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| Author(s): |
Luan Vinicio de Mattos Ferreira Silva
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica |
| Defense date: | 2024-04-04 |
| Examining board members: |
Douglas Duarte Novaes;
Ricardo Miranda Martins;
Claudio Aguinaldo Buzzi;
Francisco Braun;
Marcelo Messias
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| Advisor: | Douglas Duarte Novaes |
| Abstract | |
Since the 1960s, the boundedness of solutions of the Duffing-type equations $ \ddot{x} +g(x)=p(t)$ has been a significant research focus in dynamical systems. This is primarily attributed to mathematician John E. Littlewood, who proposed investigating conditions on the functions $ g(x) $ and $ p(t) $ to determine the boundedness of all solutions to the Duffing-type equation. In light of this, a crucial strategy is to establish the existence of invariant tori in the extended phase space of the differential equation, confining all the dynamics in the interiors of the regions delimited by them. Assuming $ p(t) $ to be a $ \sigma $-periodic function, we aim to study the existence of invariant tori for the family of second-order discontinuous differential equations \[ {\bf (\mathcal{F}):} \quad \ddot{x}+\mu\;\sgn(x)= \theta x +\varepsilon \; p(t), \] where $ \theta $ and $ \varepsilon $ are real parameters, $ \sgn(x) $ represents the usual sign function, and $ \mu\in\{-1,1\} $ is a modal parameter. We focus on cases where the unperturbed equation ($ \e=0 $) admits a ring of periodic orbits. More precisely, assuming $ \theta\neq0 $, we employ KAM theory to investigate the existence of invariant tori for $ {\bf (\mathcal{F})} $. In this case, $ p(t) $ is required to be sufficiently differentiable. For $ \theta=0 $, considering $ p(t) $ as a Lebesgue-integrable function with vanishing average, we establish the existence of invariant tori through a constructive and non-perturbative method. These results provide conditions for the boundedness of solutions that initiate either on these tori or in the interiors of the regions delimited by them, as well as conditions for the existence of periodic orbits. Finally, for the sake of completeness, we perform a Melnikov analysis on a more general class of differential equations given by $ \ddot{x}+\alpha\;\sgn(x)= \theta x +\varepsilon \; f(t,x,\dot{x})$, where $ \alpha\neq0 $ and $ f(t,x,\dot{x}) $ is a function of class $ \mathcal{C}^{1} $ and $ \si $-periodic in $ t$, aiming to detect bifurcating periodic orbits of the differential equation (AU) | |
| FAPESP's process: | 18/22398-0 - Global aspects of Filippov Systems via KAM theory |
| Grantee: | Luan Vinicio de Mattos Ferreira Silva |
| Support Opportunities: | Scholarships in Brazil - Doctorate |