the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

on stable nonconstant stationary solutions and mesoscopic patterns for fitzhugh nagumo equations in higher dimensions abstract fitzhugh nagumo equation has been studied extensively in the field of mathematical biology it has the mechanism of lateral inhibition which seems to play a big role in the pattern formation of plankton distribution we consider fitzhugh nagumo equation in high dimension and show the existence of stable nonconstant stationary solutions which have fine structures on a mesoscopic scale we construct spatially periodic stationary solutions moreover we compute the singular limit energy which suggests that the transition from planar structure to droplet pattern can occur when parameters change

chaos in one dimensional collision complexes the classical scattering of a particle by an oscillating potential well is investigated it is shown that the singularities of the scattering map s form a cantor set for this system this result is obtained from an analysis of the particle s return map which contains a smale s horseshoe and consequently a hyperbolic invariant set with chaotic dynamics the cantor set structure of the s singularities involves arbitrarily complex scattering behaviour

the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

finite time transport in aperiodic flows abstract we study the transport of particles in a general two dimensional incompressible flow in the presence of a transient eddy i e a bounded set of closed streamlines with a finite time of existence using quantities obtained from eulerian observations we provide explicit conditions for the existence of a hyperbolic structure in the flow which induces mixing between the eddy and its environment our results can be used directly to study finite time transport in numerically or experimentally generated vector fields with general time dependence

fourth order analogies to the painleve equations using the compatibility condition for the painleve equations several new fourth order ordinary differential equations odes that are analogies of the painleve equations are found the isomonodromic linear problems for these equations are given special solutions of the fourth order odes found are discussed the painleve test is applied to investigate several fourth order odes

chaotic motions near homoclinic manifolds and resonant tori in quasiperiodic perturbations of planar hamiltonian systems abstract we study chaotic dynamics of nonlinear oscillators with the form of a two frequency quasiperiodic perturbation of a planar hamiltonian system possessing a homoclinic orbit whose interior contains a one parameter family of periodic orbits in the extended phase space the unperturbed system has a three dimensional homoclinic manifold and a one parameter family of invariant 3 tori using melnikov s technique and the second order averaging method we show that chaotic motions may exist near the unperturbed homoclinic manifold and the unperturbed resonant tori these chaotic motions result from transverse intersection between the stable and unstable manifolds of normally hyperbolic invariant tori and are characterized by a generalization of the bernoulli shift we also give an example for the quasiperiodically forced duffing oscillator and demonstrate the existence chaotic motions by numerical simulation

scaling properties of a scattering system with an incomplete horseshoe we investigate a parameter dependent map describing a chaotic scattering system in parameter ranges leading to an incomplete horseshoe we construct an approximate symbolic dynamics which describes quite well the hyperbolic component of the invariant set its grammatical rules are correlated with the convergence properties of the thermodynamical formalism for the measures characterizing the invariant set

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

a formula to compute the microcanonical volume of reactive initial conditions in transition state theory we present the formal proof of a procedure to compute the phase space volume of initial conditions for trajectories that for a constant energy escape or react from a multi dimensional potential well with one or several exit entrance channels the procedure relies on a phase space formulation of transition state theory it gives the volume of reactive initial conditions as the sum over the exit entrance channels where each channel contributes by the product of the phase space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel an example is given to demonstrate the computational efficiency of the procedure

on the phase diagram for microphase separation of diblock copolymers an approach via a nonlocal cahn hilliard functional we consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal cahn hilliard functional based upon two parameters which characterize the phase diagram we give a preliminary analysis of the phase plane that is we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers in particular we identify a regime wherein the uniform disordered state is the unique global minimizer a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry and a regime of small volume fraction wherein we conjecture that small well separated approximately spherical objects are the unique global minimizer for this last regime we present an asymptotic analysis from the point of view of the energetics which will be comp

chaos in the 1 2 3 hamiltonian normal form the normal form of the hamiltonian 1 2 3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable associated with this complex unstable solution is an invariant manifold n on which the dynamics can be characterised completely one of the ingredients of n is a set of homoclinic orbits in the normal form to degree 4 the set of homoclinic orbits breaks up for certain parameter values into one homoclinic orbit this enables us to apply silnikov devaney theory to prove at this stage numerically the existence of a horseshoe map in the system with the implication of non integrability and chaos in the normal form

geometric phases reduction and lie poisson structure for the resonant three wave interaction hamiltonian lie poisson structures of the three wave equations associated with the lie algebras su 3 and su 2 1 are derived and shown to be compatible poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated these results can be applied to applications of nonlinear waves in for instance nonlinear optics some of the general structures presented in the latter part of this paper are implicit in the literature our purpose is to put the three wave interaction in the modern setting of geometric mechanics and to explore some new things such as explicit geometric phase formulas as well as some old things such as integrability in this context

