Arturo Olvera

An obstruction method for the destruction of invariant curves

Abstract An important point in a conservative dynamical system which is a parameter-depending perturbation of an integrable system is the destruction of the invariant curves. For small perturbations many invariant curves subsist according to KAM theorem. An increase in the size of perturbations produces, eventually, the destruction of the curves. The stochastic zone is then very large. In this paper we present for the standard map a geometrical approach to obtain values of the parameter for which no invariant curves exist. This is due to the existence of heteroclinic points. As it has been observed using other methods, there are several scaling properties in the phenomena which are also discussed here.

The obstruction criterion for non-existence of invariant circles and renormalization

The goal of this paper is to show that the renormalization group and the obstruction criterion can work together. We formulate a conjecture which supplements the standard renor- malization scenario for the breakdown of golden circle in twist maps. We show rigorously that if the conjecture was true then: (1) The stable manifold of the non-trivial fixed point would be part of the boundary between the existence of smooth invariant tori and hyperbolic orbits with golden mean rotation number. In partic- ular, the boundary of the set of twist maps with a torus with a golden mean rotation number would include a smooth submanifold in the space of analytic mappings. Moreover, if the conjecture was true, in the domain of universality (i.e. a small neighborhood of the non-trivial fixed point), we would have the following (2), (3), (4). (2) The obstruction criterion for non-existence of golden mean invari- ant circle (Olvera-Simo) is sharp. That is, for maps in the univer- sality class either there is a golden invariant circle or the condition in (Olvera-Simo) for non-existence of golden circles applies. (3) The criterion of (Greene-79) for existence of invariant circles if and only if there the residues of approximating orbits are finite would be valid. That is, for maps in the universality class there would be a smooth invariant circle if and only if the residue of periodic orbits approximating the circle goes to zero. (4) If there is no invariant circle, there are hyperbolic sets with golden mean rotation number. We also provide numerical evidence which suggests that the conjec- ture is true. We derive several scaling relations for observables related to the ob- struction criterion and verify them.

Estimation of the Amplitude of Resonance in the General Standard Map

This paper formulates some conjectures about the amplitude of resonance in the General Standard Map. The main idea is to expand the periodic perturbation function in Fourier series. Given any rational rotation number, we choose a finite number of harmonics in the Fourier expansion and we compute the amplitude of resonance of the reduced perturbation function of the map, using a suitable normal form around the resonance, which is valid for asymptotically small values of the perturbation parameter. For this map, we obtain a relation between the amplitude of resonance and the perturbation parameter: the amplitude is proportional to a rational power of the parameter, and so can be represented as a straight line on a log-log graph. The convex hull of these straight lines gives a lower bound for the amplitude of resonance, valid even when the perturbation parameter is of the order of 1. We find that some perturbation functions give rise a phenomenon that we call collapse of resonance; this means that the amplitude of resonance goes to zero for some value of the perturbation parameter. We find an empirical procedure to estimate this value of the parameter related to the collapse of resonance.

Hydrodynamics of an oscillating water column seawater pump. Part II: tuning to monochromatic waves

Flume experiments with a scale-model of a wave driven seawater pump in monochromatic waves are described. A tuning mechanism optimises the pump performance by keeping it at resonance with the waves. The pumping process itself was found to de-tune the system because of the reduced gravity restoring force due to spilling in the compression chamber. A perturbation analysis of the pump equations shows that performance of the system can be increased by optimising the shape of the pump intake to minimise losses due to vortex formation. An algorithm is derived, using a numerical model of the pump, which accurately determines the required volume of air in the compression chamber to induce resonance given variations in the wave frequency, the wave height and the tides. A sustainable development project to use a seawater pump to manage fisheries at a coastal lagoon in Mexico is described.

ELLIPTIC NON-BIRKHOFF PERIODIC ORBITS IN THE TWIST MAPS

We consider a perturbed twist map when the perturbation is big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant Cantorian set appears. From another point of view, the destruction of the IRC is associated with the appearance of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated with the existence of non-Birkhoff orbits. In this paper we relate the different approaches. In order to explain the creation of non-Birkhoff orbits, we provide qualitative and quantitative models. We show the existence of elliptic non-Birkhoff periodic orbits for an open set of values of the perturbative parameter. The bifurcations giving rise to the elliptic non-Birkhoff orbits and other related bifurcations are analysed. In the last section, we show a celestial mechanics example displaying the described behavior.

Mathematical accuracy of Aztec land surveys assessed from records in the Codex Vergara.

Land surveying in ancient states is documented not only for Eurasia but also for the Americas, amply attested by two Acolhua–Aztec pictorial manuscripts from the Valley of Mexico. The Codex Vergara and the Códice de Santa María Asunción consist of hundreds of drawings of agricultural fields that uniquely record surface areas as well as perimeter measurements. A previous study of the Codex Vergara examines how Acolhua–Aztecs determined field area by reconstructing their calculation procedures. Here we evaluate the accuracy of their area values using modern mathematics. The findings verify the overall mathematical validity of the codex records. Three-quarters of the areas are within 5% of the maximum possible value, and 85% are within 10%, which compares well with reported errors by Western surveyors that postdate Aztec–Acolhua work by several centuries.

Universal scalings of universal scaling exponents

In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disc boundaries, etc). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators. The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit.

Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model

Self-consistent chaotic transport is studied in a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flows and electron dynamics in plasmas. Self-consistency is incorporated through a mean-field that couples all the degrees-of-freedom. The model is formulated as a large set of N coupled standard-like area-preserving twist maps in which the amplitude and phase of the perturbation, rather than being constant like in the standard map, are dynamical variables. Of particular interest is the study of the impact of periodic orbits on the chaotic transport and coherent structures. Numerical simulations show that self-consistency leads to the formation of a coherent macro-particle trapped around the elliptic fixed point of the system that appears together with an asymptotic periodic behavior of the mean field. To model this asymptotic state, we introduced a non-autonomous map that allows a detailed study of the onset of global transport. A turnstile-type transport mechanism that allows transport across instantaneous KAM invariant circles in non-autonomous systems is discussed. As a first step to understand transport, we study a special type of orbits referred to as sequential periodic orbits. Using symmetry properties we show that, through replication, high-dimensional sequential periodic orbits can be generated starting from low-dimensional periodic orbits. We show that sequential periodic orbits in the self-consistent map can be continued from trivial (uncoupled) periodic orbits of standard-like maps using numerical and asymptotic methods. Normal forms are used to describe these orbits and to find the values of the map parameters that guarantee their existence. Numerical simulations are used to verify the prediction from the asymptotic methods.

A continuation method to study periodic orbits of the Froeschlé map

Abstract The dynamics of many Hamiltonian systems with three degrees of freedom is represented by the Froeschle map, which is symplectic and four-dimensional. In this paper we study sequences of periodic orbits approaching the invariant tori in order to obtain information about its stability. A homotopic method is used to continue the branches of periodic orbits. We found that the existence of turning points is related to the linear stability of the periodic orbit and its rotation vector.

Quadrilaterals and Bretschneider's Formula

We talk about a formula deduced by Bretschneider in the mid-nineteenth century that not only allows the computation of areas of planar quadrilaterals, regular and irregular, but also provides a more thorough understanding of the geometry of such figures. We present one version of Bretschneiders formula and talk about its scope of application and what it can say about the possible shapes of quadrilaterals.