In 2007, Modolo and colleagues derived a population density equation for a population of Izhekevich neurons. This population density equation can describe oscillations in the brain that occur in Parkinson’s disease. Numerical simulations of the population density equation showed bursting behaviour even though the individual neurons had parameters that put them in the tonic firing regime. The bursting comes from neuron interactions but the mechanism producing this behaviour was not clear. In this thesis we study numerical behaviour of the population density equation and then use a combination of analysis and numerical simulation to analyze the basic qualitative behaviour of the population model by means of a simplifying assumption: that the initial density is a Dirac function and all neurons are identical, including the number of inputs they receive, so they remain as a point mass over time. This leads to a new ODE model for the population. For the new ODE system, we define a Poincaré map and then to describe and analyze it under conditions on model parameters that are met by the typical values adopted by Modolo and colleagues. We show that there is a unique fixed point for this map and that under changes in a bifurcation parameter, the system transitions from fast tonic firing, through an interval where bursting occurs, the number of spikes decreasing as the bifurcation parameter increases, and finally to slow tonic firing. / Graduate

The main issues discussed in this paper are related to the refinement of the on-ground casualty risk computation for the specific case of INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL). The current approach, unable to predict the spacecraft attitude motion, assumes random tumbling motion as initial condition to simulate the fragmentation process. The wide experience in break-up analysis, acquired after years of practice with simulation, identified attitude to be one of the major drivers of uncertainty. The Space Debris Office (SDO) demanded a specific research in the field of the long-term propagation applied to the attitude motion and INTEGRAL offered the perfect test bench to conduct a preliminary study in this direction. In particular, observing whether environmental torques were able to trigger stable attitude motion, maintainable till re-entry, was considered to be the major challenge. The propagation of coupled orbital􀀀attitude motion for a random attitude configuration represents only one side of the coin. Indeed, chaos theory analysis constituted the other. The use of the Poincaré map in a non-canonical way managed to bring evidence for constrained motion in the angular rate motion of INTEGRAL, under gravity perturbations. Such results allowed to conduct further investigation on the overall attitude motion and estimate that the attitude configuration at the re-entry appears as precession about the maximum axis of inertia, in the majority of the cases. / De aspekter som behandlas i detta examensarbete är relaterade till skaderisken när rymdskrot som passerar atmosfären och landar på marken. Detta illustreras för en specifik satellit: INTEGRAL. Den nuvarande strategin som används i rymdindustrin är oförmögen att tillräckligt noggrant prediktera satellitens at-titydförändring vid atmosfärsinträdet och antar därför en stokastisk tumlande rörelse som initialvillkor för en analys av sönderdelningen av farkosten när den passerar atmosfären. Den erfarenhet som finns i rymdindustrin kring sönderdelningssimuleringar har identifierat att attityden är den faktor som genererar störst osäkerhet i resultaten. För att bättre förstå attitydens betydelse användes INTEGRAL i den fallstudie som presenteras i denna rapport. Specifikt studerades hur externa kraftmoment från rymdmiljön kan skapa en stabil attitydrörelse, som behålls ända till farkostens inträde i atmosfären. Propageringen av den kopplade rörelsen bana-attityd för en stokastisk attitydkonfiguration representerar endast en del av denna analys, där kaosteori representerar den andra delen. Med hjälp av Poincaré-mappning har simuleringar som indikerar en begränsad vinkelhastighet för INTEGRAL-satelliten när den utsätts för gravitationsstörning. I majoriteten av de analyserade fallen representerades attityden av en precession kring den största huvudtröghetsaxeln.