statistical properties of chaotic systems the theory of smooth dynamical systems and the theory of abstract dynamical systems ergodic theory although having the same roots have for many years developed quite independently of one another these theories have now matured to the point where they can be combined to shed light on the nature of chaotic behavior

une borne superieure pour l entropie topologique d une application rationnelle let x be a complex projective manifold and let f be a dominating rational map from x onto x we show that the topological entropy h f of f is bounded from above by the logarithm of its maximal dynamical degree

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

controlled lagrangians and the stabilization of mechanical systems ii potential shaping for pt i see ibid vol 45 p 2253 70 2000 we extend the method of controlled lagrangians cl to include potential shaping which achieves complete state space asymptotic stabilization of mechanical systems the cl method deals with mechanical systems with symmetry and provides symmetry preserving kinetic shaping and feedback controlled dissipation for state space stabilization in all but the symmetry variables potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables the approach also extends the method of controlled lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline

reflection and attachment of spirals at obstacles for the fitzhugh nagumo and beeler reuter models in this paper the fitzhugh nagumo fhn equations and a modified fhn mfhn are considered for the modified version the recovery variable v has three different time scales by considering different parameters in the local dynamics of the mfhn equations it is observed that the phenomenon of reflection and annihilation at an impermeable boundary is observed just as in the beeler reuter model the interaction of spirals obtained with the fhn mfhn and beeler reuter model and an obstacle is also considered the phenomenon of reflection of the spiral wave at a boundary changes when the boundary becomes an obstacle four properties for attachment of a spiral wave to an obstacle are presented in this work

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

tracking for fully actuated mechanical systems a geometric framework we present a general framework for the control of lagrangian systems with as many inputs as degrees of freedom relying on the geometry of mechanical systems on manifolds we propose a design algorithm for the tracking problem the notions of error function and transport map lead to a proper definition of configuration and velocity error these are the crucial ingredients in designing a proportional derivative feedback and feedforward controller the proposed approach includes as special cases a variety of results on control of manipulators pointing devices and autonomous vehicles our design provides particular insight into both aerospace and underwater applications where the configuration manifold is a lie group

a geometrical description of the discrete painleve vi and v equations we present a geometrical approach for the discrete painleve equations based on weyl groups the method relies on the bilinear formalism and assumes that the multidimensional function lives on the weight lattice of the appropriate affine weyl group the equations for the function a system of nonautonomous hirota miwa equations govern the evolution along the independent variable and the parameters of the equation the latter evolution induced by the schlesinger transformations in the present paper we analyse the case of the e 1 7 group using the geometrical description we derive the nonlinear discrete equations we find that in the case of the e 1 7 group these are the asymmetric q pvi and d pv that were recently proposed

ergodic theory on moduli spaces let m be a compact surface with x m 0 and let g be a compact lie group whose levi factor is a product of groups locally isomorphic to su 2 for example su 2 itself then the mapping class group rm of m acts on the moduli space x m of flat g bundles over m possibly twisted by a fixed central element of g when m is closed then fm preserves a symplectic structure on x m which has finite total volume on m more generally the subspace of x m corresponding to flat bundles with fixed behavior over am carries a rm invariant symplectic structure the main result is that rm acts ergodically on x m with respect to the measure induced by the symplectic structure

a classification of two dimensional integrable mappings and rational elliptic surfaces we classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces while the qrt mappings can be restricted on each fiber there exist several classes of integrable mappings which exchange fibers we also show an equivalent condition when a generalized halphen surface becomes a halphen surface of index m

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

periodic points for area preserving birational maps of surfaces it is a basic problem to count the number of periodic points of a surface mapping since the growth rate of this number as the period tends to infinity is an important dynamical invariant however this problem becomes difficult when the map admits curves of periodic points in this situation we give a precise estimate of the number of isolated periodic points for an area preserving birational map of a projective complex surface

