Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when $\int_{0}^{T} a(t) \!dt < 0$ and $\lambda > 0$ is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.Comment: 26 page

Caratterizzazione dinamica del telaio del Monster S4R

Ricerca delle caratteristiche modali del telaio del Monster S4R, condotta agli elelmenti finiti e sperimentalmente

Amplitude and frequency modulation of subthalamic beta oscillations jointly encode the dopaminergic state in Parkinson's disease.

Brain states in health and disease are classically defined by the power or the spontaneous amplitude modulation (AM) of neuronal oscillations in specific frequency bands. Conversely, the possible role of the spontaneous frequency modulation (FM) in defining pathophysiological brain states remains unclear. As a paradigmatic example of pathophysiological resting states, here we assessed the spontaneous AM and FM dynamics of subthalamic beta oscillations recorded in patients with Parkinson's disease before and after levodopa administration. Even though AM and FM are mathematically independent, they displayed negatively correlated dynamics. First, AM decreased while FM increased with levodopa. Second, instantaneous amplitude and instantaneous frequency were negatively cross-correlated within dopaminergic states, with FM following AM by approximately one beta cycle. Third, AM and FM changes were also negatively correlated between dopaminergic states. Both the slow component of the FM and the fast component (i.e. the phase slips) increased after levodopa, but they differently contributed to the AM-FM correlations within and between states. Finally, AM and FM provided information about whether the patients were OFF vs. ON levodopa, with partial redundancy and with FM being more informative than AM. AM and FM of spontaneous beta oscillations can thus both separately and jointly encode the dopaminergic state in patients with Parkinson's disease. These results suggest that resting brain states are defined not only by AM dynamics but also, and possibly more prominently, by FM dynamics of neuronal oscillations

Periodic solutions to a perturbed relativistic Kepler problem

We consider a perturbed relativistic Kepler problem \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\dfrac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right)=-\alpha\, \dfrac{x}{|x|^3}+\varepsilon \, \nabla_x U(t,x), \qquad x \in \mathbb{R}^2 \setminus \{0\}, \end{equation*} where $m, \alpha > 0$, $c$ is the speed of light and $U(t,x)$ is a function $T$-periodic in the first variable. For $\varepsilon > 0$ sufficiently small, we prove the existence of $T$-periodic solutions with prescribed winding number, bifurcating from invariant tori of the unperturbed problem.Comment: 21 pages, 2 figure