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
-
• nonlinear space-time phenomena
-
• global dynamical features and their dependence on the parameters, including network structure
-
• 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
-
• Dynamical Systems
-
• Analysis of PDEs
-
• 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
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.
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:
-
• approximations of continuous space systems by lattice models (and vice-versa)
-
• 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
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
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