almost global tracking of simple mechanical systems on a general class of lie groups we present a general intrinsic tracking controller design for fully actuated simple mechanical systems when the configuration space is one of a general class of lie groups we first express a state feedback controller in terms of a function the error function satisfying certain regularity conditions if an error function can be found then a general smooth and bounded reference trajectory may be tracked asymptotically from almost every initial condition with locally exponential convergence asymptotic convergence from almost every initial condition is referred to as almost global asymptotic stability error functions may be shown to exist on any compact lie group or any lie group diffeomorphic to the product of a compact lie group and r sup n this covers many cases of practical interest such as so n se n their subgroups and direct products we show here that for compact lie groups the dynamic configuration feedback controller obtained by composing the full state feedback law with an exponentially convergent velocity observer is also almost globally asymptotically stable with respect to the tracking error we emphasize that no invariance is needed for these results however for the special case where the kinetic energy is left invariant we show that the explicit expression of these controllers does not require coordinates on the lie group the controller constructions are demonstrated on so 3 and simulated for the axi symmetric top results show excellent performance

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

a model for fast computer simulation of waves in excitable media abstract starting from a two variable system of reaction diffusion equations an algorithm is devised for efficient simulation of waves in excitable media the spatio temporal resolution of the simulation can be varied continuously for fine resolutions the algorithm provides accurate solution of the underlying reaction diffusion equations for coarse resolutions the algorithm provides qualitative simulations at small computational cost

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

on almost global tracking for a certain class of simple mechanical systems in this paper we propose a control law for the almost global asymptotic tracking agat of a smooth reference trajectory for a fully actuated simple mechanical system sms evolving on a riemannian manifold that can be embedded in a euclidean space the existing results on tracking for an sms are either local or almost global only in the case the manifold is a lie group in the latter case the notion of a configuration error is naturally defined by the group operation and facilitates a global analysis however such a notion is not intrinsic to a riemannian manifold in this paper we define a configuration error followed by error dynamics on a riemannian manifold and then prove the agat the results are demonstrated for a spherical pendulum which is an sms on s 2 and for a particle moving on a lissajous curve in mathbb r 3

rational solutions of the painleve vi equation in this paper we classify all values of the parameters and of the painleve vi equation such that there are rational solutions we give a formula for them up to the birational canonical transformations and the symmetries of the painleve vi equation

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

an ergodic study of painleve vi an ergodic study of painleve vi is developed the chaotic nature of its poincare return map is established for almost all loops the exponential growth of the numbers of periodic solutions is also shown principal ingredients of the arguments are a moduli theoretical formulation of painleve vi a riemann hilbert correspondence the dynamical system of a birational map on a cubic surface and the lefschetz fixed point formula

asymptotic smooth stabilization of the inverted 3 d pendulum the 3 d pendulum consists of a rigid body supported at a fixed pivot with three rotational degrees of freedom it is acted on by gravity and it is fully actuated by control forces the 3 d pendulum has two disjoint equilibrium manifolds namely a hanging equilibrium manifold and an inverted equilibrium manifold the contribution of this paper is that two fundamental stabilization problems for the inverted 3 d pendulum are posed and solved the first problem asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and attitude of the 3 d pendulum the second problem asymptotic stabilization of the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3 d pendulum these control problems for the 3 d pendulum exemplify attitude stabilization problems on the configuration manifold so 3 in the presence of potential forces lyapunov analysis and nonlinear geometric methods are used to assess global closed loop properties yielding a characterization of the almost global domain of attraction for each case

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

energy and invariant measures for birational surface maps given a birational self map of a compact complex surface it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology under a very weak hypothesis on the map we show how to construct such a measure the main point in the construction is to make sense of the wedge product of two positive closed 1 1 currents we are able to do this in our case because local potentials for each current have finite energy with respect to the other our methods also suffice to show that the resulting measure is mixing does not charge curves and has nonzero lyapunov exponents

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

bilinear equations and q discrete painleve equations satisfied by variables and coefficients in cluster algebras we construct cluster algebras the variables and coefficients of which satisfy the discrete mkdv equation the discrete toda equation and other integrable bilinear equations several of which lead to q discrete painleve equations these cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation period property we will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation

dynamics on k3 surfaces salem numbers and siegel disks this paper presents the first examples of k3 surface automorphisms f x rightarrow x with siegel disks domains on which f acts by an irrational rotation the set of such examples is countable and the surface x must be non projective to carry a siegel disk these automorphisms are synthesized from salem numbers of degree 22 and trace 1 which play the role of the leading eigenvalue for f h 2 x the construction uses the torelli theorem the atiyah bott fixed point theorem and results from transcendence theory

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure