We consider the effect of a non-autonomous periodic perturbation on a 2-dof autonomous system obtained as a truncation of the Hamiltonian-Hopf normal form. Our analysis focuses on the behaviour of the splitting of the invariant 2-dimensional stable/unstable manifolds. We analyse the different changes of dominant harmonic in the splitting functions. We describe how the dominant harmonics depend on the quotients of the continuous fraction expansion of the periodic forcing frequency. We have considered different frequencies including quadratic irrationals, frequencies having continuous fraction expansion with bounded quotients and frequencies with unbounded quotients. The methodology used is general enough to systematically deal with all these frequency types. All together allow us to get a detailed description of the asymptotic splitting behaviour for the concrete perturbation considered.

We study the dynamics of a family of 4D symplectic mappings near a doubly resonant elliptic fixed point. We derive and discuss algebraic properties of the resonances required for the analysis of a Takens type normal form. In particular, we propose a classification of the double resonances adapted to this problem, including cases of both strong and weak resonances.Around a weak double resonance (a junction of two resonances of two different orders, both being larger than 4) the dynamics can be described in terms of a simple (in general non-integrable) Hamiltonian model. The non-integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder.We use a 4D generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4D symplectic map. We evaluate numerically the volume of a 4D parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively. In good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2D analog.Different numerical studies point out that double resonances play a key role to understand Arnold diffusion. This paper has to be seen, also, as a first step in this direction.

We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is ‘universal’ in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system.

We consider a family F-epsilon of area-preserving maps (APMs) with a hyperbolic point H-epsilon whose invariant manifolds form a figure-eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes (EPL), a domain typically dominated by chaotic behavior. To this end, we use the Chirikov separatrix map (SM) as a model of the return to a fundamental domain containing lobes. We obtain an explicit estimate, valid for families F-epsilon with central symmetry and close to an integrable limit, of the relative measure of the set of parameters e for which F-epsilon has EPL trajectories. To get this estimate we look for EPL of the SM with the lowest possible period. The analytical results are complemented with quantitative numerical studies of the following families F-epsilon of APMs:The SM family, and we compare our analytical results with the numerical estimates.The standard map (STM) family, and we show how the results referring to the SM model apply to the EPL visiting the lobes that the invariant manifolds of the STM hyperbolic fixed point form.The conservative Henon map family, and we estimate the number of a particular type of symmetrical EPL related to the separatrices of the 4-periodic resonant islands.The results obtained can be seen as the quantitative analogs to those in Simo and Treschev (2008) [9], although here we deal with the a priori stable situation instead. (C) 2011 Elsevier B.V. All rights reserved.

The purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically. First of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out. The well-known separatrix map, defined on a figure eight when needed, is used to determine the location of rotational invariant curves (r.i.c.) inside and outside the resonance. The interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices: to produce theoretical estimates of the width of the stochastic zone, distance to the r.i.c., existence of tiny islands close to the separatrices, …In every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations, describing the analogy when possible. After this study, we focus on the formation of larger domains without r.i.c. (e.g. Birkhoff domains). To this end we introduce the biseparatrix map model. Although this is a qualitative model, the mechanism of destruction of the “last” r.i.c., and hence the process of creation of zones without r.i.c., is clarified by means of this simple model. Several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions.

