Resonance Zones and Lobe Volumes for Volume-Preserving Maps

We study exact, volume-preserving diffeomorphisms that have heteroclinic connections between a pair of normally hyperbolic invariant manifolds. We develop a general theory of lobes, showing that the lobe volume is given by an integral of a generating form over the primary intersection, a subset of the heteroclinic orbits. Our definition reproduces the classical action formula in the planar, twist map case. For perturbations from a heteroclinic connection, the lobe volume is shown to reduce, to lowest order, to a suitable integral of a Melnikov function.Comment: ams laTeX, 8 figure

Generating Forms for Exact Volume-Preserving Maps

We study the group of volume-preserving diffeomorphisms on a manifold. We develop a general theory of implicit generating forms. Our results generalize the classical formulas for generating functions of symplectic twist maps.Comment: laTeX, 20 pages, 1 figur

Symmetry Reduction by Lifting for Maps

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.Comment: laTeX, 31 pages, 5 figure

Canonical Melnikov theory for diffeomorphisms

We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure

Quadratic Volume Preserving Maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps

A λ-lemma for Normally Hyperbolic Invariant Manifolds

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Visual Explorations of Dynamics: the Standard Map

The Macintosh application \textit{StdMap} allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program.Comment: Corrections in a couple of equations, and updated to the latest version of StdMa

A focus on selected perspectives of the NUMEN project

The use of double charge exchange reactions is discussed in view of their application to extract information that may be helpful to determinate the nuclear matrix elements entering in the expression of neutrinoless double beta decay half-life. The strategy adopted in the experimental campaigns performed at INFN - Laboratori Nazionali del Sud and in the analysis methods within the NUMEN project is briefly described, emphasizing the advantages of the multi-channel approach to nuclear reaction data analysis. An overview on the research and development activities on the MAGNEX magnetic spectrometer is also given, with a focus on the chosen technological solutions for the focal plane detector which will guarantee the performances at high-rate conditions

The NUMEN project: NUclear Matrix Elements for Neutrinoless double beta decay

The article describes the main achievements of the NUMEN project togetherwith an updated and detailed overview of the related R&D activities andtheoretical developments. NUMEN proposes an innovative technique to access thenuclear matrix elements entering the expression of the lifetime of the doublebeta decay by cross section measurements of heavy-ion induced Double ChargeExchange (DCE) reactions. Despite the two processes, namely neutrinoless doublebeta decay and DCE reactions, are triggered by the weak and strong interactionrespectively, important analogies are suggested. The basic point is thecoincidence of the initial and final state many-body wave-functions in the twotypes of processes and the formal similarity of the transition operators. Firstexperimental results obtained at the INFN-LNS laboratory for the40Ca(18O,18Ne)40Ar reaction at 270 MeV, give encouraging indication on thecapability of the proposed technique to access relevant quantitativeinformation. The two major aspects for this project are the K800Superconducting Cyclotron and MAGNEX spectrometer. The former is used for theacceleration of the required high resolution and low emittance heavy ion beamsand the latter is the large acceptance magnetic spectrometer for the detectionof the ejectiles. The use of the high-order trajectory reconstructiontechnique, implemented in MAGNEX, allows to reach the experimental resolutionand sensitivity required for the accurate measurement of the DCE cross sectionsat forward angles. However, the tiny values of such cross sections and theresolution requirements demand beam intensities much larger than manageablewith the present facility. The on-going upgrade of the INFN-LNS facilities inthis perspective is part of the NUMEN project and will be discussed in thearticle