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2014-08-06T10:24:20Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) ciclos_limites_para_a_equacao_de_abel_generalizada.pdf: 641062 bytes, checksum: e4be39606562d4f6805c21c2cceb451c (MD5) / Made available in DSpace on 2014-08-06T10:24:20Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) ciclos_limites_para_a_equacao_de_abel_generalizada.pdf: 641062 bytes, checksum: e4be39606562d4f6805c21c2cceb451c (MD5) Previous issue date: 2009-10-30 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we conducted a study on the equations of the type dx dt = nå i=0 ai(t)xi; (A) where ai 2 C1, i = 0; ;n and 0 t 1. An equation of the form (A) is called a generalized Abel equation. Our study refers to the problem proposed by C. Pugh: There is a natural number N depending only on n, such that the equation (A) has at most N limit cycles? Initially we study the problem of C. Pugh for n = 1 and n = 2, for which the equation (A) has at most one and two limit cycles, respectively. For n = 3, A. Lins Neto shows that if a3(t) does not change sign on [0;1], then the equation (A) has at most three limit cycles. Also A. Lins Neto shows that, given a natural number l, it is possible to construct an equation of the form (A) with n = 3 that has at least l limit cycles. Still for n = 3, A. Gasull and J. Llibre study the problem of C. Pugh considering that a2(t) does not change sign on [0;1], and M. J. Alvarez, A. Gasull and H. Giacomini also study the problem of C. Pugh considering that there are real numbers a and b such that aa3(t)+ba2(t) does not change sign on [0;1] and a1(t) = a0(t) = 0. Besides this, we study some more general results studied by A. Gasull and A. Guillamon. / Neste trabalho realizamos um estudo sobre as equações do tipo dx dt = nå i=0 ai(t)xi; (A) onde ai 2 C1, i = 0; ;n e 0 t 1. Uma equação da forma (A) é denominada equação de Abel generalizada. Nosso estudo se refere ao problema proposto por C. Pugh: existe um número natural N dependendo apenas de n, tal que a equação (A) possui no máximo N ciclos limites? Inicialmente estudamos o problema de C. Pugh para n=1 e n=2, para os quais a equação (A) possui, no máximo, um e dois ciclos limite, respectivamente. Para n = 3, A. Lins Neto mostra que, se a3(t) não muda de sinal em [0;1], então a equação (A) possui no máximo três ciclos limite. Além disso A. Lins Neto mostra que, dado um número natural l, é possível construir uma equação da forma (A) com n = 3 que possui no mínimo l ciclos limites. Ainda para n = 3, A. Gasull e J. Llibre estudam o problema de C. Pugh considerando que a2(t) não muda de sinal em [0;1], e M. J. Álvarez, A. Gasull e H. Giacomini também estudam o problema de C. Pugh considerando que existem números reais a e b tais que aa3(t)+ba2(t) não muda de sinal em [0;1] e a1(t) = a0(t) = 0. Além destes resultados, estudamos alguns resultados mais gerais estudados por A. Gasull e A. Guillamon.

Tato diplomová práce se zabývá matematickými modely kráčejících robotů. Dva z těchto modelů jsou vybrány a analyzovány. Pasivní model "rimless wheel" , který slouží jako základ pro další, složitější modely, je podrobně analyzován. "Compass gait" model dvounohého robota je v práci analyzován a numericky simulován v programovacím jazyce Python. Metoda pro nalezení podmínek pro pasivní chůzi robota je rovněž implementována.

La motivation industrielle et mécanique du problème sera présentée pour un problème continu: élasticité linéaire avec une contrainte unilatérale. Un système masse-ressort avec un contact unilatéral en découle par discrétisation. Le but de cette thèse est d'étudier ces systèmes à vibro-impact de N degrés de liberté avec un contact unilatéral. Le système résultant est linéaire en l'absence de contact; Il est régi par une loi d'impact autrement. L'auteur identifie les modes non linéaires qui présentent une phase de contact collant pour un modèle à deux degrés de liberté en présence d'un obstacle rigide. L'application de premier retour de Poincaré est un outil fondamental pour étudier la dynamique près de solutions périodiques. Étant donné que la section de Poincaré est un sous-ensemble de l'interface de contact dans l'espace des phases, elle peut être tangente aux orbites pour les contacts rasants et conduire à une singularité en « racine carrée » déjà connue en Mécanique. Cette singularité est revisitée dans un cadre mathématique rigoureux. Elle implique la discontinuité du temps de premier retour. Enfin, l’instabilité des modes linéaire rasants est abordée. / The mechanical motivation is presented for a PDE with a constraint. The purpose of this thesis is to study N degree-of-freedom vibro-impact systems with an unilateral contact. The resulting system is linear in the absence of contact; it is governed by an impact law otherwise. The author identifies some nonlinear modes that display a sticking phase. The First Return Map is a fundamental tool to explore periodic solutions. Since the Poincaré section is a subset of the contact interface in the phase-space, it can be tangent to orbits which yields the well-known square-root singularity. This singularity is here revisited in a rigorous mathematical framework. Moreover, the study of this singularity implies a more important singularity: the discontinuity of the first return time. Finally, the square-root dynamics near the linear grazing modes which may lead to the instability of these linear grazing modes is studied.

Les gyroscopes laser à gaz constituent une solution technique de haute performances dans les problématiques de navigation inertielle. Néanmoins, pour de très faibles vitesses de rotation, les petites imperfections des miroirs de la cavité optique font que les deux faisceaux contra-propageant sont verrouillés en phase. En conséquence, les mesures en quadrature de leur différence de phase ne permettent plus de remonter directement aux vitesses de rotation à l'intérieur d'une zone autour de zéro, dite zone aveugle statique, ou, si l'on utilise une procédure d'activation mécanique, dite zone aveugle dynamique. Ce travail montre qu'il est néanmoins possible, en utilisant des méthodes issues du filtrage et de l'estimation, de remonter aux vitesses de rotation mêmes si ces dernières sont en zone aveugle. Pour cela, on part d'une modélisation physique de la dynamique que l'on simplifie par des techniques de perturbations singulières pour en déduire une généralisation des équations de Lamb. Il s'agit de quatre équations différentielles non-linéaires qui décrivent la dynamique des intensités et des phases des deux faisceaux contra-propageant. Une étude qualitative par perturbations régulières, stabilité exponentielle des points d'équilibre et applications de Poincaré permet de caractériser les zones aveugles statiques et dynamiques en fonction des imperfections dues aux miroirs. Il est alors possible d'estimer en ligne avec un observateur asymptotique fondé sur les moindre carrés récursifs ces imperfections en rajoutant aux deux mesures en quadrature celles des deux intensités. La connaissance précise de ces imperfections permet alors de les compenser dans la dynamique de la phase relative, et ainsi d'estimer les rotations en zone aveugle. Des simulations numériques détaillées illustrent l'intérêt de ces observateurs pour augmenter la précision des gyroscopes à gaz. / Gaz ring laser gyroscopes provide a high performance technical solution for inertial navigation. However, for very low rotational speeds, the mirrors imperfections of the optical cavity induce a locking phenomena between the phases of the two counter-propagating Laser beams. Hence, the measurements of the phase difference can no longer be used when the speed is within an area around zero, called lock-in zone, or,if a procedure of mechanical dithering is implemented, dithering lock-in zone. Nevertheless, this work shows that it is possible using filtering and estimation methods to measure the speed even within the lock-in zones. To achieve this result, we exploit a physical modeling of the dynamics that we simplify, using singular perturbation techniques, to obtain a generalization of Lamb's equations. There are four non-linear differential equations describing the dynamics of the intensities and phases of the two counter-propagating beams. A qualitative study by regular perturbation theory, exponential stability of the equilibrium points and Poincaré maps allows a characterisation of the lock-in zones according to the mirrors imperfections. It is then possible to estimate online, with an asymptotic observer based on recursive least squares, these imperfections by considering the additional measurements of the beam intensities. Accurate knowledge of these imperfections enables us to compensate them in the dynamic of the relative phase, and thus to estimate rotational speeds within the lock-in zones. Detailed numerical simulations illustrate the interest of those observers to increase the accuracy of gas ring laser gyroscopes.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Based on the article “Hyperbolic Main Cycles on Hypersurface of R4”, Garcia, see [4], we will study the bending lines in the vicinity of a main loop, closed bending line, a hypersurface immersed in R4. For this, we will define the Poincaré transformation associated with the cycle and calculate its derivative. With this analysis, we show under what conditions we can become hyperbolic, with a small deformation in the immersion, a major cycle given. Finally, we will build an example of a hypersurface containing a hyperbolic primary cycle, based on the article “Surfaces Around Closed Main Curvature Lines, an Inverse Problem." Garcia, Mello and Sotomayor, see [5]. / Tomando como base o artigo “Hyperbolic Principal Cycles on Hyper-surface of R4", de Garcia, ver [4], estudaremos as linhas de curvatura na vizinhança de um ciclo principal, linha de curvatura fechada, de uma hipersuperfície imersa no R4. Para isso, definiremos a transformação de Poincaré associada ao ciclo e calcularemos a sua derivada. Com essa análise, mostraremos sob quais condições podemos tornar hiperbólico, com uma pequena deformação na imersão, um ciclo principal dado. E por fim, construiremos um exemplo de uma hipersuperfície contendo um ciclo principal hiperbólico, baseando-nos no artigo “Surfaces Around Closed Principal Curvature Lines, an Inverse Problem." de Garcia, Mello e Sotomayor, ver [5].

