Research

L. Tsimring

Research interests

Rigorous mathematical analysis of network dynamics and of the dynamics of (deterministic) collective systems - inspired by modelling in Physics, Biology and other sciences. 


More specific questions of interest

  1. nonlinear space-time phenomena

  2. global dynamical features and their dependence on the parameters, including network structure

  3. dynamical quantifiers and their dependence on the system size/number of individuals


These issues require to employ techniques from various areas of mathematics, most notably

  1. Dynamical Systems

  2. Analysis of PDEs

  3. Probabilities


Other interests: Dynamical Systems with holes, Complex Networks, Modelling of Gene Rearrangement, etc

Main achievements (recents)

Kuramoto model: Full proof of nonlinear asymptotic stability of steady states in the continuum limit of the (deterministic) Kuramoto model. This includes a complete description of the phenomenology across parameters ranges and also a justification of the Ott-Antosen ansatz.

Collaborators: H. Dietert, D. Gérard-Varet and G. Giacomin

Papers: A37, A41, A44

Phase transitions in deterministic setting: Numerical evidence, analytical and computer-assisted proofs of loss of ergodicity by symmetry breaking (ie. existence of asymmetric Lebesgue ergodic components) in expanding systems of coupled maps. This model can be regarded as a (purely) deterministic analogue of the Ising model.

Papers: A34, A45, A46

Main achievements (previous)

Fronts in bistable systems: Description of traveling waves in discrete time systems with continuous and/or discrete physical support. (NB: Discrete time is convenient when modeling periodically forced systems, or simply when updating is only periodically known.) In particular, the results justify two long-standing assumptions in the literature:

  1. approximations of continuous space systems by lattice models (and vice-versa)

  2. continuous dependence on fronts velocity on system characteristics

In addition, typical features of front velocity in lattice systems have been established, such as mode-locking parameter dependence on orientation in multi-dimensional lattices.

These results have been later extended to fronts between periodic patterns in bistable space-time discrete systems (see here for more information).

Collaborator: R. Coutinho

Papers: A1, A2, A3, A5, A6, A7, A15, A31, B4, B6

Coupled piecewise expanding coupled maps: Description of topological and geometrical properties of chaotic systems of interacting maps (Coupled Map Lattices): structural stability of infinite dimensional repellers, density of traveling waves, scaling properties, estimates of the dynamical complexity (e.g. topological entropy and escape rate) and their extensive vs. intensive features.

Collaborators: V. Afraimovich, J-B. Bardet, R. Coutinho & P. Guiraud

Papers: A9, A14, A20, A25, A30, B2

Modeling of regulatory networks in System Biology and Immunology: My contributions in this domain are both fundamental and applied.

On one hand, motivated by delay effects, I have introduced discrete time version of basic genetic regulatory models, in the form of multi-dimensional piecewise contractions.

Besides, I have described the phase transition to extensive clustering in a model of globally coupled degrade-and-fire oscillators (for more details, see here).

On the other hand, I introduced a model, and proceeded to benchmarking against data, of the regulation of the V(D)J recombination machinery of beta genes in T-cells. This provides the first self-consistent explanation of the allelic exclusion process, whose disfunction is suspected to be involved in auto-immune diseases.

Collaborators: A. Blumenthal, R. Coutinho, E. Farcot, P. Ferrier, S. Jaeger, R. Lima & L. Tsimring, etc

Papers: A8, A17, A22, A24, A28, A29, A32, A35, A37, A40, B3, B8