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

on the organisation of transient chaos application to irregular scattering it is shown how the local structure of chaotic repellers being responsible for transient chaotic behaviour is deduced from the properties of hyperbolic periodic orbits relations between static and dynamical multifractal spectra with respect to the natural invariant measure on the repeller are derived for invertible maps of the plane the results obtained for maps with unit jacobian apply to hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

on the geometry of transport in phase space i transport in k degree of freedom hamiltonian systems 2 k abstract over the past several years the effect of geometrical structures such as kam tori resonance zones and cantori on phase space transport in two degree of freedom hamiltonian systems has been studied in three degree of freedom systems and higher these structures no longer form complete or partial barriers to transport in the sense that they or their associated stable and unstable manifolds are no longer codimension one in the level set of the hamiltonian in this paper we show that in the 2 k 1 dimensional level sets of the hamiltonian of a class of k degree of freedom hamiltonian systems the 2 k 2 dimensional stable and unstable manifolds of normally hyperbolic invariant 2 k 3 dimensional spheres form partial barriers to transport under certain conditions we show that lobes and turnstiles can be formed from such geometrical structures in a way that is exactly analogous to the situation for two degree of freedom hamiltonian systems considering a foliation of the level set of the hamiltonian into disjoint regions separated by such partial barriers we can then give exact formulas for the transport of volumes of phase space throughout the different regions based on the dynamics of these generalized lobes and turnstiles on the other hand despite the fact that these partial barriers are codimension one for k degree of freedom systems k 3 they may not intersect in such a way so as to divide the phase space

a proof of crystallization in two dimensions many materials have a crystalline phase at low temperatures the simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type where y x 2 is the position of particle x and v r is the pair interaction energy of two particles which are placed at distance r due to the mermin wagner theorem it can t be expected that at finite temperature this system exhibits long range ordering we focus on the zero temperature case and show rigorously that under suitable assumptions on the potential v which are compatible with the growth behavior of the lennard jones potential the ground state energy per particle converges to an explicit constant e where e is the minimum of a simple function on 0 furthermore if suitable dirichlet or periodic boundary conditions are used then the minimizers form a triangular lattice to the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the euclidean symmetry group consisting of rotations and translations

a degenerate bifurcation to chaotic scattering in a multicentre potential many scattering systems can be described as scattering of a point particle off a multicentre potential in this paper we present a two centre system which shows either regular or chaotic scattering depending on the kinetic energy i e the velocity of the incoming particle the transition points to chaotic scattering can be derived analytically by linearization of the poincare map at one of these transition velocities there is a degenerate bifurcation where the invariant set contains a parabolic surface and where the time delay statistics is algebraic with power sigma 2

cantor set structures in the singularities of classical potential scattering a classical mechanical system is analysed which exhibits complicated scattering behaviour in the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular though in the complement of this cantor set the deflection function is regular one can choose impact parameter intervals leading to arbitrarily complicated trajectories the authors show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow this invariant set contains infinitely many unstable localised orbits periodic and aperiodic ones the stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function

reduction of the semisimple 1 1 resonance the method of averaging is often used in hamiltonian systems of two degrees of freedom to find periodic orbits such periodic orbits can be reconstructed from the critical points of an associated reduced hamiltonian on a reduced space this paper details the construction of the reduced space and the reduced hamiltonian for the semisimple 1 1 resonance case the reduced space will be a 2 sphere in r3 and the reduced differential equations will be euler s equations restricted to this sphere the orbit projection from the energy surface in phase space to this sphere will be the hopf map the results of the paper are related to problems in physics on degeneracies due to symmetries of classical two dimensional harmonic oscillators and their quantum analogues for the hydrogen atom

heteroclinic dynamics in a model of faraday waves in a square container we study periodic orbits associated with heteroclinic bifurcations in a model of the faraday system for containers with square cross section and single frequency forcing these periodic orbits correspond to quasiperiodic surface waves in the physical system the heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable hamiltonian limit some of which persist in the presence of small damping the dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate poincare maps with predictions that agree with numerical computations the results suggest a great richness of possible dynamics of faraday waves even in simple geometries and with single frequency forcing

