Curriculum Vitˆ · Curriculum Vitˆ { Nils Berglund 3 List of publications Monograph [25]N....

Curriculum Vitæ

Nils BERGLUNDBorn 14 December 1969 in Lausanne, SwitzerlandSwedish citizenFrench and german (mother tongues), english

Professional address

MAPMO - CNRS, UMR 7349Federation Denis PoissonBatiment de Mathematiques, Route de ChartresUniversite d’Orleans, BP 6759F-45067 Orleans Cedex [email protected]://www.univ-orleans.fr/mapmo/membres/berglund

Education and career

Education

• 2004: Habilitation in Mathematics, University of Toulon, “Equations differentiellesstochastiques singulierement perturbees”.Referees: Gerard Ben Arous, Anton Bovier, Lawrence E. Thomas.Jury: Jean-Francois Le Gall, Etienne Pardoux, Pierre Picco, Claude-Alain Pillet.

• 1998: PhD at Institut de Physique Theorique, ETH Lausanne, “Adiabatic DynamicalSystems and Hysteresis”.Adviser: Herve Kunz.Referees: Jean-Philippe Ansermet, Serge Aubry, Oscar E. Lanford III.

• 1993: Diploma in Physics at ETH Lausanne.Diploma thesis: “Billiards magnetiques”. Direction: Herve Kunz.

• 1988–1993: Physics studies at ETH Lausanne.

Positions held

• Since September 2007: Full professor (Professeur des universites), University of Orleans.Member of MAPMO (CNRS, UMR 7349, Federation Denis Poisson).

• September 2001 – August 2007: Assistant professor (Maıtre de Conference), Mathe-matics department, University of Toulon. Member of CPT (CNRS, UMR 6207).

• October 2000 – September 2001: Postdoc with teaching, Departement Mathematik,ETH Zurich.

• April – September 2000: TMR fellow at Weierstraß-Institut fur Angewandte Analysisund Stochastik (WIAS), Berlin.

• April 1999 – March 2000: Postdoc, Swiss National Fund, Georgia Institute of Tech-nology, Atlanta.

• September 1998 – March 1999: TMR fellow at Weierstraß-Institut fur AngewandteAnalysis und Stochastik (WIAS), Berlin.

• 1994 – 1998: Assistant at Institut de Physique Theorique, ETH Lausanne.

2 February 22, 2012

Award

“Prix des doctorats 1999” of ETH Lausanne.

Funded projects

• Local correspondent for ANR project MANDy, Mathematical Analysis of NeuronalDynamics, Budget for Orleans: 25’000.– Euro (September 2009 – August 2012)

• Principal investigator of research project “ACI Jeunes Chercheurs, Modelisation sto-chastique de systemes hors equilibre”, French Ministry of Research, Budget: 70’000.–Euro (August 2004 – April 2008).

Invited stays, Sabbaticals

• One-month stays at CRC 701, Universitat Bielefeld (2007, 2008, 2009, 2010, 2011).• One-month stays at the Weierstraß-Institut fur Angewandte Analysis und Stochastik

(WIAS), Berlin (2001, 2002, 2003, 2004, 2005, 2006).• March – September 2006: Sabbatical (Chercheur delegue au CNRS), at Centre de

Physique Theorique Marseille-Luminy (CPT, UMR 6207).• March – September 2005: Sabbatical (Chercheur delegue au CNRS), WIAS Berlin

and CPT Marseille.• April 1999: Invited stay with Jacques Laskar, Bureau des Longitudes, Paris.

Peer review activities

Referee for• Annales Institut Henri Poincare• Chaos• Comm. Nonlinear Science Numerical Simul.• Discrete Contin. Dyn. Syst. A• Electron. J. Differential Equations• Inverse Problems• J. Dynam. Diff. Eq.• J. Phys. A• J. Nonlinear Science• J. Theoretical Probability• Mathematics Computers Simulation• New J. Phys.• Nonlinear Analysis• Nonlinearity• Physica A• Physica D• SIAM J. Appl. Math.• SIAM J. Scient. Comp.• Stochastic Process. Appl.• Syst. Contr. Letters

I have also reviewed grant proposals for NSF (United States) and NWO (Netherlands).

Curriculum Vitæ – Nils Berglund 3

List of publications

Monograph

[25] N. Berglund, B. Gentz, Noise-Induced Phenomena in Slow–Fast Dynamical Systems.A Sample-Paths Approach (Springer, Probability and its Applications, 276+xivpages, 2005)http://www.univ-orleans.fr/mapmo/membres/berglund/book.html

Preprints / In Preparation

[24] N. Berglund, B. Gentz, Sharp estimates for metastable lifetimes in parabolic SPDEs:Kramers’ law and beyond, Preprint (2012), arXiv/1202.0990, 62 pages, submitted.

[23] N. Berglund, Kramers’ law: Validity, derivations and generalisations, Preprint (2011),arXiv/1106.5799, 24 pages, submitted.

[22] N. Berglund, D. Landon, Mixed-mode oscillations and interspike interval statisticsin the stochastic FitzHugh–Nagumo model, Preprint (2011), arXiv/1105.1278, 33pages, submitted.

[21] N. Berglund, B. Gentz, On the noise-induced passage through an unstable periodicorbit II: General case. In preparation.

Refereed papers

[20] N. Berglund, B. Gentz, C. Kuehn, Hunting French Ducks in a Noisy Environment,J. Differential Equations 252:4786–4841 (2012).

[19] N. Berglund, B. Gentz, The Eyring-Kramers law for potentials with nonquadraticsaddles, Markov Processes Relat. Fields 16:549–598 (2010).

[18] N. Berglund, B. Gentz, Anomalous behavior of the Kramers rate at bifurcations inclassical field theories, J. Phys. A: Math. Theor. 42:052001 (9 pages) (2009).

[17] J.-P. Aguilar, N. Berglund, The effect of classical noise on a quantum two-levelsystem, J. Math. Phys. 49:102102 (23 pages) (2008).

[16] N. Berglund, B. Fernandez, B. Gentz, Metastability in interacting nonlinear stochas-tic differential equations I: From weak coupling to synchronisation, Nonlinearity20:2551–2581 (2007).

[15] N. Berglund, B. Fernandez, B. Gentz, Metastability in interacting nonlinear stochas-tic differential equations II: Large-N regime, Nonlinearity 20:2583–2614 (2007).

[14] N. Berglund, B. Gentz, Universality of first-passage- and residence-time distributionsin nonadiabatic stochastic resonance, Europhys. Lett. 70:1–7 (2005).

[13] N. Berglund, B. Gentz, On the noise-induced passage through an unstable periodicorbit I: Two-level model, J. Statist. Phys. 114:1577–1618 (2004).

