Finite-dimensional reduction, Inertial Manifolds, and Homoclinic structures in dissipative PDEs
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
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People |
ORCID iD |
| Dmitry Turaev (Principal Investigator) |
Publications
Asaoka M
(2021)
Fast growth of the number of periodic points arising from heterodimensional connections
in Compositio Mathematica
Berger P
(2019)
On Herman's positive entropy conjecture
in Advances in Mathematics
Capinski M
(2018)
Computer assisted proof of the existence of the Lorenz attractor in the Shimizu-Morioka system
in Nonlinearity
Chen S
(2022)
Planar Schrödinger-Poisson system with critical exponential growth in the zero mass case
in Journal of Differential Equations
Duca A
(2020)
Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall
in Journal of Mathematical Physics
Gonchenko A
(2017)
On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled rotators
in Physica D: Nonlinear Phenomena
Gonchenko S
(2017)
On three types of dynamics and the notion of attractor
in Proceedings of the Steklov Institute of Mathematics
Gonchenko S
(2021)
Wild pseudohyperbolic attractor in a four-dimensional Lorenz system
in Nonlinearity
Li D
(2020)
Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors
in Nonlinearity
| Description | The following results have been published so far. 1. We have shown that the violation of ergodicity in the fast dynamics of a slow-fast Hamiltonian system can drive the whole system to equilibrium. This provides a completely new view on the classical ergodicity problem. 2.New type of Feynman-like formulas for solutions of Schroedinger equation has been proposed. 3. The existence of Lorenz attractor has been rigorously established for the Shimizu-Morioka model that serves as a normal form for a large set of bifurcations of codimension 3. The result implies the proof of the existence of persistently chaotic behaviour (with no stability zones) for a a wide spectrum of models from various applications. 4. A new theoretical framework for explaining the numerically discovered phenomenon of the attractor-repeller merger is proposed and the universality of the corresponding "mixed dynamics" phenomenon has been proven. The result constitutes a discovery of a new type of chaotic behaviour typical for non-conservative time-reversible systems and, in general, systems with dissipation and external forcing. |
| Exploitation Route | The results provide novel theoretical framework for the analysis of dynamics for large classes of systems. They were reported at a series of conferences and are published in leading journals in the field. The results can be used both for the theoretical study of dynamical systems and can be directly used in applied problems. |
| Sectors | Other |
| Description | The impact of this work is recorded against grant ref EP/P024920/1' |
| Description | Alexey Kazakov |
| Organisation | National Research University Higher School of Economics |
| Country | Russian Federation |
| Sector | Academic/University |
| PI Contribution | There were three visits by Alexey Kazakov from Higher School of Economics, Nizhny Novgorod. We are conducting joint research on wild pseudohyperbolic attractors. |
| Collaborator Contribution | We are conducting joint research on wild pseudohyperbolic attractors. |
| Impact | We have discovered wild pseudohyperbolic attractor in a normal form of 4-times degenerate equilibrium state or a periodic orbit. |
| Start Year | 2018 |