the decay of a normally hyperbolic invariant manifold to dust in a three degrees of freedom scattering system we study a prototypical three degrees of freedom chaotic scattering system depending on a perturbation parameter for one limiting case of the parameter the system has a conserved quantity and can be reduced to a two degrees of freedom system and for the other limiting case the chaotic invariant set is completely hyperbolic and forms a dust for the reducible case the dynamics of the poincare map is directed by a normally hyperbolic invariant manifold and this four dimensional map can be understood with the help of a cartesian product of a circle with a continuous stack of reduced two dimensional maps as the perturbation sets in for small values of the perturbation parameter the normally hyperbolic invariant manifold persists and only for larger values of the perturbation it turns into dust we give details of this decay and describe the corresponding changes of singularities in the scattering functions an important tool for the study of the development scenario of the normally hyperbolic invariant manifold is the restriction of the poincare map to this subset itself it behaves like a two dimensional perturbed twist map we explain how it can be approximated numerically by a combination of the full dimensional poincare map with a projection

asymptotic analysis of a class of three degree of freedom hamiltonian systems near stable equilibria abstract a conservative near integrable hamiltonian dynamical system is examined which to leading order consists of three uncoupled harmonic oscillators with constant frequencies in the ratio 1 2 for certain rational formally the problem considered can arise by perturbing any three degree of freedom hamiltonian near a stable equilibrium point so that the hamiltonian consists of a power series expansion in a small parameter where successive terms are homogeneous polynomials of increasing degree in the coordinates and the momenta the special case of two exact simultaneous resonances one in the first perturbation term and one in the second is examined and explicit asymptotic solutions are obtained the solution procedure involves reducing the original hamiltonian to two degrees of freedom using one integral of the motion then transforming to standard form to find two additional adiabatic invariants by near identity averaging canonical transformations a specific example is studied numerically to verify the asymptotic validity of the results over long times

periodic patterns for a model involving short range and long range interactions abstract we consider a physical model where the total energy is governed by surface tension and repulsive screened coulomb potential on the 3 dimensional space we obtain different periodic equilibrium patterns i e stationary sets for this energy under some volume constraints the patterns bifurcate smoothly from straight lamellae lattices of round solid cylinders and lattices of round balls

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

chaotic scattering off the magnetic dipole classical scattering with singularities of cantor set type can be observed if unstable localised orbits exist whose homoclinic structures are transported to infinity by the hamiltonian flow an electron moving in the field of a magnetic dipole is a simple example of physical relevance to demonstrate this transport mechanism in the asymptotic plane spanned by impact parameter and incoming direction the deflection function is singular for initial conditions leading to captured orbits using the method of poincare sections the authors find a correspondence between this set of singularities and the stable manifolds of localised orbits the scattering data which are measured in the asymptotic region of free motion provide information about chaotic motion in a finite part of the position space

lie symmetries and invariants for the time dependent generalizations of the equation r c1rnl c2rmr 0 in this comment the authors use an invertible point transformation for the study of time dependent equations derived from the equation r c1rn l c2rmr 0 this procedure generalizes some results obtained recently by leach and gorringe 1990

transcendents defined by nonlinear fourth order ordinary differential equations three ordinary differential equations are considered the general solutions of these equations are shown to be the essentially transcendental functions with respect to their initial conditions irreducibility of these equations is discussed

complete poincare sections and tangent sets trying to extend a local definition of a surface of a section and the corresponding poincare map to a global one one can encounter severe difficulties we show that global transverse sections often do not exist for hamiltonian systems with two degrees of freedom as a consequence we present a method to generate the so called w section which by construction will be intersected by almost all orbits depending on the type of tangent set in the surface of the section we distinguish five types of w sections the method is illustrated by a number of examples most notably the quartic potential and the double pendulum w sections can also be applied to higher dimensional hamiltonian systems and to dissipative systems

reconstruction of the chaotic set from classical cross section data we study a paradigm of a chaotic scattering system with one open and one closed degree of freedom we show how the development scenario of the chaotic invariant set is reflected in the changes of the rainbow singularities in the classical cross section this allows us to reconstruct the actual development stage of the chaotic set from the knowledge of the cross section we do this by following the rainbow structure of the cross section along a curve in parameter space which connects the case we are interested in with a known case for example the case of a complete horseshoe we conclude with a discussion about the generality of our procedure and of its application to various systems