[12] N. Berglund, B. Gentz, Geometric singular perturbation theory for stochastic differ-ential equations, J. Differential equations 191:1–54 (2003).

4 February 22, 2012

[11] N. Berglund, B. Gentz, Metastability in simple climate models: Pathwise analysis ofslowly driven Langevin equations, Stoch. Dyn. 2:327–356 (2002).

[10] N. Berglund, B. Gentz, Beyond the Fokker–Planck equation: Pathwise control ofnoisy bistable systems, J. Phys. A 35:2057–2091 (2002).

[9] N. Berglund, B. Gentz, The effect of additive noise on dynamical hysteresis, Nonlin-earity 15:605–632 (2002).

[8] N. Berglund, B. Gentz, A sample-paths approach to noise-induced synchronization:Stochastic resonance in a double-well potential, Ann. Appl. Probab. 12:1419–1470(2002).

[7] N. Berglund, B. Gentz, Pathwise description of dynamic pitchfork bifurcations withadditive noise, Probab. Theory Related Fields 122:341–388 (2002).

[6] N. Berglund, T. Uzer, The averaged dynamics of the hydrogen atom in crossed electricand magnetic fields as a perturbed Kepler problem, Found. Phys. 31:283–326 (2001).

[5] N. Berglund, Control of dynamic Hopf bifurcations, Nonlinearity 13:225–248 (2000).

[4] N. Berglund, H. Kunz, Memory Effects and Scaling Laws in Slowly Driven Systems,J. Phys. A 32:15–39 (1999).

[3] N. Berglund, H. Kunz, Chaotic Hysteresis in an Adiabatically Oscillating DoubleWell, Phys. Rev. Letters 78:1691–1694 (1997).

[2] N. Berglund, A. Hansen, E.H. Hauge, J. Piasecki, Can a Local Repulsive PotentialTrap an Electron?, Phys. Rev. Letters 77:2149–2153 (1996).

[1] N. Berglund, H. Kunz, Integrability and Ergodicity of Classical Billiards in a Mag-netic Field, J. Stat. Phys. 83:81–126 (1996).

Book chapters

[26] N. Berglund, B. Gentz, Stochastic dynamic bifurcations and excitability, in C. Laing,G. Lord (Eds.), Stochastic Methods in Neuroscience (Oxford University Press, Ox-ford, 2009)

[27] N. Berglund, Bifurcations, scaling laws and hysteresis in singularly perturbed sys-tems, in B. Fiedler, K. Groger, J. Sprekels (Eds.), Equadiff ’99 (World Scientific,Singapore, 2000)

[28] N. Berglund, K.R. Schneider, Control of dynamic bifurcations, in D. Aeyels, F.Lamnabhi-Lagarrigue, A. van der Schaft (Eds.), Stability and Stabilization of Non-linear Systems (Lecture Notes in Contr. and Inform. Sciences 246, Springer, Berlin,1999)

Proceedings

[29] N. Berglund, Stochastic dynamical systems in neuroscience, Oberwolfach Reports8:– (2011). Proceedings of the miniworkshop Dynamics of Stochastic Systems andtheir Approximation, Oberwolfach, Germany, August 2011.


[30] N. Berglund, Dynamic bifurcations: hysteresis, scaling laws and feedback control,Prog. Theor. Phys. Suppl. 139:325–336 (2000). Proceedings of the conference “Let’sface chaos through nonlinear dynamics”, Maribor, Slovenia, June/July 1999.

[31] N. Berglund, Classical Billiards in a Magnetic Field and a Potential, NonlinearPhenomena in Complex Systems 3:61–70 (2000). Proceedings of the conference“Let’s face chaos through nonlinear dynamics”, Maribor, Slovenia, June/July 1996.

[32] N. Berglund, On the Reduction of Adiabatic Dynamical Systems near EquilibriumCurves, Preprint (1998), mp-arc/98-574. Proceedings of the conference “CelestialMechanics, Separatrix Splitting, Diffusion”, Aussois, France, June 1998.

Diploma thesis, Phd thesis, habilitation thesis

[33] N. Berglund, Equations differentielles stochastiques singulierement perturbees,Habilitation , University of Toulon (2004).http://www.univ-orleans.fr/mapmo/membres/berglund/hdr.html

[34] N. Berglund, Adiabatic Dynamical Systems and Hysteresis,Phd thesis 1800, ETH Lausanne (1998).http://www.univ-orleans.fr/mapmo/membres/berglund/these.html

[35] N. Berglund, Billards magnetiques, Diploma thesis, ETH Lausanne (1993).

Lecture notes

See http://www.univ-orleans.fr/mapmo/membres/berglund/teaching.html

6 February 22, 2012

Teaching

• 2007-2012: University of Orleans (lectures):– Master of mathematics, 2nd year: Martingales and stochastic calculus– Master of mathematics, 2nd year: Stochastic processes– Master of mathematics, 2nd year: Financial mathematics– Master of mathematics, 1st year: Financial mathematics– Master of economics, 1st year: Data analysis– Master of economics, 1st year: Quantitative data analysis– Bachelor of economics, 3rd year: Linear optimisation– Bachelor of mathematics, 3rd year: Probability and statistics

• 2001-2007: University of Toulon:– Lectures:

∗ Master of mathematics, 2nd year: Mathematical physics∗ Master of mathematics, 1st year: Probability theory∗ Master of mathematics, 1st year: Mathematical physics∗ Bachelor of economics, 3rd year: Probability∗ Engineering school, 3rd year: Stochastic processes∗ Engineering school, 2nd year: Financial mathematics∗ Bachelor in mathematics/computer science, 1st year: Algebra∗ Bachelor in physics, 1st year: Matrices

– Exercise classes:

∗ Master in mathematics, 1st year: Algebra∗ Bachelor in economics, 3rd year: Probability∗ Bachelor in mathematics/computer science, 2nd year: Analysis, geometry,

series∗ Bachelor in biology, 2nd year: Probability and statistics∗ Bachelor in biology, 1st year: Calculus

• 2000-2001: ETH Zurich (lectures):– Geometrische Theorie dynamischer Systeme– Storungstheorie dynamischer Systeme

• 1994-1998: ETH Lausanne (exercise classes):

– General mechanics– General relativity and cosmology– Nonlinear phenomena and chaos– Electrodynamics

Thesis supervision

• PhD students:

– Sebastien Dutercq (since 2011)– Damien Landon (since 2008)– Jean-Philippe Aguilar (2005–2008)

• Diploma students:

– Sebastien Dutercq (2011)


– Kanal Hun (2009)– Damien Landon (2008)– Antoine Loyer (2005, in collaboration with Barbara Gentz, Berlin)– Jean-Philippe Aguilar (2005)– Evelyne Vas (2004, in collaboration with Emmanuel Buffet, Dublin)

Doctoral committees

• Julian Tugaut (Universite Nancy 1, July 2010): referee• Benedicte Aguer (Universite Lille 1, June 2010): referee• Guillaume Voisin (Orleans, December 2009): president of jury• Laurent Marin (Orleans, November 2009)

8 February 22, 2012

Research interests

• Probability theory: Stochastic differential equations, stochastic partial differentialequations, stochastic exit problem, metastability, large deviations, potential theory

• Dynamical Systems: Singular perturbation theory, dynamic bifurcations, mixed-mode oscillations, spatially extended systems, Hamiltonian systems, symplectic maps

• Applications: Statistical physics, classical and quantum open systems, neuro-science, climate models

My research interests mainly lie at the intersection of different fields, such as probabilitytheory and dynamical systems. These intersections lead to interesting new approachesand results. For intance, with Barbara Gentz I have developped new methods for thequantitative analysis of stochastic differential equations with several timescales, based ona combination of stochastic analysis and singular perturbation theory.

My current research projects mainly deal with the following themes:

• Metastability in spatially extended systems: Characterisation of metastabletransitions (transition time, optimal transition path) in systems of coupled oscillatorsand stochastic partial differential equations. These problems require the extensionof known results to infinite-dimensional situations and situations with bifurcations.

• Applications in the neurosciences: The creation of action potentials (“spiking”)in neurons is often modeled by slow–fast ordinary and stochastic differential equa-tions. A particularly interesting feature is the phenomenon of excitability, in whichsmall deterministic or stochastic perturbations can trigger spikes. The distributionof interspike time intervals is still poorly understood in irreversible cases.

International collaborations

• Herve Kunz (Lausanne), Alex Hansen, Eivind Hauge (Trondheim), Jaroslav Piasecki(Warschau): Billiards.

• Herve Kunz (Lausanne): adiabatic dynamical systems.• Klaus Schneider (Berlin): control theory.• Turgay Uzer (Atlanta): Hamiltonian systems.• Barbara Gentz (Berlin/Bielefeld): stochastic differential equations.• Bastien Fernandez (Marseille): spatially extended systems.• Christian Kuehn (Vienna): fast–slow dynamical systems.

These collaborations have been partly funded by the Swiss National Fund, the Euro-pean Union (TMR Project), the Forschungsinstitut Mathematik (FIM) at ETH Zurich,and the European Science Foundation (ESF).


Administration

Committees

• Since 2012: President of appointment committee, mathematics, Orleans.• 2010: External member of appointment committee, University of Poitiers.• 2010: External member of appointment committee, University Paris 7.• Since 2009: Elected member of appointment committee, mathematics, Orleans.• Since 2008: Elected member of institute committee, MAPMO–CNRS.• Since 2008: Member of thesis committee Orleans–Tours.• 2004-2007: Member of appointment committee, mathematics, University of Toulon.• 2005-2007: External member of appointment committee, mathematics, University Aix-

Marseille I.• 2005-2007: Nominated member of institute committee, CPT-CNRS.• 2004-2007: Responsible for teaching attribution, department of mathematics, Univer-

sity of Toulon.

Organisation of seminars and conferences

• “Third meeting of the GDR Quantum Dynamics”, Orleans (February 2011).• “Fourth Workshop on Random Dynamical Systems”, Bielefeld (November 2010), with

Barbara Gentz.• Conference “Stochastic Models in Neuroscience”, CIRM, Marseille-Luminy (January

2010), with Simona Mancini and Michele Thieullen (Paris 6).• Responsible for institute seminar, MAPMO (since 2008), with Simona Mancini and

Aurelien Alvarez (since 2011).• Conference “Maths and billiards” (Orleans, March 2008), with Stephane Cordier

(Orleans) and Emmanuel Lesigne (Tours).• Closing meeting of the ACI Jeunes Chercheurs “Modelisation stochastique de systemes

hors equilibre” (Orleans, February 2008).• “Journees de Probabilites 2007”(La Londe, September 2007), with K. Bahlali, Y.

Lacroix (Toulon), P. Picco (Marseille).• Minisymposium “Recent developments in the stochastic exit problem” at the interna-

tional conference ICIAM07 (Zurich, July 2007), with Samuel Herrmann (Nancy).• International conference QMath9 (Giens, September 2004), with J. Asch, J.-M. Bar-

baroux, J.-M. Combes, J.-M. Ghez (Toulon), A. Joye (Grenoble).• Colloquium “Equations Differentielles Stochastiques”, University of Toulon (2003),

with K. Bahlali, P. Briet (Toulon), P. Picco (Marseille).• “Seminaire de Dynamique Quantique et Classique”, CPT (2002).

10 February 22, 2012

Talks and conferences (since 1999)

Some slides are available athttp://www.univ-orleans.fr/mapmo/membres/berglund/resevents.html

Invited talks at conferences

• 3.2012. “Critical Transitions in Complex Systems”, Imperial College London, Sample-path behaviour of noisy systems near critical transitions (Invitation: Jeroen Lamb,Greg Pavliotis, Martin Rasmussen).

• 12.12.2011. EPSRC Symposium Workshop “Multiscale Systems: Theory and Ap-plications”, Warwick Mathematics Institute, Canards, mixed-mode oscillations andinterspike distributions in stochastic systems (Invitation: Martin Hairer, Greg Pavli-otis, Andrew Stuart).

• 2.11.2011. “Stochastic Dynamics in Mathematics, Physics and Engineering”, ZiF,Bielefeld, Does noise create or suppress mixed-mode oscillations? (Invitation: BarbaraGentz, Max-Olivier Hongler, Peter Reimann).

• 22.8.2011. “Dynamics of Stochastic Systems and their Approximation”, Mathematis-ches Forschungszentrum Oberwolfach, Stochastic dynamical systems in neuroscience(Invitation: Evelyn Buckwar, Barbara Gentz, Erika Hausenblas).

• 21.7.2011. ICIAM 2011, Vancouver, On the Interspike Time Statistics in the Stochas-tic FitzHugh–Nagumo Equation (Invitation: Jinqiao Duan, Barbara Gentz).

• 26.1.2011. “Inhomogeneous Random Systems”, Institut Henri Poincare, Paris, TheKramers law: Validity, derivations and generalizations (Invitation: Christian Maes).

• 2.12.2010. “Open quantum systems”, Institut Fourier, Grenoble, Metastability in aclass of parabolic SPDEs (Invitation: Alain Joye).

• 19.11.2009. 3rd Workshop “Random Dynamical Systems”, Universitat Bielefeld,Metastability for Ginzburg-Landau-type SPDEs with space-time white noise (Invita-tion: Barbara Gentz).

• 14.5.2009. “Deterministic and Stochastic Modeling in Computational Neuroscienceand Other Biological Topics”, CRM, Barcelona, Stochastic models for excitable systems(Invitation: Antony Guillamon).

• 12.3.2009. “Journees Systemes Ouverts”, Institut Fourier, Grenoble, Metastabilityfor the Ginzburg–Landau equation with space-time noise (Invitation: Alain Joye).

• 18.11.2008. 2nd Workshop “Random Dynamical Systems”, Universitat Bielefeld,Metastability in systems with bifurcations (Invitation: Barbara Gentz).

• 19.6.2008. Workshop “Applications of spatio-temporal dynamical systems in biol-ogy”, Laboratoire Dieudonne, Nice, Metastability in a system of coupled nonlineardiffusions (Invitation: Elisabeth Pecou).

• 30.11.2007. Workshop “Random Dynamical Systems”, Universitat Bielefeld, Meta-stable lifetimes and optimal transition paths in spatially extended systems (Invitation:Barbara Gentz).

• 23.2.2007. Workshop “Multiscale Analysis and Computations in Stochastic Differ-ential Equation Modelling 2007”, University of Sussex, Noise-induced phenomena inslow–fast dynamical systems (Invitation: Greg Pavliotis).

• 12.10.2006. Conference “Chaos and Complex Systems”, Novacella, Desynchronisa-tion of coupled bistable oscillators perturbed by additive noise (Invitation: Marco Rob-nik).

• 27.6.2006. AIMS conference “Dynamical Systems and Differential Equations”, Poi-tiers, Metastability and stochastic resonance in slow–fast systems with noise (Invita-


tion: Jianzhong Su).• 9.6.2005. Stochastic Partial Differential Equations and Climatology, ZIF, Universitat

Bielefeld, Geometric singular perturbation theory for stochastic differential equations(Invitation: Friedrich Gotze, Yuri Kondratiev, Michael Rockner).

• 31.5.2005. Period of Concentration, Stochastic climate models, Max-Planck Institut,Leipzig, Geometric singular perturbation theory applied to stochastic climate models(Invitation: Peter Imkeller).

• 19.5.2005. Period of Concentration, Metastability, ageing, and anomalous diffusion,Max-Planck Institute, Leipzig, Metastability in irreversible diffusion processes andstochastic resonance (Invitation: Anton Bovier).

• 22.6.2004. Coupled Map Lattices 2004 , Institut Henri Poincare, Paris, Special sessionon effect of noise on synchronization (Invitation: Bastien Fernandez).

• 28.4.2004. European Geophysical Union meeting (EGU04), Nice, Universal propertiesof noise-induced transitions (Invitation: Daniel Scherzer).

• 31.5.2003. 4th European Advanced Studies Conference, Novacella, On the noise-induced passage through an unstable periodic orbit (Invitation: Marko Robnik).

Invited talks at seminars

• 7.2.2012. Seminaire de Probabilites, Institut de Mathematiques de Toulouse, Oscil-lations multimodales et chasse aux canards stochastiques (Invitation: Clement Pelle-grini).

• 5.5.2011. Groupe de travail modelisation, LPMA, Paris 6, Oscillations multimodalesdans les equations differentielles stochastiques (Invitation: Gianbattista Giacomin).

• 15.3.2011. Groupe de travail Neuromathematiques et modeles de perception, IHP,Paris, Geometric singular perturbation theory for stochastic differential equations withapplications to neuroscience (Invitation: Alessandro Sarti).

• 14.3.2011. Groupe de travail mathematiques et neurosciences, IHP, Paris, Chasse auxcanards en environnement bruite (Invitation: Michele Thieullen).

• 11.2.2011. Laboratoire Dieudonne, Nice, Chasse aux canards en environnement bruite(Invitation: Jean-Marc Gambaudo, Bruno Marcos).

• 18.1.2011. Institut du Fer a Moulin, Paris, Quantifying the effect of noise on oscil-latory patterns (Invitation: Jacques Droulez).

• 12.11.2010. Seminaire de probabilites, EPF Lausanne, Metastability in a class ofparabolic SPDEs (Invitation: Charles Pfister).

• 18.6.2010. IGK-Seminar, Bielefeld, Stochastic differential equations in neuroscience(Invitation: Sven Wiesinger).

• 1.4.2010. Groupe de travail systemes dynamiques, LJLL, Paris 6, Systemes lents-rapides perturbes par un bruit — Applications dans les neurosciences (Invitation:Jean-Pierre Francoise, Alain Haraux).

• 26.6.2009. Journee en l’honneur d’Herve Kunz, EPF Lausanne, Metastabilite dans desequations de Ginzburg–Landau avec bruit blanc spatiotemporel (Invitation: VincenzoSavona).

• 27.11.2008. Groupe de Travail Modelisation de l’Evolution du Vivant, Ecole Poly-technique, Paris/Palaiseau, Equations differentielles stochastiques lentes–rapides (In-vitation: Sylvie Meleard).

• 27.10.2008. Theoretical Physics Seminar, KU Leuven, Kramers rate theory at bifur-cations (Invitation: Christian Maes).

• 24.6.2008. Seminaire de physique mathematique, Lille, L’effet d’un bruit classique


sur un systeme quantique a deux niveaux (Invitation: Stephan de Bievre).• 27.5.2008. Seminaire de probabilites, Paris 6 & 7, Metastabilite dans un systeme de

diffusions couplees (Invitation: Jean Bertoin).• 10.3.2008. Seminaire de probabilites Nord-Ouest, Angers, Metastabilite dans un sys-

teme de diffusions couplees (Invitation: Loıc Chaumont).• 13.6.2007. Seminar on Stochastic Analysis, University of Warwick, Metastability in

a chain of coupled nonlinear diffusions (Invitation: Martin Hairer).• 18.4.2007. Berliner Kolloquium Wahrscheinlichkeitstheorie, WIAS Berlin, Metastabi-

lity in a chain of coupled nonlinear diffusions (Invitation: Anton Bovier).• 2.4.2007. Seminaire de Probabilites, Universite Rennes 1, Metastabilite dans un sys-

teme de diffusions non-lineaires couplees (Invitation: Ying Hu, Florent Malrieu).• 10.1.2007. Oberseminar Math. Statistik & Wahrscheinlichkeitstheorie, Universitat

Bielefeld, Metastability in a system of interacting nonlinear diffusions (Invitation:Barbara Gentz).

• 7.12.2006. Seminaire de probabilites, Nancy, Synchronisation de diffusions couplees(Invitation: Samy Tindel).

• 15.11.2006. Seminar on Stochastic processes, Zurich, Metastability in a system ofinteracting diffusions (Invitation: Erwin Bolthausen).

• 10.3.2006. Seminaire de probabilites, LATP Marseille, Synchronisation de diffusionscouplees (Invitation: Laurent Miclo).

• 18.11.2003. Institut Fourier, Grenoble, Passage sous l’effet d’un bruit a travers uneorbite periodique instable (Invitation: Alain Joye).

• 29.10.2003. WIAS Berlin, On the noise-induced passage through an unstable periodicorbit (Invitation: Barbara Gentz).

• 8.11.2002. LATP, Marseille, Theorie geometrique des perturbations singulieres pourequations differentielles stochastiques (Invitation: Yvan Velenik).

• 26.6.2002. Humboldt-Universitat zu Berlin, Geometric singular perturbation theoryfor stochastic differential equations (Invitation: Peter Imkeller).

• 8.5.2001. EPFL, Resonance stochastique et hysterese (Invitation: Charles Pfister).• 21.3.2001. CPT Marseille, Resonance stochastique et synchronisation induite par

bruit (Invitation: Claude-Alain Pillet).• 14.11.2000. Institut Fourier, Grenoble, L’effet de bruit additif sur les bifurcations

dynamiques (Invitation: Alain Joye).• 18.10.2000. WIAS Berlin, Dynamic pitchfork bifurcations with additive noise (Invi-

tation: Barbara Gentz).• 5.9.2000. Universitat Augsburg, The effect of additive noise on dynamic pitchfork

bifurcations (Invitation: Fritz Colonius).• 23.9.1999. University of Victoria, BC, Hysteresis and scaling laws in dynamic bifur-

cations (Invitation: Florin Diacu).• 6.5.1999. Bureau des Longitudes, Paris, Bifurcations dynamiques (Invitation: Jac-

ques Laskar).• 24.2.1999. Universitat Augsburg, Control of dynamic Hopf bifurcations (Invitation:

Fritz Colonius).

Minicourses

• October 2011. CIRM, Marseille Luminy, 5 lectures, Bifurcations in stochastic sys-tems with multiple timescales (Invitation: Olivier Faugeras).

• April 2011. IGK, Bielefeld, 5 lectures, Stochastic differential equations with multiple


timescales (Invitation: Barbara Gentz).

Talks at local seminars

• 14.12.2010. Evaluation AERES, Orleans, Quantifying the effect of noise on oscilla-tory patterns.

• 10/12.2008. Groupe de travail de probabilites, Orleans, Metastablite.• 15.11.2007. Journee de la Federation Denis Poisson, Orleans, EDS lentes-rapides,

metastablite et resonance stochastique.• 2/3.2007. Groupe de travail Systemes Ouverts, CPT Marseille, Grandes deviations

et processus d’exclusion I-III .• 10.11.2006. Open Systems Days, Marseille, Metastabilite dans une chaıne classique

bruitee.• 17.11.2003. Colloquium Equations Differentielles Stochastiques, Universite de Tou-

lon, Modeles climatiques stochastiques.• 14.11.2003. Seminaire de Travail en Physique Mathematique, CPT Marseille, L’eva-

sion stochastique a travers une orbite periodique instable.• 2.10.2003. Colloquium Equations Differentielles Stochastiques, Universite de Toulon,

Mouvement Brownien, integration stochastique.• 13.6.2003. Seminaire de Travail en Physique Mathematique, CPT Marseille, Equa-

tions differentielles stochastiques singulierement perturbees.• 6.12.2002. Seminaire de Travail en Physique Mathematique, CPT Marseille, Equa-

tions differentielles singulierement perturbees.• 25.1.2002. Colloquium du Groupe 3, CPT Marseille, Atome d’hydrogene en champs

croises.• 6.6.2001. Seminar on Stochastic Processes, ETH/Universitat Zurich, Stochastic res-

onance and hysteresis.• 30.11.2000. Talks in Mathematical Physics, ETH Zurich, Slowly time-dependent sys-

tems with additive noise: delay and resonance phenomena.• 10.12.1999. Seminaire de Physique Theorique, EPFL, Atome d’hydrogene en champs

croises.• 11.1999. Physics Colloquia, Georgia Tech, Adiabatic Dynamical Systems and Hys-

teresis.

Other talks at conferences

• 9.6.2009. Journees de Probabilites, Poitiers: Metastabilite dans des EDPS du typeGinzburg–Landau.

• 4.9.2008. Journees de Probabilites, Lille: Metastabilite et loi de Kramers pour lesdiffusions dans des potentiels a selles non-quadratiques.

• 4.6.2007. CCT07, Marseille: Metastability in interacting nonlinear stochastic differ-ential equations.

• 20.9.2006. Journees de Probabilites, CIRM, Marseille: Synchronisation dans les dif-fusions couplees.

• 5.9.2005. Journees de Probabilites, Nancy: Distribution des temps de residence pourla resonance stochastique.

• 13.9.2004. QMath9, Giens: On the stochastic exit problem — irreversible case.• 8.9.2004. Journees de Probabilites, CIRM, Marseille: Le probleme de sortie d’une

diffusion d’un domaine borne par une orbite periodique instable: Au-dela des grandesdeviations.


• 6.7.2002. Let’s face chaos through nonlinear dynamics, Maribor: Stochastic reso-nance: Some new results.

• 5.6.2000. Second Nonlinear Control Network (NCN) Workshop, Paris: Effect of noiseand open loop control on dynamic bifurcations.

• 2.8.1999. Equadiff’99, Berlin: Bifurcations, scaling laws and hysteresis in singularlyperturbed systems.

• 30.6.1999. Let’s face chaos through nonlinear dynamics, Maribor: Dynamic bifurca-tions: hysteresis, scaling laws, and feedback control .

• 15.3.1999. Nonlinear Control Network (NCN) Workshop “Stability and Stabiliza-tion”, Ghent: Effect of noise and open loop control on dynamic bifurcations.

Other participation in conferences

• 8–10.2.2012. 4eme Rencontre du GDR Dynamique Quantique, IMT Toulouse.• 24–25.1.2012. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 3–6.8.2011. Limit theorems in Probability, Statistics and Number Theory, Bielefeld.• 20–24.6.2011. Journees de Probabilites, Nancy.• 25–27.11.2010. Quantum dynamics, CIRM, Marseille.• 14–16.10.2010. Workshop on Probabilistic Techniques in Statistical Mechanics, Berlin.• 24–26.3.2010. Second meeting of the GDR “Quantum Dynamics”, Dijon.• 26–27.1.2010. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 1–8.6.2008. Hyperbolic dynamical systems, Erwin Schodinger Insitute, Wien.• 22–23.1.2008. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 5–9.11.2007. Entropy, turbulence and transport, Institut Henri Poincare, Paris.• 9–13.7.2007. ICIAM 07, Zurich.• 23–24.1.2007. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 15–19.1.2007. Vth Meeting of the GDRE Mathematics and Quantum Physics, CIRM,

Marseille.• 4–29.7.2005. Mathematical Statistical Physics, Les Houches.• 16–19.3.2005. Workshop random spatial models from physics and biology, Berlin.• 18–23.4.2004. Equilibrium and Dynamics of Spin Glasses, Ascona.• 3–28.3.2003. Statistical Mechanics and Probability Theory, CIRM, Marseille.• 27–31.5.2002. Aspects Mathematiques des Systemes Aleatoires et de la Mecanique

Statistique, CIRM, Marseille.• 9–11.7.2001. Workshop on stochastic climate models, Chorin.• 2–7.7.2001. Stochastic analysis, Berlin.• 11–17.3.2001. Stochastics in the sciences, Oberwolfach.• 5–9.7.1999. ICIAM 99, Edinburgh.

