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In this work we study the splitting distance of a rapidly perturbed pendulum H(x, y, t) = 1 2 y 2 + (cos(x) - 1) + µ(cos(x) - 1)g t e with g(t ) = P |k|>1 g [k] e ikt a 2p-periodic function and µ, e 1. Systems of this kind undergo exponentially small splitting and, when µ 1, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided g [±1] 6= 0. Our study focuses on the case g [±1] = 0 and it is motivated by two main reasons. On the one hand the general understanding of the splitting, as current results fail for a perturbation as simple as g(t ) = cos(5t ) + cos(4t ) + cos(3t ). On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency p/q in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with g(t ) = sin p · t e + cos q · t e , where, for most p, q ¿ Z the perturbation satisfies g [±1] 6= 0. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton-Jacobi formalism. The leading exponentially small term appears at order µ n , where n is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it., Preprint

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1, . . . , L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0 < \mu \leqslant 1. In particular we show that L3 cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries \mu is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order \sqrt{\mu }, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to \sqrt{\mu } . Thus, classical perturbative methods (i.e. the Melnikov-Poincaré method) can not be applied., Peer Reviewed, Postprint (published version)

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, , called the Lagrange points. The Lagrange point is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper and its sequel [10], we provide an asymptotic formula for the distance between the one dimensional stable and unstable invariant manifolds of when the ratio between the masses of the primaries µ is small. It implies that cannot have one-round homoclinic orbits. If the mass ratio µ is small, the hyperbolic eigenvalues are weaker than the elliptic ones by factor of order . This implies that the distance between the invariant manifolds is exponentially small with respect to µ and, therefore, the classical Poincaré–Melnikov method cannot be applied. In this first paper, we approximate the RPC3BP by an averaged integrable Hamiltonian system which possesses a saddle center with a homoclinic orbit and we analyze the complex singularities of its time parameterization. We also derive and study the inner equation associated to the original perturbed problem. The difference between certain solutions of the inner equation gives the leading term of the distance between the stable and unstable manifolds of . In the sequel [10] we complete the proof of the asymptotic formula for the distance between the invariant manifolds., Peer Reviewed, Postprint (author's final draft)

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies., Peer Reviewed, Preprint

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequencies, Preprint

In 1964, V. I. Arnol'd proved the existence of nearly-integrable Hamiltonian systems which have global instabilities (global chaotic behaviour). This phenomenon is nowadays termed under the name "Arnol'd diffusion". One of the key ideas that he used is to "travel" along invariant manifolds of the Hamiltonian system. The purpose of this project is to understand the Arnol'd instability mechanism and study new ones using different invariant objects.

The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied., Preprint

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt2-1$. We show that the oincare-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisffies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon, Preprint

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector $\omega/v\epsilon$, with $\epsilon=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincar´e–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies., Preprint

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases., Postprint (published version)

We study the splitting of invariant manifolds of whiskered t ori with two or three frequencies in nearly-integrable Hamiltonian systems. We consider 2-dimensional tori with a frequency vector ω = (1 , Ω) where Ω is a quadratic irrational number, or 3-dimensional tori with a frequency v ector ω = (1 , Ω , Ω 2 ) where Ω is a cubic irrational number. Applying the Poincar ́e–Melnikov method, we find exponentia lly small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associa ted to the invariant torus, showing that such estimates depend strongly on the arithmetic properties of the frequen cies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfille d in 24 cases, which allows us to provide asymptotic estimate s in a simple way. In the cubic case, we focus our attention to th e case in which Ω is the so-called cubic golden number (the real root of x 3 + x − 1 = 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubi c cases., Preprint

In this paper we prove the breakdown of a heteroclinic connection in the analytic versal unfoldings of the generic Hopf-zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon “beyond all orders”. In this paper we provide a formula for the distance between the corresponding stable and unstable one-dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings, Peer Reviewed, Postprint (published version)

The key point of our approach is to write the invariant manifolds in terms of generating functions, which are solutions of the Hamilton-Jacobi equation. In some examples, we show that it is enough to analyse the phase portrait of the Riccati equation without solving it explicitly. Finally, we consider an analogous problem in a perturbative situation. If the invariant manifolds of the unperturbed loop coincide, we have a problem of splitting of separatrices. In this case, the Riccati equation is replaced by a Mel0nikov potential defined as an integral, providing a condition for the existence of a perturbed loop and its transversality. This is also illustrated with a concrete example., Preprint