multi bump orbits homoclinic to resonance bands we establish the existence of several classes of multi bump orbits homoclinic to resonance bands for completely integrable hamiltonian systems subject to small amplitude hamiltonian or dissipative perturbations each bump is a fast excursion away from the resonance band and the bumps are interspersed with slow segments near the resonance band the homoclinic orbits which include multi bump silnikov orbits connect equilibria and periodic orbits in the resonance band the main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands

trellises formed by stable and unstable manifolds in the plane a trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diseomorphism of a 2 manifold this paper describes and classifies some trellises the set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold each homoclinic point is assigned a type number which is constant on its orbit combinatorial properties of trellises are studied using type numbers and the pair of linear orderings trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones introduction in chapter 33 of new methods of celestial mechanics poincare describes the figure formed by the stable and unstable manifolds of a hyperbolic fixed point of a transformation of the upper half plane as follows when we try to represent the figure formed by these two curves and their intersections each of which corresponds to a doubly asymptotic solution these intersections form a type of trellis tissue or grid with infinitely serrated mesh neither of the two curves must ever 6ut across itself again but must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times the complexity of this figure will be striking and i shall not even try to draw it nothing is more suitable for providing us with an idea of the complex nature of the three body problem and of all the problems of dynamics in general the aim of this paper is to understand figures of the type poincare described so well if p is a hyperbolic periodic point of a diffeomorphism of a 2 manifold then the figure formed by its stable and unstable manifolds will be called the trellis of p whenever these manifolds have nonempty intersection figure 1 illustrates the beginning development of such a trellis similarly the figure formed by the stable and unstable manifolds of a cycle of hyperbolic periodic points is called the trellis of the cycle a cycle of hyperbolic periodic points of a diffeomorphism is a finite collection p of periodic points having a cyclic permutation p p such that the unstable manifold of p intersects the stable manifold of p for each p g p it is natural to conjecture that in some cases ergodic zones are the closures of trellises anosov transformations of the torus provide examples where this is the case 1 one may also conjecture that strange attractors are sometimes the closures of the unstable manifolds of cycles of hyperbolic periodic points for certain values of the parameters a and b the henon map of the plane defined by t x y 1 y ax by may provide examples received by the editors may 7 1985 1980 mathematics subject classification primary 58f13 secondary 58f05 58f11 1986 american mathematical society 0002 9947 86 1 00 25 per page

the last remake of the mixmaster universe model we review the existing evidence on the non integrability of the mixmaster universe model we show how a local painleve analysis can be used to study the possible existence of essential singularities in agreement with recent studies we find that the mixmaster model possesses critical essential singularities the multivalued character of which is incompatible with integrability our analysis is complemented by a numerical study of the spectrum of the stretching numbers of the system we show that the zero energy case is characterized by a vanishing maximal lyapunov characteristic number

multi pulse jumping orbits and homoclinic trees in a modal truncation of the damped forced nonlinear schro dinger equation abstract in this paper we prove the existence of multi pulse orbits homoclinic to a slow manifold in a two mode truncation of the damped forced nonlinear schrodinger equation first suggested by bishop et al these orbits are jumping i e the corresponding solutions keep switching in time between neighborhoods of the two charateristic breathers of the integrable limit in the case of no damping we find multi pulse smale horseshoes in the two mode model while in the dissipative case we establish the existence of structurally stable multi pulse heterolinic connections between two unstable equilibria the orbits we construct are not amenable to melnikov type perturbation methods in both the hamiltonian and the dissipative case we find homoclinic trees which describe the repeated bifurcations of multi pulse solutions to illustrate the theoretical predictions we also present visualizations of these complicated structures

amalgamations of the painleve equations new hierarchies of nonlinear ordinary differential equations that are generalizations of the painleve equations are presented these hierarchies are shown to contain the painleve equations as special cases emphasis is on the sixth order ordinary differential equations special solutions for one of them are expressed via the general solutions of the p1 p2 equations and special cases of the p3 and p5 equations special solutions of another sixth order ode are determined by the general solutions of the p2 p4 p3 and p5 equations five of six painleve equations can be considered as special cases of these sixth order odes isomonodromic linear problems to solve the cauchy problems of hierarchies by the inverse monodromy transform are given

method for deriving rational solutions of some nonlinear evolution equations maps for nonlinear evolution equations are discussed an algorithmic method for deriving rational solutions is presented this approach is illustrated by examples which have solutions in the form of two truncated expansions