Popularisation

• 22.6.2011 and 29.6.2011. Centre Galois, Orleans: Probabilites et Physique Statis-tique.


Description of Research Activities

Past Research Activities

Hamiltonian Systems

With Herve Kunz, as well as Alex Hansen, Eivind Hauge and Jarek Piasecki, I haveanalysed dynamical systems of the billiard type. These systems are described by symplecticmaps of the plane, which can display a great variety of behaviours, from integrable tochaotic.

In particular, we have used methods from perturbation theory (Kolmogorov–Arnold–Moser theory, Nekhoroshev theory) to describe the dynamics of billiards in magneticfields [BK96] and in electromagnetic fields [BHHP96, Ber00b].

With Turgay Uzer, I have studied the dynamics of atoms in electromagnetic fields,using methods of Celestial Mechanics (Delaunay variables, averaging) [BU01].

Singular perturbation theory

With Herve Kunz, I have studied slow–fast ordinary differential equations, and systemswith slowly time-dependent parameters. These are described by singularly perturbedequation of the form

εx = f(x, t) , x ∈ R n

where ε is a small parameter.Using methods from analytic and geometric singular perturbation theory [Fen79], we

have analysed in particular the following aspects:• The dynamics in a neighbourhood of equilibrium branches t 7→ x?(t), f(x?(t), t) = 0.

Fenichel’s geometrical theory shows the existence of invariant manifolds of the formx = x(t, ε), x(t, ε) = x?(t) +O(ε). We have shown, in particular, that these manifoldscan be approximated by asymptotic expansion with exponential accuracy.

• The assumptions for the existence of invariant manifolds break down in the vicin-ity of bifurcation points, that is in points (x?(t0), t0) in which the Jacobian matrix∂xf(x?(t0), t0) admits eigenvalues with vanishing real part. One speaks in this case ofa dynamic bifurcation. We have developed a method allowing to determine the scalingbehaviour of solutions in a neighbourhood of these points, in a very general setting, us-ing the bifurcation point’s Newton polygon [BK99, Ber98a, Ber98b, Ber00d, Ber00a].This method also allows to describe the scaling behaviour of hysteresis cycles [Ber98a].

• In a mechanical system, we have proved the existence of chaotic hysteresis, which canbe described by geometrical methods [BK97].With Klaus Schneider, I have applied these methods to problems of feedback control

of dynamic bifurcations [Ber00c, BS99].

Stochastic differential equations

With Barbara Gentz, I have studied the pathwise behaviour of solutions of stochasticdifferential equations with two time scales, of the form

dxt =1εf(xt, yt, ε)dt+

σ√εF (xt, yt, ε)dWt , x ∈ R n ,

dyt = g(xt, yt, ε)dt+ σ′G(xt, yt, ε)dWt , y ∈ Rm ,


where ε, σ and σ′ are small parameters. Such equations describe systems with fast andslow variables, perturbed by random noise. They play an important role in applications inPhysics (e.g. hysteresis in ferromagnets, Josephson junctions), in Climatology (models ofthe thermohaline circulation, Ice Ages), and biology (neural dynamics, generic regulatorynetworks).

Our recently published monograph [BG06] describes a new, constructive method for thequantitative description of the sample-path behaviour of such equations. This approachcombines methods from singular perturbation theory (Fenichel’s geometric theory [Fen79],asymptotic expansions) with methods from stochastic analysis (large deviations [FW98],the exit problem, properties of martingales). A large part of the book is dedicated toapplications of this approach to concrete problems from Physics, Climatology and Biology.

The main steps of our approach are the following:• A slow manifold of the systems is a set of the form {(x?(y), y) ∈ R n+m : f(x?(y), y) =

0, y ∈ D0}. It is stable if all eigenvalues of the Jacobian matrix ∂xf(x?(y), y) havenegative real parts, bounded away from zero, for all y ∈ D0. In this case one constructsa neighbourhood B(h) of the manifold, where hmeasures the size of the neighbourhood,such that an estimate of the form

C−(t, ε) e−κ−h2/2σ2

6 Px?(y0),y0

{τB(h) < t

}6 C+(t, ε) e−κ+h2/2σ2

holds for the first-exit time τB(h) of paths from B(h). This implies that paths arelikely to spend exponentially long time spans in B(h), as long as h is chosen somewhatlarger than σ.

• In case the slow manifold in unstable, one can show that sample paths typically leavea vicinity of the manifold in a time of order ε|log σ|.

• The slow manifold admits bifurcation points whenever eigenvalues of the Jacobianmatrix ∂xf(x?(y), y) have a vanishing real part at some y ∈ D0. In such cases, theslow manifold can fold back on itself, or split into several manifolds. The analysis ofsuch cases proceeds in two steps. First, one reduces the problem to a lower-dimensionalone by projection on the centre manifold. Next one carries out a local analysis of thereduced dynamics. Using informations on the deterministic dynamics, partitions ofthe time interval, and the strong Markov property, one can describe the behaviour oftypical paths. Depending on the situations, paths choose to follow one of the new slowmanifolds, our jump to a remote manifold.In addition to the general theory [BG03], we have described in particular the following

phenomena and applications:• Dynamic bifurcations with noise: Pitchfork bifurcation [BG02d], saddle–node [BG02b]

and Hopf bifurcation [BG06, Chapter 5]. A remarkable fact is that the noise canfacilitate transitions between manifolds, so that these can happen earlier as in thedeterministic case.

• Stochastic resonance: [BG02e, BG04, BG05a, BG05b]. This phenomenon appearswhen a deterministic periodic perturbation is combined with the noise, yielding al-most periodic transitions between stable manifolds, which would be impossible in theabsence of noise. The analysis of the distribution of waiting times between transitionsrequired an extension of the classical Wentzell–Freidlin theory [FW98] to systems withperiodic orbits.

• Hysteresis: Effect of noise on scaling laws of the area of hysteresis loops [BG02b].• Models from Physics: In particular from Solid State Physics [BG02a].• Climate models: Models of Ice Ages and the Gulf Stream [BG02c].