This work focusses on analysing the stability of the MSFR – a molten salt reactor with a fast neutron spectrum. The investigations are based on a model, which was published and studied by the Politecnico di Milano using a linear approach. Since linear methods can only provide stability information to a limited extent, this work continues the conducted investigations by applying nonlinear methods. In order to examine the specified reactor model, the system equations were implemented, adjusted and verified using MATLAB code. With the help of the computational tool MatCont, a so-called fixed-point solution was tracked and its stability monitored during the variation of selected control parameters. It was found that the considered fixed point does not change its stability state and remains stable. Coexisting fixed points or periodic solutions could not be detected. Therefore, the analysed MSFR model is considered to be a stable system, in which the solutions always tend towards a steady state.:1. Introduction 2. Molten Salt Reactor Technology 2.1. Introduction 2.2. Historical Development 2.3. Working Principle of Molten Salt Reactors 2.4. Molten Salt Coolants 2.5. Advantages and Drawbacks 2.6. Classification 2.7. Molten Salt Fast Reactor Design 3. Stability Characteristics of Dynamical Systems 3.1. Introduction 3.2. Dynamical Systems 3.3. Stability Concepts 3.3.1. Introduction 3.3.2. Lagrange Stability (Bounded Stability) 3.3.3. Lyapunov Stability 3.3.4. Poincaré Stability (Orbital Stability) 3.4. Fixed-Point Solutions 3.4.1. Stability Analysis of Fixed-Point Solutions 3.4.2. Bifurcations of Fixed-Point Solutions 3.5. Periodic Solutions 3.5.1. Stability Analysis of Periodic Solutions 3.5.2. Bifurcations of Periodic Solutions 4. Analysed Reactor System 4.1. Introduction 4.2. Specified Reactor Model 4.3. Implementation and Verification of the Linearised System of Equations 4.3.1. Linearised System of Delayed Differential Equations 4.3.2. Comparison with Reference Plots 4.3.3. Adaptation of Parameter Values 4.4. Implementation and Verification of the Nonlinear System of Equations 4.4.1. Nonlinear System of Delayed Differential Equations 4.4.2. Delayed Neutron Precursor Equation Adjustments 4.4.3. Salt Temperature Equation Adjustments 4.4.4. Nonlinear System of Ordinary Differential Equations 4.4.5. Verification of the Nonlinear System of Ordinary Differential Equations 5. Conducted Stability Analyses 5.1. Introduction 5.2. Nonlinear Stability Analysis 5.2.1. Implementation 5.2.2. Results 5.2.3. Interpretation 5.3. Linear Stability Analysis 5.3.1. Comparison Between the Linearised and Nonlinearised MSFR System of Equations 5.3.2. Stability Investigations Using a Linear Criterion 5.4. MatCont Reliability Test Using an MSBR Model 6. Conclusions and Recommendations for Future Studies / Im Fokus dieser Arbeit steht die Stabilitätsanalyse des MSFR – eines Flüssigsalzreaktors mit schnellem Neutronenspektrum. Als Grundlage wurde ein Modell verwendet, das am Politecnico di Milano erstellt und dort mittels linearer Methoden untersucht wurde. Da lineare Betrachtungen nur eingeschränkte Stabilitätsaussagen treffen können, erweitert diese Arbeit die Untersuchungen um die nichtlineare Stabilitätsanalyse. Zur Untersuchung des vorgegebenen Reaktormodells wurden die Systemgleichungen in MATLAB übertragen und verifiziert. Mithilfe der Rechensoftware MatCont wurde eine sogenannten Fixpunkt-Lösung des Modells unter der Variation ausgewählter Parameter verfolgt und deren Stabilität überprüft. Es hat sich gezeigt, dass der betrachtete Fixpunkt seinen Stabilitätszustand dabei nicht verändert und stabil bleibt. Koexistierende Fixpunkte oder periodische Lösungen konnten nicht nachgewiesen werden. Daher gilt das betrachtete MSFR-Modell als ein stabiles System, dessen Lösungen immer auf einen stationären Zustand zulaufen.:1. Introduction 2. Molten Salt Reactor Technology 2.1. Introduction 2.2. Historical Development 2.3. Working Principle of Molten Salt Reactors 2.4. Molten Salt Coolants 2.5. Advantages and Drawbacks 2.6. Classification 2.7. Molten Salt Fast Reactor Design 3. Stability Characteristics of Dynamical Systems 3.1. Introduction 3.2. Dynamical Systems 3.3. Stability Concepts 3.3.1. Introduction 3.3.2. Lagrange Stability (Bounded Stability) 3.3.3. Lyapunov Stability 3.3.4. Poincaré Stability (Orbital Stability) 3.4. Fixed-Point