classical chaotic scattering periodic orbits symmetries multifractal invariant sets the infinite set of periodic orbits of a chaotic system is investigated their globally exact symbolic dynamics is related to birkhoff s method of iterated symmetry lines which produces symmetric periodic orbits of various classes a complete picture emerges as to what part of the periodic orbits can be obtained by means of the birkhoff lines in addition the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits

an application of the modular function in nonlocal variational problems using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model we present a mathematical account of a hexagonal pattern selection observed in di block copolymer melts

using the extended melnikov method to study the multi pulse global bifurcations and chaos of a cantilever beam the aim of this paper is to investigate the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended melnikov method in the resonant case first the extended melnikov method for studying the shilnikov type multi pulse homoclinic orbits and chaos in high dimensional nonlinear systems is briefly introduced in the theoretical frame then this method is utilized to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the cantilever beam how to employ this method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications is demonstrated through this example finally the results of numerical simulation are given and also show that the shilnikov type multi pulse chaotic motions can occur for the nonlinear non planar oscillations of the cantilever beam which verifies the analytical prediction

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

third order differential equations with fixed critical points abstract the singular point analysis of third order ordinary differential equations which are algebraic in y and y is presented some new third order ordinary differential equations that pass the painleve test as well as the known ones are found

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

reactivity boundaries for chemical reactions associated with higher index and multiple saddles reactivity boundaries that divide the origin and destination of trajectories are of crucial importance to reveal the mechanism of reactions which was recently found to exist robustly even at high energies for index 1 saddles phys rev lett 105 048304 2010 here we revisit the concept of the reactivity boundary and propose a more general definition that can involve a single reaction associated with a bottleneck composed of higher index saddles and or several saddle points with different indices where the normal form theory based on expansion around a single stationary point does not work we numerically demonstrate the reactivity boundary by using a reduced model system of the h 5 cation where the proton exchange reaction takes place through a bottleneck composed of two index 2 saddle points and two index 1 saddle points the cross section of the reactivity boundary in the reactant region of the phase space reveals which initial conditions are effective in making the reaction happen and thus sheds light on the reaction mechanism

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

one generalization of the second painleve hierarchy new integrable hierarchy of ordinary differential equations is presented this hierarchy is shown to generalize the p2 hierarchy while on the other hand this hierarchy gives a special case of the third painleve equation the isomonodromic linear problem is used to find this new hierarchy the painleve approach to investigate the fourth order ordinary differential equation of this hierarchy is applied

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

homoclinic and heteroclinic orbits of reversible vectorfields under perturbation on demontre l existence de liens homocline heterocline entre les orbites periodiques hyperboliques d une classe de systemes hamiltoniens admettant un nombre suffisant de symetries sous une reduction isoenergetique le probleme se reduit a un probleme comprenant des champs vectoriels reversibles avec perturbations dependantes du temps admettant les memes symetries de renversement

topological aspects of chaotic scattering in higher dimensions we investigate the topological properties of the chaotic invariant set associated with the dynamics of scattering systems with three or more degrees of freedom we show that the separation of one degree of freedom from the rest in the asymptotic regime a common property in a large class of scattering models defines a gate which is a dynamical object with phase space separating invariant manifolds the manifolds form an invariant set causing singularities in the scattering process the codimension one property of the manifolds ensures that the fractal structure of the invariant set can be studied by scattering functions defined over simple one dimensional families of initial conditions as usually done in two degree of freedom scattering problems it is found that the fractal dimension of the invariant set is not due to the gates but to interior hyperbolic periodic orbits

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

invariant manifolds and the initialization problem for some atmospheric equations abstract an invariant manifold of quasi geostrophic states is shown to exist for a finite dimensional system of atmospheric equations there is another set of scalings for the parameters with respect to the rossby number which leads to the same limiting geostrophic equations but such that there need not be an invariant manifold also discussed are computational methods

nonlinear differential equations of the second third and fourth order with exact solutions abstract we present the class of nonlinear ordinary differential equations of the second third and fourth order in a polynomial form with exact solutions this class has exact solutions in the form of solitary waves kinks and one periodic solutions

