Periodic Reporting for period 1 - RBT (Random Bifurcation Theory)
Okres sprawozdawczy: 2022-04-04 do 2024-04-03
This project aims at developing the bifurcation theory of random dynamical systems in new directions beyond the traditional setting that are highly relevant in applied sciences. Despite huge developments in stochastic differential equations and their applications in both science and economics, bifurcations in this context are only poorly understood, although qualitative changes in the behaviour of random dynamical systems are omnipresent due to nonlinearities. Research on bifurcations in random systems has flourished in the last decade, with main contributions on local bifurcations coming from the applicant and the host, as explained below. The proposed research builds upon these foundations to address not only open questions in local bifurcation theory, but also develop a fundamental theory for global bifurcations of random dynamical systems.
In the frame of the project, an extensive work in developing the qualitative theory of dynamical systems in the random/nonautonomous setting was carried out. This work is highly motivated by a poor understand of the bifurcation theory in the context of stochastic differential equations despite of their important applications in both science and economics.
The project outcome relating towards directly the objective of the action is to develop a thorough understanding of the local and global aspects of random bifurcation theory presented in a comprehensive research monograph “Thai Son Doan, Spectral Theory of Nonautonomous Dynamical Systems and Applications, Springer 2025”.
Additionally, the progress of the project also partly supports and motivates the beneficiary to establish a systematic result in the local and global assignment of spectrum of nonautonomous control systems published in high-ranking mathematical journals:
1. Pham The Anh, Artur Babiarz, Adam Czornik, Thai Son Doan , Proportional local assignability of the dichotomy spectrum of one-sided discrete time-varying linear systems, SIAM Journal on Control and Optimization Vol. 60 (2022), pp. 1294—1319.
2. Artur Babiarz, Le Viet Cuong, Adam Czornik, Thai Son Doan, Necessary and sufficient conditions for assignability of dichotomy spectrum of one-sided discrete time-varying linear systems, IEEE-Transactions on Automatic Control, Volume 67 (2022), Issue 4, 2039-2043.
3. Pham The Anh, Adam Czornik, Thai Son Doan, Stefan Siegmund, Proportional Local Assignability of Dichotomy Spectrum of One-sided Continuous Time-Varying Linear Systems, Journal of Differential Equations Volume 309 (2022), 176-195.
A major work in communication and dissemination of the project was a 2 week workshop "Modelling challenges in a changing climate and environment” at Vietnam organized by the beneficiary and senior researchers at the dynamical systems group at Imperial College London: http://math.ac.vn/conference/MCCCE2025/index.php?lang=vi(odnośnik otworzy się w nowym oknie). The workshop attracted approximately 70 participants and provided a forum for both foreigner experts and local experts in mixed research fields dynamical systems, mathematical modelling and data analysis.
The project outcome relating towards directly the objective of the action is to develop a thorough understanding of the local and global aspects of random bifurcation theory presented in a comprehensive research monograph “Thai Son Doan, Spectral Theory of Nonautonomous Dynamical Systems and Applications, Springer 2025”.
Additionally, the progress of the project also partly supports and motivates the beneficiary to establish a systematic result in the local and global assignment of spectrum of nonautonomous control systems published in high-ranking mathematical journals:
1. Pham The Anh, Artur Babiarz, Adam Czornik, Thai Son Doan , Proportional local assignability of the dichotomy spectrum of one-sided discrete time-varying linear systems, SIAM Journal on Control and Optimization Vol. 60 (2022), pp. 1294—1319.
2. Artur Babiarz, Le Viet Cuong, Adam Czornik, Thai Son Doan, Necessary and sufficient conditions for assignability of dichotomy spectrum of one-sided discrete time-varying linear systems, IEEE-Transactions on Automatic Control, Volume 67 (2022), Issue 4, 2039-2043.
3. Pham The Anh, Adam Czornik, Thai Son Doan, Stefan Siegmund, Proportional Local Assignability of Dichotomy Spectrum of One-sided Continuous Time-Varying Linear Systems, Journal of Differential Equations Volume 309 (2022), 176-195.
A major work in communication and dissemination of the project was a 2 week workshop "Modelling challenges in a changing climate and environment” at Vietnam organized by the beneficiary and senior researchers at the dynamical systems group at Imperial College London: http://math.ac.vn/conference/MCCCE2025/index.php?lang=vi(odnośnik otworzy się w nowym oknie). The workshop attracted approximately 70 participants and provided a forum for both foreigner experts and local experts in mixed research fields dynamical systems, mathematical modelling and data analysis.
[A] Local random bifurcation theory.
Local bifurcation theory was studied from various aspects. Firstly, the structural, regularity and explicit formula of spectral (Lyapunov, Sacker-Sell and Bohl) theory of nonautonomous/random dynamical systems have been investigated in Chapter 1, Chapter 3 and Chapter 4 of “Thai Son Doan, Spectral Theory of Nonautonomous Differential Equations, Springer, 2025”. Finally, using a technique from normal form theory several results on linearization and partial linearization of nonautonomous differential equations are established in Chapter 2 of “Thai Son Doan, Spectral Theory of Nonautonomous Differential Equations, Springer, 2025”.
[B] Global random bifurcation theory.
A creation of chaotic attractors in random dynamical systems, a a topological approach to study changes in such attractors were investigated to scalar or planar stochastic differential equations under additive noise in Chapter 5 of “Thai Son Doan, Spectral Theory of Nonautonomous Differential Equations, Springer, 2025”.
[C] Applications.
Motivated from the understand critical transition of climate models by using transfer operators, a theoretical aspect in describing stability of random compact operator was established in “Thai Son Doan, Genericity of Lyapunov spectrum of bounded random compact operators on infinite-dimensional Hilbert spaces, arXiv:2303.14359 44 pages”. There is still a work in linking between this theoretical findings and applications to a real climate models. This work is still in progress.
Local bifurcation theory was studied from various aspects. Firstly, the structural, regularity and explicit formula of spectral (Lyapunov, Sacker-Sell and Bohl) theory of nonautonomous/random dynamical systems have been investigated in Chapter 1, Chapter 3 and Chapter 4 of “Thai Son Doan, Spectral Theory of Nonautonomous Differential Equations, Springer, 2025”. Finally, using a technique from normal form theory several results on linearization and partial linearization of nonautonomous differential equations are established in Chapter 2 of “Thai Son Doan, Spectral Theory of Nonautonomous Differential Equations, Springer, 2025”.
[B] Global random bifurcation theory.
A creation of chaotic attractors in random dynamical systems, a a topological approach to study changes in such attractors were investigated to scalar or planar stochastic differential equations under additive noise in Chapter 5 of “Thai Son Doan, Spectral Theory of Nonautonomous Differential Equations, Springer, 2025”.
[C] Applications.
Motivated from the understand critical transition of climate models by using transfer operators, a theoretical aspect in describing stability of random compact operator was established in “Thai Son Doan, Genericity of Lyapunov spectrum of bounded random compact operators on infinite-dimensional Hilbert spaces, arXiv:2303.14359 44 pages”. There is still a work in linking between this theoretical findings and applications to a real climate models. This work is still in progress.