• Application in Biology: Models of neurons, such as the Hodgkin–Huxley equations, inwhich noise is responsible for the phenomenon of excitability [BG06, Chapter 6].

• Quantum dynamics: The effect of classical noise on certain quantum systems has beenstudied in the PhD thesis of Jean-Philippe Aguilar [AB08].

Effect of noise on spatially extended systems

With Barbara Gentz and Bastien Fernandez I have examined spatially extended systemswith stochastic perturbations, in the framework of the research project “Modelisationstochastique de phenomenes hors equilibre” (supported by the French Ministry of Researchby a contract of 70’000 Euros). Such systems model in particular the influence of differentheat reservoirs, which influence transport properties through physical systems [EPRB99].

In [BFG07a, BFG07b], we examine the dynamics of a chain of bistable coupled units,described by a diffusion on R Z /NZ of the form

dxσi (t) = f(xσi (t))dt+γ

2[xσi+1(t)− 2xσi (t) + xσi−1(t)

]dt+ σdWi(t) .

Here bistability is created by the form of f , e.g. the choice f(x) = x − x3 causes thesystem to hesitate between values +1 and −1. On the other hand, the coupling term withnearest neighbours compels the units to cooperate, yielding a tendency of local systemsto synchronise. The system has two fundamental states, given by I+ = (1, 1, . . . , 1) andI− = −I+. The noise triggers transitions between these fundamental states, and possiblyother existing metastable states.

With the help of Wentzell–Freidlin theory, we were able to prove the following:

• At weak coupling γ, the system has 2N local (metastable) equilibrium states. Transi-tions between these states are similar to those of an Ising model with Glauber dynam-ics, indeed they occur through the growth of a droplet of one phase within the phasewith opposite sign [dH04].

• At strong coupling (of the order N2), by contrast, the system synchronises: Only thetwo metastable equilibrium states I± are left. During transitions between these twometastable states, the elements of the chain keep the same position in their respectivelocal potential with high probability.

• The most interesting apect is the transition between these two regimes, which involvesa sequence of symmetry-breaking bifurcations. The description of these bifurcationsinvolves the so-called equivariant bifurcation theory, which takes the system’s sym-metries into account. In particular in the limit of large chain length N , we were ableto determine the bifurcation diagram with the help of the theory of symplectic maps,allowing for a precise determination of the time scales and the geometry of transitionpaths.

In all these regimes, Wentzell–Freidlin theory yields the expected transition time be-tween metastable states on the level of large deviations. Results by Bovier and cowork-ers [BEGK04, BGK05] allow to go beyond Wentzell–Freidlin exponential asymptotics, andto determine prefactors of metastable transition times (by proving the so-called Kramersformula, see for instance [Ber11]).


Current and planned research activities

Metastability

We continue our work on metastability in systems of coupled stochastic differential equa-tions. As the particular system described above depends on parameters, it undergoesbifurcations whenever the determinant of the Hessian of the potential vanishes. Theclassical Kramers formula fails in these cases, since it would predict vanishing or diverg-ing transition times. We have extended the Kramers formula to situations with bifur-cations [BG10, BG09a], by taking into account the effect of higher-order terms of thepotential at critical points.

Another extension of the Kramers formula is to the infinite-dimensional case of stochas-tic partial differential equations. In [BG12], we have obtained sharp estimates on expectedtransition times for a class of one-dimensional parabolic equations of the form

dut(x) =[∆ut(x)− U ′(ut(x))

]dt+

√2εdW (t, x) .

This result uses spectral Galerkin approximations of the system, and required a control ofcapacities with error bounds uniform in the dimension.

We are currently working on an extension of the results on the chain of coupled particlesto systems with asymmetric local dynamics f(x), and to systems with conservation laws(PhD thesis of Sebastien Dutercq). A spatial modulation of the noise intensity also yieldsto physically relevant questions. We further plan to examine similar questions for systemswhose local dynamics is not overdamped, but involves oscillators – these systems frequentlyoccur in biology (coupled neurons) [PRK01].

Neuroscience

Several models for action-potential generation in neurons, such as the Hodgkin–Huxley,FitzHugh–Nagumo and Morris–Lecar equations involve systems of slow–fast differentialequations. For appropriate parameter values, these systems display the phenomenon ofexcitability: They admit a stable equilibrium point, which, however, is such that smalldeterministic or random perturbations cause the system to make a big excursion in phasespace, corresponding to a spike. An important question is to determine the statistics of in-terspike intervals. In certain, effectively one-dimensional cases, the interspike distributionis close to an exponential one [BG09b]. In multidimensional cases, however, one numeri-cally observes different distributions, involving for instance clusters of spikes [MVE08]. Amajor difficulty in the mathematical analysis of these equations stems from the fact thatthey are far from being reversible.

Building on methods in [BG06], we have started developing a mathematical descriptionof the effect of noise on excitable systems. This work is part of the ANR project MANDy,Mathematical Analysis of Neuronal Dynamics (principal investigator: Michele Thieullen,University Paris 6). The PhD thesis of Damien Landon addresses the problem of char-acterising the interspike distribution for the FitzHugh–Nagumo model. Using propertiesof substochastic Markov chains and quasistationary distributions, we have been able todescribe this distribution in the regime where the excitable stationary point is a focus. Inparticular, we have proved that the distribution is asymptotically geometric [BL11].

An interesting feature of three-dimensional models is that they can produce mixed-mode oscillations, which are patterns alternating large- and small-amplitude oscillations(see [DGK+12] for a review). In collaboration with Barbara Gentz and Christian Kuehn


(Vienna), we have started a systematic study of the effect of noise on mixed-mode oscil-lations in these systems. First results in [BGK12] quantify the effect of noise on maskingsmall-amplitude oscillations, and on decreasing the number of small oscillations betweenspikes. We are currently working on extending the description by a study of Poincaresections for the stochastic system.

Bibliography

[AB08] Jean-Philippe Aguilar and Nils Berglund, The effect of classical noise on a quantumtwo-level system, Journal of Mathematical Physics 49 (2008), 102102 (23 pages).

[BEGK04] Anton Bovier, Michael Eckhoff, Veronique Gayrard, and Markus Klein, Metastabilityin reversible diffusion processes. I. Sharp asymptotics for capacities and exit times, J.Eur. Math. Soc. (JEMS) 6 (2004), no. 4, 399–424.

[Ber98a] Nils Berglund, Adiabatic dynamical systems and hysteresis, Ph.D. thesis, EPFL, 1998.

[Ber98b] , On the reduction of adiabatic dynamical systems near equilibrium curves,Preprint mp-arc/98-574. Proceedings of the international workshop Celestial Mechan-ics, Separatrix Splitting, Diffusion, Aussois, France, 1998.

