in planar fast-slow maps

Authors: Engel, M. and Jardon-Kojakhmetov, H.
Note: preprint
Date: Jan 2020
Download: arXiv

A random dynamical systems perspective on isochronicity for stochastic oscillations

Authors: Engel, M. and Kuehn, C.
Note: preprint
Date: Nov 2019
Download: arXiv

Discretized fast-slow systems near pitchfork singularities

Authors: Arcidiacono, L. and Engel, M. and Kuehn, C.
Journal: Journal of Difference Equations and Applications, Vol. 25, No. 7, pp. 1024-1051
Date: Aug 2019
DOI: https://doi.org/10.1080/10236198.2019.1647185
Download: external

Discretized fast-slow systems with canard points in two dimensions

Authors: Engel, M. and Kuehn, C. and Petrera, M. and Suris, Y.
Note: preprint
Date: Jul 2019
Download: arXiv

Conditioned Lyapunov exponents for random dynamical systems

Authors: Engel, M. and Lamb, J.S.W. and Rasmussen, M.
Journal: Transactions of the American Mathematical Society 372(9), 2019
Date: May 2019
DOI: https://doi.org/10.1090/tran/7803
Download: external

Discretized fast-slow systems near transcritical singularities

Authors: Engel, M. and Kuehn, C.
Journal: Nonlinearity, Vol. 32, No. 7, 2365-2391
Date: May 2019
DOI: https://doi.org/10.1088/1361-6544/ab15c1
Download: external

On fast-slow consensus networks with a dynamic weight

Authors: Jardón-Kojakhmetov, Hildeberto and Kuehn, Christian
Note: preprint
Date: Apr 2019
Download: arXiv

Duck traps: two-dimensional critical manifolds in planar systems

Authors: Kuehn, C. and Münch, C.
Journal: Dynamical Systems: An International Journal, Vol. 34, No. 4, pp. 584-612,
Date: Feb 2019
DOI: https://doi.org/10.1080/14689367.2019.1575337
Download: external

A Markov jump process modelling animal group size statistics

Authors: Degond, P. and M.Engel and J.Liu and Pego, R.
Journal: Communications in Mathematical Sciences
Date: Jan 2019

A survey on the blow-up method for fast-slow systems

Authors: Jardon-Kojakhmetov, H. and Kuehn, C.
Note: Preprint
Date: 2019
Download: arXiv

Bifurcation analysis of a stochastically driven limit cycle

Authors: Engel, M. and Lamb, J.S.W. and Rasmussen, M.
Journal: Communications in Mathematical Physics 365(3), 2019
Date: Jan 2019
DOI: 10.1007/s00220-019-03298-7
Download: external arXiv

Multiscale dynamics of an adaptive catalytic network model

Author: Kuehn, C.
Journal: Math. Model. Nat. Pheno., 14(4):402
Date: 2019
DOI: 10.1051/mmnp/2019015
Download: external

New results on integrability of the Kahan-Hirota-Kimura discretizations

Authors: Petrera, Matteo and Suris, Yuri B.
In Collection: Nonlinear systems and their remarkable mathematical structures. Vol. 1, CRC Press, Boca Raton, FL
Date: 2019
Download: external

Hopf bifurcation with additive noise

Authors: Doan, Thai Son and Engel, Maximilian and Lamb, Jeroen S.W. and Rasmussen, Martin
Journal: Nonlinearity 31 (2018), no. 10, 4567–4601
Date: Aug 2018
DOI: 10.1088/1361-6544/aad208
Download: external arXiv

Multiscale geometry of the Olsen model and non-classical relaxation oscillations

Authors: Kuehn, C. and Szmolyan, P.
Journal: J. Nonlinear Sci., 25(3):583--629
Date: 2015
DOI: 10.1007/s00332-015-9235-z
Download: external

Normal hyperbolicity and unbounded critical manifolds

Author: Kuehn, Christian
Journal: Nonlinearity, 27(6):1351-1366
Date: May 2014
DOI: 10.1088/0951-7715/27/6/1351
Download: external

On integrability of Hirota-Kimura type discretizations

Authors: Petrera, M. and Pfadler, A. and Suris, Yu.
Journal: Regular and Chaotic Dynamics, 16(3):245-289
Date: Jun 2011
DOI: 10.1134/S1560354711030051
Download: external

A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics

Author: Kuehn, C.
Journal: Physica D, 240(12):1020--1035
Date: 2011
DOI: 10.1016/j.physd.2011.02.012
Download: external

Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system

Authors: Guckenheimer, J. and Kuehn, C.
Journal: SIAM J. Appl. Dyn. Syst., 9:138--153
Date: 2010


Books
PDE Dynamics

Author: Kuehn, Christian
Date: 2019
ISBN: 978-1-61197-565-9; 978-1-61197-566-6
Download: external

Multiple time scale dynamics

Author: Kuehn, Christian
Date: 2015
DOI: 10.1007/978-3-319-12316-5
ISBN: 978-3-319-12315-8; 978-3-319-12316-5
Download: external


Team+

Prof. Dr. Christian Kühn   +

Projects: B10
University: TU München
E-Mail: ckuehn[at]ma.tum.de
Website: http://www-m8.ma.tum.de/personen/kuehn/


Prof. Dr. Yuri Suris   +

Projects: B02, B10
University: TU Berlin
E-Mail: suris[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~suris/


Dr. Maximilian Engel   +

Projects: B10
University: TU München
E-Mail: maximilian.engel[at]tum.de