The dynamics of a Rayleigh-Benard convection problem in a cubical cavity at moderate values of the Rayleigh number (Ra <= 10(5)) and a Prandtl number of Pr = 0.71 (with extensions to Pr = 0.75 and 0.80) was investigated. The cubical cavity was heated from below and had perfectly conducting sidewalls and uniform temperature distributions on the two horizontal walls. A system of ordinary differential equations with a dimension of typically N approximate to 000 was obtained when the conservation equations were discretized by means of a Galerkin method. Previous knowledge of the bifurcation diagram of steady solutions, reported in the literature, was used to identify the origin of several branches of periodic orbits that were continued with Ra. Half a dozen of such periodic orbits were found to be stable within narrow ranges of Ra (at most, some 5000 units wide). An attracting two-torus, restricted to a very narrow region of Ra, was also identified. It was found that the instabilization of periodic orbits quite often resulted into the development of complex dynamics such as the creation of homoclinic and heteroclinic orbits. Instances of both types of global bifurcations were analyzed in some detail.One particular instance of chaotic dynamics (a strange attractor) was also identified. Chaotic dynamics has been found at Pr = 0.71 in a flow invariant subspace, which can be interpreted as a fixed-point subspace in terms of equivariant theory; this subspace is not attracting. However, some regions of attracting chaotic dynamics for moderate Rayleigh numbers (9 x 10(4) <= Ra <= 10(5)) were found at values of Pr slightly above 0.71. The role of a particular homoclinic solution found at Pr = 0.71 in the generation of these chaotic regions was analyzed. (C) 2011 Elsevier B.V. All rights reserved.

This paper studies the dynamical mechanisms potentially involved in the so-called atmospheric low-frequency variability, occurring at midlatitudes in the Northern Hemisphere. This phenomenon is characterised by recurrent non-propagating and temporally persistent flow patterns, with typical spatial and temporal scales of 6000–10 000 km and 10–50 days, respectively. We study a low-order model derived from the 2-layer shallow-water equations on a β -plane channel. The main ingredients of the low-order model are a zonal flow, a planetary scale wave, orography, and a baroclinic-like forcing. A systematic analysis of the dynamics of the low-order model is performed using techniques and concepts from dynamical systems theory. Orography height ( h 0 ) and magnitude of zonal wind forcing ( U 0 ) are used as control parameters to study the bifurcations of equilibria and periodic orbits. Along two curves of Hopf bifurcations an equilibrium loses stability ( U 0 ≥ 12.5 m / s ) and gives birth to two distinct families of periodic orbits. These periodic orbits bifurcate into strange attractors along three routes to chaos: period doubling cascades, breakdown of 2-tori by homo- and heteroclinic bifurcations, or intermittency ( U 0 ≥ 14.5 m / s and h 0 ≥ 800 m ). The observed attractors exhibit spatial and temporal low-frequency patterns comparing well with those observed in the atmosphere. For h 0 ≤ 800 m the periodic orbits have a period of about 10 days and patterns in the vorticity field propagate eastward. For h 0 ≥ 800 m , the period is longer (30–60 days) and patterns in the vorticity field are non-propagating. The dynamics on the strange attractors are associated with low-frequency variability: the vorticity fields show weakening and strengthening of non-propagating planetary waves on time scales of 10–200 days. The spatio-temporal characteristics are “inherited” (by intermittency) from the two families of periodic orbits and are detected in a relatively large region of the parameter plane. This scenario provides a characterisation of low-frequency variability in terms of intermittency due to bifurcations of waves.

Fluid particle trajectories for the Rayleigh-Benard problem in a cube with perfectly conducting lateral walls have been investigated. The velocity and temperature fields of the stationary flow solutions have been obtained by means of a parameter continuation procedure based on a Galerkin spectral method. The rich dynamics of the resulting fluid particle paths has been studied for three branches of stationary solutions and different values of the Rayleigh number within the range 10(4) <= Ra <= 1.5 x 10(5) at a Prandtl number equal to 130. The stability properties and bifurcations of fixed points, which play a key role in the global dynamics, have been analyzed. Main periodic orbits and their stability character have also been determined. Poincare maps reveal that regions of chaotic motion and regions of regular motion coexist inside the cavity. The boundaries of these three-dimensional regions have been determined. The metric entropy gives an indication of the mixing properties of the large chaotic zone. (C) 2008 Elsevier B.V. All rights reserved.