Solutions 3.4.1. Stability Analysis of Fixed-Point Solutions 3.4.2. Bifurcations of Fixed-Point Solutions 3.5. Periodic Solutions 3.5.1. Stability Analysis of Periodic Solutions 3.5.2. Bifurcations of Periodic Solutions 4. Analysed Reactor System 4.1. Introduction 4.2. Specified Reactor Model 4.3. Implementation and Verification of the Linearised System of Equations 4.3.1. Linearised System of Delayed Differential Equations 4.3.2. Comparison with Reference Plots 4.3.3. Adaptation of Parameter Values 4.4. Implementation and Verification of the Nonlinear System of Equations 4.4.1. Nonlinear System of Delayed Differential Equations 4.4.2. Delayed Neutron Precursor Equation Adjustments 4.4.3. Salt Temperature Equation Adjustments 4.4.4. Nonlinear System of Ordinary Differential Equations 4.4.5. Verification of the Nonlinear System of Ordinary Differential Equations 5. Conducted Stability Analyses 5.1. Introduction 5.2. Nonlinear Stability Analysis 5.2.1. Implementation 5.2.2. Results 5.2.3. Interpretation 5.3. Linear Stability Analysis 5.3.1. Comparison Between the Linearised and Nonlinearised MSFR System of Equations 5.3.2. Stability Investigations Using a Linear Criterion 5.4. MatCont Reliability Test Using an MSBR Model 6. Conclusions and Recommendations for Future Studies

The first part of this Thesis deals with the analysis of piecewise-smooth mechanical systems and the definition of special stability criteria in presence of non-conservative follower forces. To illustrate the peculiar stability properties of this kind of dynamical system, a reference 2 d.o.f. structure has been considered, composed of a rigid bar, with one and constrained to slide, without friction, along a curved profile, whereas the other and is subject to a follower force. In particular, the curved constraint is assumed to be composed of two circular profiles, with different and opposite curvatures, defining two separated subsystems. Due to this jump in the curvature, located at the junction point between the curved profiles, the entire mechanical structure can be modelled by discontinuous equations of motion, the differential equations valid in each subsystem can be combined, leading to the definition of a piecewise-smooth dynamical system. When a follower force acts on the structure, an unexpected and counterintuitive behaviour may occur: although the two subsystems are stable when analysed separately, the composed structure is unstable and exhibits flutter-like exponentially-growing oscillations. This special form of instability, previously known only from a mathematical point of view, has been analysed in depth from an engineering perspective, thus finding a mechanical interpretation based on the concept of non-conservative follower load. Moreover, the goal of this work is also the definition of some stability criteria that may help the design of these mechanical piecewise-smooth systems, since classical theorems cannot be used for the investigation of equilibrium configurations located at the discontinuity. In the literature, this unusual behaviour has been explained, from a mathematical perspective, through the existence of a discontinuous invariant cone in the phase space. For this reason, starting from the mechanical system described above, the existence of invariant cones in 2 d.o.f. mechanical systems is investigated through Poincaré maps. A complete theoretical analysis on piecewise-smooth dynamical systems is presented and special mathematical properties have been discovered, valid for generic 2~d.o.f. piecewise-smooth mechanical systems, which are useful for the characterisation of the stability of the equilibrium configurations. Numerical tools are implemented for the analysis of a 2~d.o.f. piecewise-smooth mechanical system, valid for piecewise-linear cases and extendible to the nonlinear ones. A numerical code has been developed, with the aim of predicting the stability of a piecewise-linear dynamical system a priori, varying the mechanical parameters. Moreover, “design maps” are produced for a given subset of the parameters space, so that a system with a desired stable or unstable behaviour can easily be designed. The aforementioned results can find applications in soft actuation or energy harvesting. In particular, in systems devoted to exploiting the flutter-like instability, the range of design parameters can be extended