We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and suffcient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop., Preprint

The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic xed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coe cients involved in the expansion have a resurgent origin, as shown in, Peer Reviewed, Postprint (published version)

A sequence of \inner equations" attached to certain perturbations of the McMillan map was considered in [5], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [5]. The present work relies on ideas from resur- gence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Ecalle's alien derivations to measure the discrepancy between di erent Borel-Laplace sums., Peer Reviewed, Postprint (published version)

This paper gives an account of two articles which illustrate the use of Resurgence theory for estimating the diference between two complex invariant manifolds associated with the area-preserving H?enon map for the first one, and with a Hamiltonian system which stems from the rapidly forced pendulum for the second one., Peer Reviewed

És difícil dibuixar una frontera, dins la Teoria de Sistemas Dinàmics, entre lleis de conservació i simetries doncs, sovint, les seves característiques es confonen. Un clar exemple d'aquest fenómen el constitueixen els sistemes Hamiltonians i els sistemes reversibles. Breument, un sistema dinàmic es diu temps-reversible (o, per nosaltres, simplement reversible) si és invariant sota l'acció d'un difeomorfisme involutiu a l'espai i una inversió en el sentit del temps. és en aquest marc on cal situar aquesta memòria. Concretament, ens centrem en dos punts molt particulars: la Teoria de Formes Normals i el fenómen del trencament de separatrius, tots dos introduïts per Poincaré a la seva tesi (1890). Respecte al primer d'aquests punts, en aquesta tesi s'introdueix el concepte de Pseudo Forma Normal (breument PNF), inspirat en idees d'en Moser, i que permet transformar, sota certes hipotesis, un sistema analític en un d'equivalent d'aspecte el més simple possible. Aquesta PNF és una generalització de la coneguda Forma Normal de Birkhoff amb la qual coincideix si el sistema considerat és Hamiltonià o reversible. Com a conseqüència, s'obté, en determinats casos, l'equivalència local entre aquests dos tipus de sistemes. Aquesta PNF pot esdevenir una eina útil per estudiar la dinàmica d'un sistema analític a l'entorn d'un equilibri (un punt, una òrbita periòdica o un tor). El segon punt, l'escissió de separatrius, fa referència a l'intersecció transversal de varietats invariants procedent del trencament d'una certa connexió homoclínica a l'afegir al sistema una petita pertorbació. Un dels motius d'interés sobre aquest fenòmen és que és un dels principals causants de comportament estocàstic en sistemes Hamiltonians. Un problema relacionat amb aquest trencament de separatrius és el de mesurar-lo, sigui a partir del càlcul de l'angle amb el que es troben aquestes varietats per primer cop, per l'àrea que tanquen entre elles, etc. El mètode habi, It is difficult, in the Theory of Dynamical Systems, to draw a boundary line between conservation laws and symmetries because often their effects on the dynamics are very similar. This is the case of the Hamiltonian and the Reversible systems. Briefly, a dynamical system is called time-reversible (or, simply, reversible) if it is invariant under the action of an involutive spatial diffeomorphism and a reversion in time's arrow. This is the frame where this work must be placed. Precisely, we focus our attention in two particular points: the Theory of Normal Forms and the phenomenon of the splitting of separatrices, both introduced by Poincare in his thesis (1890). Regarding the first one of this topics, we introduce the concept of Pseudo-normal Form (PNF in short). It comes from ideas of Moser and allows to transform, under suitable conditions, an analytic system around an equilibrium in another equivalent one having a quite simple form. This PNF is a generalization of the celebrated Birkhoff Normal Form and both coincide if the system is Hamiltonian or reversible. Consequently, the local equivalence between both types of systems is derived in some cases. This PNF can become a useful tool to study the dynamics of an analytic system in a neighborhood of an equilibrium (a fixed point, a periodic orbit or a torus). The second topic, the splitting of separatrices, is related to the transversal intersection of invariant manifolds derived from the splitting of a given homoclinic connection when some small perturbation is considered. One of the reasons that makes this phenomenon interesting is that it seems to be one of the main causes of the stochastic behavior in Hamiltonian systems. One problem related to this splitting of separatrices becomes to measure it, studying, for instance, some angle the form when they meet for the first time, the area of the first lobe, etc. The standard method to estimate this size is the celebrated Poincaré, Postprint (published version)