[Ber00a] , Bifurcations, scaling laws and hysteresis in singularly perturbed systems, In-ternational Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci.Publishing, River Edge, NJ, 2000, pp. 79–81.

[Ber00b] , Classical billiards in a magnetic field and a potential, Nonlinear Phenom.Complex Syst. 3 (2000), no. 1, 61–70.

[Ber00c] , Control of dynamic Hopf bifurcations, Nonlinearity 13 (2000), no. 1, 225–248.

[Ber00d] , Dynamic bifurcations: hysteresis, scaling laws and feedback control, Prog.Theor. Phys. Suppl. 139 (2000), 325–336.

[Ber11] , Kramers’ law: Validity, derivations and generalisations, arXiv:1106.5799,2011.

[BFG07a] Nils Berglund, Bastien Fernandez, and Barbara Gentz, Metastability in interactingnonlinear stochastic differential equations: I. From weak coupling to synchronization,Nonlinearity 20 (2007), no. 11, 2551–2581.

[BFG07b] , Metastability in interacting nonlinear stochastic differential equations II:Large-N behaviour, Nonlinearity 20 (2007), no. 11, 2583–2614.

[BG02a] Nils Berglund and Barbara Gentz, Beyond the Fokker–Planck equation: Pathwise con-trol of noisy bistable systems, J. Phys. A 35 (2002), no. 9, 2057–2091.

[BG02b] , The effect of additive noise on dynamical hysteresis, Nonlinearity 15 (2002),no. 3, 605–632.

[BG02c] , Metastability in simple climate models: Pathwise analysis of slowly drivenLangevin equations, Stoch. Dyn. 2 (2002), 327–356.

[BG02d] , Pathwise description of dynamic pitchfork bifurcations with additive noise,Probab. Theory Related Fields 122 (2002), no. 3, 341–388.

[BG02e] , A sample-paths approach to noise-induced synchronization: Stochastic reso-nance in a double-well potential, Ann. Appl. Probab. 12 (2002), 1419–1470.

[BG03] , Geometric singular perturbation theory for stochastic differential equations,J. Differential Equations 191 (2003), 1–54.

21

22 BIBLIOGRAPHY

[BG04] , On the noise-induced passage through an unstable periodic orbit I: Two-levelmodel, J. Statist. Phys. 114 (2004), 1577–1618.

[BG05a] , On the noise-induced passage through an unstable periodic orbit II: The generalcase, In preparation, 2005.

[BG05b] , Universality of first-passage and residence-time distributions in non-adiabaticstochastic resonance, Europhys. Letters 70 (2005), 1–7.

[BG06] , Noise-induced phenomena in slow–fast dynamical systems. a sample-paths ap-proach, Probability and its Applications, Springer-Verlag, London, 2006.

[BG09a] , Anomalous behavior of the Kramers rate at bifurcations in classical field the-ories, J. Phys. A: Math. Theor 42 (2009), 052001.

[BG09b] , Stochastic methods in neuroscience, Carlo Laing and Gabriel Lord eds,ch. Stochastic dynamic bifurcations and excitability, pp. 64–93, Oxford UniversityPress, 2009.

[BG10] , The Eyring–Kramers law for potentials with nonquadratic saddles, MarkovProcesses Relat. Fields 16 (2010), 549–598.

[BG12] , Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ lawand beyond, arXiv:1202.0990, 2012.

[BGK05] Anton Bovier, Veronique Gayrard, and Markus Klein, Metastability in reversible diffu-sion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc. (JEMS)7 (2005), no. 1, 69–99.

[BGK12] Nils Berglund, Barbara Gentz, and Christian Kuehn, Hunting french ducks in a noisyenvironment, J. Differential Equations (2012), in press.

[BHHP96] Nils Berglund, Alex Hansen, Eivind H. Hauge, and Jaroslaw Piasecki, Can a localrepulsive potential trap an electron?, Phys. Rev. Letters 77 (1996), 2149–2153.

[BK96] Nils Berglund and Herve Kunz, Integrability and ergodicity of classical billiards in amagnetic field, J. Statist. Phys. 83 (1996), no. 1-2, 81–126.

[BK97] , Chaotic hysteresis in an adiabatically oscillating double well, Phys. Rev. Letters78 (1997), 1691–1694.

[BK99] , Memory effects and scaling laws in slowly driven systems, J. Phys. A 32 (1999),no. 1, 15–39.

[BL11] Nils Berglund and Damien Landon, Mixed-mode oscillations and interspike intervalstatistics in the stochastic FitzHugh-Nagumo model, arXiv:1105.1278, 2011.

[BS99] Nils Berglund and Klaus R. Schneider, Control of dynamic bifurcations, Stability andstabilization of nonlinear systems (Ghent, 1999), Lecture Notes in Control and Inform.Sci., vol. 246, Springer, London, 1999, pp. 75–93.

[BU01] Nils Berglund and Turgay Uzer, The averaged dynamics of the hydrogen atom in crossedelectric and magnetic fields as a perturbed Kepler problem, Found. Phys. 31 (2001),no. 2, 283–326, Invited papers dedicated to Martin C. Gutzwiller, Part III.

[DGK+12] M. Desroches, J. Guckenheimer, C. Kuehn, B. Krauskopf, H. Osinga, and M. Wech-selberger, Mixed-mode oscillations with multiple time scales, SIAM Review (2012), inpress.

[dH04] F. den Hollander, Metastability under stochastic dynamics, Stochastic Process. Appl.114 (2004), no. 1, 1–26.

[EPRB99] J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Non-equilibrium statistical mechanicsof anharmonic chains coupled to two heat baths at different temperatures, Comm. Math.Phys. 201 (1999), no. 3, 657–697.

BIBLIOGRAPHY 23

[Fen79] Neil Fenichel, Geometric singular perturbation theory for ordinary differential equa-tions, J. Differential Equations 31 (1979), no. 1, 53–98.

[FW98] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, seconded., Springer-Verlag, New York, 1998.

[MVE08] Cyrill B. Muratov and Eric Vanden-Eijnden, Noise-induced mixed-mode oscillations ina relaxation oscillator near the onset of a limit cycle, Chaos 18 (2008), 015111.

[PRK01] Arkady Pikovsky, Michael Rosenblum, and Jurgen Kurths, Synchronization, a universalconcept in nonlinear sciences, Cambridge Nonlinear Science Series, vol. 12, CambridgeUniversity Press, Cambridge, 2001.

Curriculum Vitˆ · Curriculum Vitˆ { Nils Berglund 3 List of publications Monograph [25]N....

Documents

Transcript of Curriculum Vitˆ · Curriculum Vitˆ { Nils Berglund 3 List of publications Monograph [25]N....