Scattering parabolic solutions for the spatial N-centre problem

For the $N$-centre problem in the three dimensional space, $$\ddot x = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{\alpha+2}}, \qquad x \in \mathbb{R}^3 \setminus \{c_1,\ldots,c_N\},$$ where $N \geq 2$, $m_i > 0$ and $\alpha \in [1,2)$, we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions

Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation

We deal with the singularly perturbed Nagumo-type equation $$\epsilon^2 u'' + u(1-u)(u-a(s)) = 0,$$ where $\epsilon > 0$ is a real parameter and $a: \mathbb{R} \to \mathbb{R}$ is a piecewise constant function satisfying $0 < a(s) < 1$ for all $s$. We prove the existence of chaotic, homoclinic and heteroclinic solutions, when $\epsilon$ is small enough. We use a dynamical systems approach, based on the Stretching Along Paths method and on the Conley-Wazewski's method

Design and development of a quarter car test rig

Il presente lavoro di tesi ha come obiettivo la progettazione di un banco prova per un quarto di veicolo e la realizzazione di un generatore di segnale in grado di inviare segnali di ingresso ad un attuatore idraulico il quale sarà utilizzato per eccitare la ruota in modo da simulare il profilo stradale. La fase di progettazione è stata svolta utilizzando il software Solidworks. In seguito sono state eseguite simulazioni per l’analisi strutturale e di frequenza di alcune parti del banco tramite l’utilizzo del software Ansys. Terminata la fase di progetto, il modello Solidworks è stato importato in ambiente Simulink utilizzando i blocchi di modellazione della piattaforma Simscape/SimMechanics, in modo da effettuare un'analisi dinamica del modello. L’ultima parte dello studio riguarda la realizzazione di un generatore di segnale in grado di ricevere il segnale di feedback proveniente dal servo controller dell’attuatore. Il generatore è stato realizzato utilizzando il micro controllore Arduino Uno. Tale dispositivo, grazie alle sue potenzialità, ha permesso la generazione di un segnale sinusoidale a diverse ampiezza e frequenze in modo da coprirne un certo campo di valori in base alla richiesta. Inoltre tale sistema è in grado di ricevere il segnale di feedback dal servo controller dello shaker in modo tale da leggerne il valore e monitorarlo in tempo reale sul PC. I risultati di questo studio mostrano che il Quarter car test rig progettato è una piattaforma in grado di studiare il comportamento dinamico dei sistemi sospensivi, la cui struttura si rende capace di poter testare diverse tipologie di sospensioni e pesi di veicolo, rappresentando un solido punto di partenza per una futura realizzazione fisica del banco

Real-time value-driven diagnosis

Diagnosis is often thought of as an isolated task in theoretical reasoning (reasoning with the goal of updating our beliefs about the world). We present a decision-theoretic interpretation of diagnosis as a task in practical reasoning (reasoning with the goal of acting in the world), and sketch components of our approach to this task. These components include an abstract problem description, a decision-theoretic model of the basic task, a set of inference methods suitable for evaluating the decision representation in real-time, and a control architecture to provide the needed continuing coordination between the agent and its environment. A principal contribution of this work is the representation and inference methods we have developed, which extend previously available probabilistic inference methods and narrow, somewhat, the gap between probabilistic and logical models of diagnosis

A note on the existence of multiple solutions for a class of systems of second order ODEs

Abstract We prove the existence of multiple solutions for some systems of second order ODEs with Dirichlet boundary conditions. Such systems are obtained by coupling scalar ODEs with different growth conditions. The proof relies on a global continuation technique

Multiple positive solutions to elliptic boundary blow-up problems

We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem $$\left\{\begin{array}{ll} \Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array} \right.$$ where $g$ is a function superlinear at zero and at infinity, $a^+$ and $a^-$ are the positive/negative part, respectively, of a sign-changing function $a$ and $\mu > 0$ is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight function $a$. The proof is based on a careful shooting-type argument for the equivalent singular ODE problem. As a further application of this technique, the existence of multiple positive radial homoclinic solutions to $$\Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, \qquad x \in \mathbb{R}^N,$$ is also considered

Unbounded Solutions to Systems of Differential Equations at Resonance

We deal with a weakly coupled system of ODEs of the type xj\u2032\u2032+nj2xj+hj(x1,\u2026,xd)=pj(t),j=1,\u2026,d,with hj locally Lipschitz continuous and bounded, pj continuous and 2 \u3c0-periodic, nj 08 N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h1, \u2026 , hd are assumed