by using piecewise-smooth instead of smooth structures, since unstable flutter-like behaviour is possible also when each subsystem is actually stable. The second part of this Thesis deals with the numerical analysis of an elastic cloak for transient flexural waves in Kirchhoff-Love plates and the design of special metamaterials for this goal. In the literature, relevant applications of transformation elastodynamics have revealed that flexural waves in thin elastic plates can be diverted and channelled, with the aim of shielding a given region of the ambient space. However, the theoretical transformations which define the elastic properties of this “invisibility cloak” lead to the presence of a strong compressive prestress, which may be unfeasible for real applications. Moreover, this theoretical cloak must present, at the same time, high bending stiffness and a null twisting rigidity. In this Thesis, an orthotropic meta-structural plate is proposed as an approximated elastic cloak and the presence of the prestress has been neglected in order to be closer to a realistic design. With the aim of estimating the performance of this approximated cloak, a Finite Element code is implemented, based on a sub-parametric technique. The tool allows the investigation of the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. Moreover, the Finite Element code is extended to investigate a meta-plate interacting with a Winkler foundation, to analyse how the substrate modulus transforms in the cloak region. This second topic of the Thesis may find applications in the realization of approximated invisibility cloaks, which can be employed to reduce the destructive effects of earthquakes on civil structures or to shield mechanical components from unwanted vibrations.

Nous nous intéressons à des équations différentielles stochastiques obtenues en perturbant par un bruit blanc des équations différentielles ordinaires admettant N orbites périodiques asymptotiquement stables. Nous construisons une chaîne de Markov à temps discret et espace d’états continu appelée application de Poincaré aléatoire qui hérite du comportement métastable du système. Nous montrons que ce processus admet exactement N valeurs propres qui sont exponentiellement proches de 1 et nous donnons des expressions pour ces valeurs propres et les fonctions propres associées en termes de fonctions committeurs dans les voisinages des orbites périodiques. Nous montrons également que ces valeurs propres sont bien séparées du reste du spectre. Chacune de ces valeurs propres exponentiellement proche de 1 est également reliée à un temps d’atteinte de ces voisinages. De plus, les N valeurs propres exponentiellement proches de 1 et fonctions propres à gauche et à droite associées peuvent être respectivement approchées par des valeurs propres principales, des distributions quasi-stationnaires, et des fonctions propres principales à droite de processus tués quand ils atteignent ces voisinages. Les preuves reposent sur une représentation de type Feynman–Kac pour les fonctions propres, la transformée harmonique de Doob, la théorie spectrale des opérateurs compacts et une propriété de type équilibré détaillé satisfaite par les fonctions committeurs. / We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting N asymptotically stable periodic orbits. We construct a discrete-time,continuous-space Markov chain, called a random Poincaré map, which encodes the metastable behaviour of the system. We show that this process admits exactly N eigenvalues which are exponentially close to 1,and provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committorfunctions of neighbourhoods of periodic orbits. We also provide a bound for the remaining part of the spectrum. The eigenvalues that are exponentially close to 1 and the right and left eigenfunctions are well-approximated by principal eigenvalues, quasistationary distributions, and principal right eigenfunctions of processes killed upon hitting some of these neighbourhoods. Each eigenvalue that is exponentially close to 1is also related to the mean exit time from some metastable neighborhood of the periodic orbits. The proofsrely on Feynman–Kac-type representation formulas for eigenfunctions, Doob’s h-transform, spectral theory of compact operators, and a recently discovered detailed balance property satisfied by committor functions.