On the response of nonlinear dynamical systems to step input

• Thesis title in Czech: Odezva nelineárních dynamických systémů na skokové buzení
• Thesis title in English: On the response of nonlinear dynamical systems to step input
• Key words: nelineární obyčejné diferenciální rovnice; Colombeau algebra; skoková nespojitost; viskoelasticita
• English key words: nonlinear ordinary differential equations; Colombeau algebra; jump discontinuity; viscoelasticity
• Academic year of topic announcement: 2016/2017
• Thesis type: Bachelor's thesis
• Thesis language: angličtina
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. Mgr. Vít Průša, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 25.10.2016
• Date of assignment: 25.10.2016
• Confirmed by Study dept. on: 14.02.2017
• Date and time of defence: 21.06.2017 00:00
• Date of electronic submission: 19.05.2017
• Date of submission of printed version: 19.05.2017
• Date of proceeded defence: 21.06.2017
• Opponents: RNDr. Karel Tůma, Ph.D.

Guidelines

The aim of the bachelor thesis is to investigate the response of simple systems governed by nonlinear ordinary differential equations to step input. Since the governing differential equations are nonlinear, and the input is a function with a jump discontinuity, the appropriate mathematical tool for the analysis of the response is the Colombeau algebra. Consequently, the first step is to get familiar with the Colombeau algebra, see the recommended literature. Once the fundamentals of Colombeau algebra are mastered, it is possible to proceed with the analysis of the response of some simple systems to a step input. The choice of the model systems will be motivated by the applications in rheology of viscoelastic fluids, see the recommended literature. In particular, the detailed analysis of the impact of the choice of specific stress rate on the response of a class of viscoelastic materials will be of interest.

References

Wineman, A. S. and K. R. Rajagopal (2000). Mechanical response of polymers—an introduction. Cambridge: Cambridge University Press.
Colombeau, J.-F. (1992). Multiplication of distributions, Volume 1532 of Lecture Notes in Mathematics. Berlin: Springer-Verlag. A tool in mathematics, numerical engineering and theoretical physics.
Průša, V. and K. R. Rajagopal (2016). On the response of physical systems governed by nonlinear ordinary differential equations to step input. Int. J. Non-Linear Mech. 81, 207–221.
Řehoř, M., V. Průša, and K. Tůma (2016). On the response of nonlinear viscoelastic materials in creep and stress relaxation experiments in the lubricated squeeze flow setting. Phys. Fluids 28 (10).
Szabó, L. and M. Balla (1989). Comparison of some stress rates. Int. J. Solids and Structures 25 (3), 279–297.
Filippov, A. F. (1988). Differential equations with discontinuous right hand sides, Volume 18 of Mathematics and its Applications (Soviet Series). Dordrecht: Kluwer Academic Publishers Group. Translated from the Russian.

Preliminary scope of work

V řadě aplikací je předmětem zájmu odezva nelineárních dynamických systému na skokové buzení. Typickým příkladem jsou například testy tečení/napěťové relaxace pro viskoelastické materiály. Matematický popis chování takovýchto systémů vyžaduje zobecnění pojmu derivace tak, aby bylo možné derivovat nespojité funkce a zároveň provádět nelineární operace. Vhodným nástrojem je proto takzvaná Colombeau algebra, což je jedno z možných rozšíření klasické lineární teorie distribucí. Colombeau algebra totiž, mimo jiné, nabízí nástroje pro popis podivných objektů typu součin Dirac distribuce a Heaviside funkce.

Bakalářská práce by byla zaměřena na studium odezvy jednoduchých systémů nelineárních obyčejných diferenciálních rovnic s pomocí Colombeau algebry.

Preliminary scope of work in English

In many application one is interested in the response of nonlinear dynamical systems to a step input. Typical examples thereof are creep/stress-relaxation tests in the mechanics of viscoelastic materials. Mathematical description of such systems requires one to generalise the notion of the derivative, in such a way that it allows to simultaneously deal with nonlinearity and discontinuity. Consequently, the suitable mathematical tool is the Colombeau algebra, which is a generalisation of the classical theory of distributions to the nonlinear setting. Colombeau algebra, among others, allows one to work with strange objects such as the multiplication of the Dirac distribution and the Heaviside function.

The thesis will be focused on Colombeau algebra type approach to the analysis of the response of some simple systems governed by nonlinear ordinary differential equations.