Active control of ionized boundary layers

The challenging problems, in the field of control of chaos or of transition to chaos, lie in the domain of infinite-dimensional systems. Access to all variables being impossible in this case and the controlling action being limited to a few collective variables, it will not in general be possible to drive the whole system to the desired behaviour. A paradigmatic problem of this type is the control of the transition to turbulence in the boundary layer of fluid motion. By analysing a boundary layer flow for an ionized fluid near an airfoil, one concludes that active control of the transition amounts to the resolution of an generalized integro-differential eigenvalue problem. To cope with the required response times and phase accuracy, electromagnetic control, whenever possible, seems more appropriate than mechanical control by microactuators.Comment: 10 pages Latex, 5 ps-figure

Structure-generating mechanisms in agent-based models

The emergence of dynamical structures in multi-agent systems is analysed. Three different mechanisms are identified, namely: (1) sensitive-dependence and convex coupling, (2) sensitive-dependence and extremal dynamics and (3) interaction through a collectively generated field. The dynamical origin of the emergent structures is traced back either to a modification, by interaction, of the Lyapunov spectrum or to multistable dynamics.Comment: 27 pages Latex, 10 figures (.gif

Network dependence of strong reciprocity

Experimental evidence suggests that human decisions involve a mixture of self-interest and internalized social norms which cannot be accounted for by the Nash equilibrium behavior of Homo Oeconomicus. This led to the notion of strong reciprocity (or altruistic punishment) to capture the human trait leading an individual to punish norm violators at a cost to himself. For a population with small autonomous groups with collective monitoring, the interplay of intra- and intergroup dynamics shows this to be an adaptive trait, although not fully invasive of a selfish population. However, the absence of collective monitoring in a larger society changes the evolution dynamics. Clustering seems to be the network parameter that controls maintenance and evolution of the reciprocator trait.Comment: 14 pages Late

Quantum collision states for positive charges in an octahedral cage

One-electron energy levels are studied for a configuration of two positive charges inside an octahedral cage, the vertices of the cage being occupied by atoms with a partially filled shell. Although ground states correspond to large separations, there are relatively low-lying states with large collision probabilities. Electromagnetic radiation fields used to excite the quantum collisional levels may provide a means to control nuclear reactions. However, given the scale of the excitation energies involved, this mechanism cannot provide an explanation for the unexplained ``cold fusion'' events.Comment: 12 pages Latex, 5 eps figure

Dynamics of networks and applications

A survey is made of several aspects of the dynamics of networks, with special emphasis on unsupervised learning processes, non-Gaussian data analysis and pattern recognition in networks with complex nodes.Comment: Talk at the Heraeus Seminar "Scientific Applications of Neural Nets", to appear in the proceedings (Springer LNP

Clustering and synchronization with positive Lyapunov exponents

Clustering and correlation effects are frequently observed in chaotic systems in situations where, because of the positivity of the Lyapunov exponents, no dimension reduction is to be expected. In this paper, using a globally coupled network of Bernoulli units, one finds a general mechanism by which strong correlations and slow structures are obtained at the synchronization edge. A structure index is defined, which diverges at the transition points. Some conclusions are drawn concerning the construction of an ergodic theory of self-organization.Comment: 10 pages Latex, 2 ps figure

A Stochastic Process for the Dynamics of the Turbulent Cascade

Velocity increments over a distance r and turbulent energy dissipation on a box of size r are well described by the multifractal models of fully developed turbulence. These quantities and models however, do not involve time-correlations and therefore are not a detailed test of the dynamics of the turbulent cascade. If the time development of the turbulent cascade, in the inertial range, is related to the lifetime of the eddies at different length scales, the time correlations may be described by a stochastic process on a tree with jumping kernels which are a function of the ultrametric (tree) distance. We obtain the solutions of the Chapman-Kolmogorov equation for such a stochastic process, with jumping kernels depending on the ultrametric distance, but with an arbitrarily specified invariant probability measure. We then show how to use these solutions to compute the time correlations in the turbulent cascade.Comment: 11 pages, LATeX,2 fig available at ftp://cpt.univ-mrs.fr/pub/preprints/93/dynamical-systems/93-P.296

A consistent measure for lattice Yang-Mills

The construction of a consistent measure for Yang-Mills is a precondition for an accurate formulation of non-perturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus has been constructed for a theory of non-abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.Comment: 15 pages Latex, 2 figures. arXiv admin note: substantial text overlap with arXiv:1504.0779

Multistability in dynamical systems

In neuroscience, optics and condensed matter there is ample physical evidence for multistable dynamical systems, that is, systems with a large number of attractors. The known mathematical mechanisms that lead to multiple attractors are homoclinic tangencies and stabilization, by small perturbations or by coupling, of systems possessing a large number of unstable invariant sets. A short review of the existent results is presented, as well as two new results concerning the existence of a large number of stable periodic orbits in a perturbed marginally stable dissipative map and an infinite number of such orbits in two coupled quadratic maps working on the Feigenbaum accumulation point.Comment: 11 pages Latex, to appear in Dynamical Systems: From Crystal to Chaos, World Scientific, 199

The geometry of noncommutative space-time

Stabilization, by deformation, of the Poincar\'{e}-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.Comment: 12 pages Late