A Swinging Atwood Machine (SAM ) is built and some experimental results concerning its dynamic behaviour are presented. Experiments clearly show that pulleys play a role in the motion of the pendulum, since they can rotate and have non-negligible radii and masses. Equations of motion must therefore take into account the inertial momentum of the pulleys, as well as the winding of the rope around them. Their influence is compared to previous studies. A preliminary discussion of the role of dissipation is included. The theoretical behaviour of the system with pulleys is illustrated numerically, and the relevance of different parameters is highlighted. Finally, the integrability of the dynamic system is studied, the main result being that the Machine with pulleys is non-integrable. The status of the results on integrability of the pulley-less Machine is also recalled.

We study the effect of a small dissipative radial perturbation acting on a one parameter family of area preserving diffeomorphisms. This is a specific type of dissipative perturbation. The interest is on the global effect of the dissipation on a fixed domain around an elliptic fixed/periodic point of the family, rather than on the effects around a single resonance. We describe the local/global bifurcations observed in the transition from the conservative to a weakly dissipative case: the location of the resonant islands, the changes in the domains of attraction of the foci inside these islands, how the resonances disappear, etc. The possible ω-limits are determined in each case. This topological description gives rise to three different dynamical regimes according to the size of dissipative perturbation. Moreover, we determine the conservative limit of the probability of capture in a generic resonance from the interpolating flow approximation, hence assuming no homoclinics in the resonance. As a paradigm of weakly dissipative radial maps, we use a dissipative version of the Hénon map.

We consider a one-parameter family of area preserving maps in a neighbourhood of an elliptic fixed point. As the parameter evolves hyperbolic and elliptic periodic orbits of different periods are created. The exceptional resonances of order less than 5 have to be considered separately. The invariant manifolds of the hyperbolic periodic points bound islands containing the elliptic periodic points. Generically, these manifolds split. It turns out that the inner and outer splittings are different under suitable conditions. We provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as a function of geometric properties. The numerical agreement is illustrated using mainly the Henon map as an example.

Low-frequency variability is investigated in a low-order atmospheric model derived from the 2-layer shallow water equations on a β-plane channel with bottom topography. • Low-frequency variability in our model: – variability on time scales from 10 to 200 days on a zonal wavenumber 3 (Benzi & Speranza 1989); – irregular weakening and amplification of nonpropagating planetary waves; – related to strange attractors through bifurcations of periodic orbits; – dominant time scales and spatial patterns: inherited from periodic orbits; – persistent in parameter space. • New scenario: available theories (Charney & DeVore 1979, Legras & Ghil 1985, Crommelin et al. 2004) rely on multiple equilibria and unrealistic zonal windspeeds.

One of the basic problems in dynamics, and specially in predi cting close approaches of NEOs relies on the fact that, even if the physical laws of motion are known with a reasonable app roximation, the set of initial data at a given epoch are not. Typically they are known to be in a given box in the phase space , maybe with some probability density, either known analytically (e.g., a normal multivariate distribution) o r in some experimental way. The problem is how to transport these data to a future epoch. A typical procedure consists in sampling the domain and transport every initial point independently. This is a c ostly process, specially to transport probability distrib ut ons. A solution to this problem will be presented. It is based on th e so-called Taylor methods based on the use of higher order variational equations or, in mathematical jargon, je t transport. In turn, this transport can be done using Taylor integration methods. It is extremely powerful, flexible, ac curate and efficient. We have applied this methodology to the particular case of (9 9942) Apophis, a NEO that will experience the most significant close approach with the Earth in the next 20 years . We will discuss how the initial uncertainties evolve along time and if we are able to determine the existence or not of pos sible future collisions. INTRODUCTION The asteroid (99942) Apophis represents an interesting cas e of Near Earth Object with no negligible probability of collision with the Earth. Among the known objects, it will get to the most significant distance with respect to the Earth in the next 20 years. Several studies have actually warded off t he threat of an impact in 2029, but it is not clear what this approach will imply for the future behaviour of the asteroid . It is well known that as long as the asteroid stays far from the planets, its motion can be approximated by the Keplerian laws, which do not hold at a close approach with the Earth–Moo n system. As we will see, to model the motion of Apophis we consider a restricted N−body problem. Our purpose is to analyse the motion of Apophis taking into account the uncertainties given by the observations using high orde r variational equations. We will use a Taylor method for the integration of the ODE and the jet transport methodology to t ransport a box of data along time. First, we will focus on the neighbourhood of April 13, 2029 to identify up to which order the variational equations are necessary, the minimum distance reached and the role of t he Moon in this passage. Clearly, we will also obtain information about the shape of the covered space and the rang of velocity swept. Then we will try to propagate the region found by increasing the order of the jet transport, ai ming to characterise the second foreseen approach that will take place between 2036 and 2037. MODEL We consider the motion of the asteroid to be sufficiently well d scribed by a restricted N−body model. That is, Apophis moves driven by the gravitational force exerted by the Sun, t he nine planets and Moon ( N = 11). The major bodies attract each other, but they are not affected at all by the asteroid. W e remark that the Moon is taken into account, because we are interested in close approaches of Apophis with respect to th e rbit of the Earth. Then the equations of motion are:

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.

The bifurcation diagram of steady convective flow patterns inside a cubical cavity with adiabatic lateral walls heated from below and filled with silicone oil (Pr = 130) was determined for values of the Rayleigh number (Ra) up to 1.5 x 10(5). A continuation procedure based on the Galerkin spectral method was used to determine the steady convective solutions as a function of Ra. Bifurcations leading to either new steady or time-dependent solutions were identified and new steady solution branches were also continued. A total of fifteen steady solutions were tracked and the stability analysis predicted that six flow patterns were stable and that two, three, or even four of these patterns coexisted over certain ranges of Ra in the studied domain. Predicted flow patterns and transitions are in agreement with flow visualizations previously reported in the literature. The variation of the Nusselt number (Nu) as a function of Pr was investigated for three of the stable flow patterns identified: a x or y roll, a diagonal oriented roll and a pattern formed by four connected half rolls. It was found that whereas the Nusselt changes within the region 0.71 < or = Pr < or = 10 it tends to an asymptotic value with increasing Pr.

In this paper we study the existence of analytic families of reducible linear quasi-periodic differential equations in matrix Lie algebras. Under suitable conditions we show, by means of a Kolmogorov-Arnold-Moser (KAM) scheme, that a real analytic quasi-periodic system close to a constant matrix can be modified by the addition of a time-free matrix that makes it reducible to constant coefficients. If the system depends analytically on external parameters, then this modifying term is also analytic. As a major application, we prove the analyticity of resonance tongue boundaries in Hill's equation with a small quasi-periodic forcing. Several consequences for the spectrum of Schrodinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schrodinger operators with a small real analytic and quasi-periodic potential has all spectral gaps open and, therefore, it is a Cantor set. Some other applications are included for linear quasi-periodic systems on so(3, R) and sp(n, R).

A class of strange attractors is described, occurring in a low-dimensional model of general atmospheric circulation. The differential equations of the system are subject to periodic forcing, where the period is one year - as suggested by Lorenz in 1984. The dynamics of the system is described in terms of a Poincare map, computed by numerical means. It is conjectured that certain strange attractors observed in the Poincare map are of quasi-periodic Henon-like type, i.e., they coincide with the closure of the unstable manifold of a quasi-periodic invariant circle of saddle type. A route leading to the formation of such strange attractors is presented. It involves a finite number of quasi-periodic period doubling bifurcations, followed by the destruction of an invariant circle due to homoclinic tangency.

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ1, is positive. Concerning the next Lyapunov exponent, Λ2, there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ2| < ρ, where ρ is some tolerance ranging between 10-5 and 10-6). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.

The goal of this paper is to show some properties of real analytic Hamiltonian systems of the form H = 1/2y(2) + V(x, t), where V is periodic in x and t. We prove that this system has invariant tori of rotational type for sufficiently large values of vertical bar y vertical bar. The measure of the set of such tori is shown to be exponentially close to full measure as y -> +/-infinity.