Regions of nonexistence of invariant tori for spin-orbit models

The spin-orbit problem in celestial mechanics describes the motion of an oblate satellite moving on a Keplerian orbit around a primary body. We apply the conjugate points criterion for the nonexistence of rotational invariant tori. We treat both the conservative case and a case including a dissipative effect modeling a tidal torque generated by internal nonrigidity. As a by-product of the conjugate points criterion we obtain a global view of the dynamics, thanks to the introduction of a tangent orbit indicator, which allows us to discern the dynamical character of the motion.     

Quasi-genericity of bifurcations to high dimensional invariant tori for maps

We consider a family of maps in a Banach spaceE near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle, the other part of the spectrum being strictly inside the unit disc. We focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantT ¹-circles into a family of invariantT ²-tori. We show that this problem needs a 3 dimensional parameter unfolding and that, for the complete maps, bifurcation occurs at points _,, where is the rotation number on the non-normally hyperbolicT ¹-circle, ande ^±2i are the eigenvalues of the constant matrix conjugated to the non-contracting part of the linearization on the normal fiber bundle overT ¹. Making some non-resonance and diophantine assumptions on (, ) leading to a positive measure Cantor set inT ², we show that in paraboloïdal regions of the 3 dim. parameter space we have clean bifurcations as for the truncated normal form. The complement of these regions forms a set of bubbles such as the ones obtained by Chenciner in [Chen] for a codimension 2 problem for maps in ². The main tool here is a generalization for a matrix function onT ¹, close to a constant, of the quasi-conjugacy to a constant, modulo a minimum of additional parameters (moved quasi-conjugacy). For the infinite dimensional case we use aC decoupling result on the angular dependent linear parts into a contraction, still angular dependent, and another part quasi-conjugated to a constant matrix. This type of analysis applies for a wide range of problems, where truncated normal forms of the maps give bifurcations fromT ⁿ toT ⁿ⁺¹ tori, and this needs a (n+1)-dimensional parameter unfolding.We gratefully acknowledge the DRET (contrat 86/1445) who supported one of the authors (J.L.) during this work. This research has been also supported by the E.E.C. contract No. ST 2J-0316-C (EDB) on Mathematical problems in nonlinear Mechanics     

Dual invariant theory. Renormalization. Models

The problem of renormalization of a dual-Invariant theory of massive charged vector particles is considered for the case of electromagnetic interactions. A model of weak interactions is suggested, in which particles described by the dual-invariant theory play the role of intermediate vector bosons. Renormalizability and unitarity of the model are shown.     

Renormalization group invariant average momentum of non-singlet parton densities

We compute, within the Schrödinger functional scheme, a renormalization group invariant renormalization constant for the first moment of the non-singlet parton distribution function. The matching of the results of our non-perturbative calculation with the ones from hadronic matrix elements allows us to obtain eventually a renormalization group invariant average momentum of non-singlet parton densities, which can be translated into a preferred scheme at a specific scale.     

Renormalization scheme invariant predictions for deep-inelastic scattering and determination ofΛ QCD

I discuss theoretical aspects of the renormalisation scheme (RS) ambiguity problem and the approaches to its solution from the point of view of QCD phenomenology and the scale Λ determination. I advocate the method of RS-invariant perturbation theory (RSIPT) as a sound basis for describing experiment in QCD. To this end the method is developed for the non-singlet structure functions (SF) of deep-inelastic scattering and recent high precision data on SF's are analysed in a RS-invariant way. It is shown that RSIPT leads to a more accurate and reliable determination of the QCD scale Λ, which is consistent with theoretical assumption about better convergence of RS-invariant perturbative series.     

A dynamical system with a strange attractor and invariant tori

This paper describes a simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities and the unusual property that it is exhibits conservative behavior for some initial conditions and dissipative behavior for others. The conservative regime has quasi-periodic orbits whose amplitude depend on the initial conditions, while the dissipative regime is chaotic. Thus a strange attractor coexists with an infinite set of nested invariant tori in the state space.     

The transition to explosive solitons and the destruction of invariant tori

We investigate the transition to explosive dissipative solitons and the destruction of invariant tori in the complex cubic-quintic Ginzburg-Landau equation in the regime of anomalous linear dispersion as a function of the distance from linear onset. Using Poncaré sections, we sequentially find fixed points, quasiperiodicity (two incommesurate frequencies), frequency locking, two torus-doubling bifurcations (from a torus to a 2-fold torus and from a 2-fold torus to a 4-fold torus), the destruction of a 4-fold torus leading to non-explosive chaos, and finally explosive solitons. A narrow window, in which a 3-fold torus appears, is also observed inside the chaotic region.     

Renormalization and destruction of 1/gamma2 tori in the standard nontwist map

Extending the work of del-Castillo-Negrete, Greene, and Morrison [Physica D 91, 1 (1996); 100, 311 (1997)] on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator.     

Strange attractor for the renormalization flow for invariant tori of hamiltonian systems with two generic frequencies

We analyze the stability of invariant tori for Hamiltonian systems with two degrees of freedom by constructing a transformation that combines Kolmogorov-Arnold-Moser theory and renormalization-group techniques. This transformation is based on the continued fraction expansion of the frequency of the torus. We apply this transformation numerically for arbitrary frequencies that contain bounded entries in the continued fraction expansion. We give a global picture of renormalization flow for the stability of invariant tori, and we show that the properties of critical (and near critical) tori can be obtained by analyzing renormalization dynamics around a single hyperbolic strange attractor. We compute the fractal diagram, i.e., the critical coupling as a function of the frequencies, associated with a given one-parameter family.     

The invariant charges of the Nambu-Goto theory in WKB-approximation: Renormalization

We discuss the one loop renormalization of the reparametrization invariant non-local conserved charges of the Nambu-Goto string theory. In addition we show the stability of a special well-known state under the corresponding infinitesimal symmetry transformations — at least in the WKB-approximation.Work supported by Deutsche Forschungsgemeinschaft     

Study of a dynamical system with a strange attractor and invariant tori

A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.     

Renormalization scheme invariant analysis of the DIS structure functionsF 2 andF L

A renormalization scheme invariant analysis of the deep inelastic scattering structure functionF ₂ andF _L is performed. Expressions for the moments are given in this approach. We find a significant improvement of the agreement withR=σ _L /σ _T -SLAC data with respect to the conventional perturvative \((\overline {MS} )\) analysis. Higher twist corrections are also required confirming previous evidence.