transport rates of a class of two dimensional maps and flows abstract a method is developed for estimating the transport rates of phase space areas for a class of two dimensional diffeomorphisms and flows the class of diffeomorphisms we considered are defined by the topological structure of their stable and unstable manifolds and hence are universal we show how to estimate the transport rates for a class of diffeomorphisms found by easton and for an extension of this class of diffeomorphisms which is found via a perturbation of the topology of the stable and unstable manifolds this is done by introducing symbolic dynamics and transfer matrices which in turn relate transport phenomena in phase space to markov processes in a precise manner in addition to the transport rates we use the transfer matrices to obtain estimates for the topological entropy averaged stretching rates and the elongation rate of the unstable manifold the flows we consider are two dimensional time periodic flows which can be reduced via poincare section to the extended family of maps we develop an analytical method based on chirikov s whisker map to classify a given flow according to the structure of its manifolds in its poincare section this allows the techniques developed here for maps to be directly applied to time periodic flows

visualizing the perturbation of partial integrability in this paper we explain how the perturbation of partial integrability can be visualized by the restriction of the poincare map to an invariant two dimensional subset of its full domain the most appropriate surface is a normally hyperbolic invariant manifold built up by stack construction over the most prominent hyperbolic fixed point of the reduced poincare map for the partially integrable case as an example we present the numerical results for the motion of an electron in a perturbed magnetic dipole field

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

the chaotic saddle of a three degrees of freedom scattering system reconstructed from cross section data the topic of this paper is hyperbolic chaotic scattering in a three degrees of freedom system we generalize how shadows in the domain of the doubly differential cross section are found they are traced out by the appropriately filtered unstable manifolds of the periodic trajectories in the chaotic saddle these shadows are related to the rainbow singularities in the doubly differential cross section as a result of this relation we discover a method of how to recognize in the cross section a smoothly deformed image of the chaotic saddle allowing the reconstruction of the symbolic dynamics of the chaotic saddle its topology and its scaling factors

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

on the spectral analysis of trajectories in near integrable hamiltonian systems spectral properties associated with the deformation of tori and the transition to chaos in near integrable hamiltonian systems are studied information about the construction of tori is provided by studying the evolution of the integrals of the unperturbed system when a perturbation is added the authors show that the low band of the power spectrum converges exponentially for regular trajectories but they pass abruptly to 1 f alpha divergence when chaos occurs these results are valid for systems of two or more degrees of freedom and provide a clear distinction between regular and chaotic motion

a wiener filter state space flux optimal control against escape from a potential well escape from a potential well often identified with system failure is reformulated for a class of two dimensional dynamical systems as exiting the safe region of the system state space bounded by the separatrix derived from the potential well a control against system failure based on minimizing the state space flux out of this safe region is considered for systems subject to weak external forcing with available control power exceeding a certain threshold the finite lag state space flux optimal control against failure is shown to involve a wiener filter a non wiener solution is proposed for available power below this threshold these results hold for wide sense stationary forcing with any statistics and any spectrum

n pulse homoclinic orbits in perturbations of resonant hamiltonian systems in this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable two degree of freedom resonant hamiltonian systems our energy phase method applies to both hamiltonian and dissipative perturbations and reveals families of multi pulse solutions which are not amenable to melnikov type methods as an example we study a two mode approximation of the nonlinear nonplanar oscillations of a parametrically forced inextensional beam in this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically

standing pulse solutions to fitzhugh nagumo equations reaction diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences a localized pattern is a stable non constant stationary solution usually located far away from neighborhoods of bifurcation induced by turing s instability in the study of fitzhugh nagumo equations we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial this amounts to seeking a homoclinic orbit for a corresponding hamiltonian system and we utilize a variational formulation which involves a nonlocal term such a functional is referred to as helmholtz free energy in modeling microphase separation in diblock copolymers while its global minimizer does not exist in our setting of dealing with standing pulse the homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions in contrast with the known results for positive standing pulses found in the literature a new technique is attempted by seeking a standing pulse solution with a sign change

normal forms for three dimensional parametric instabilities in ideal hydrodynamics we derive and analyze several low dimensional hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics the equations depend on two parameters where is the strength of a system symmetry breaking perturbation and is a detuning parameter in many cases the resulting equations are completely integrable and have an interesting hamiltonian structure our work is motivated by three dimensional instabilities of rotating columnar fluid flows with circular streamlines such as the burger vortex subjected to precession elliptical distortion or off center displacement

parabolic resonances in 3 degree of freedom near integrable hamiltonian systems abstract perturbing an integrable 3 degree of freedom d o f hamiltonian system containing a normally parabolic 2 torus which is m resonant m 1 or 2 creates a parabolic m resonance m pr prs of different types are either persistent or of low co dimension hence they appear robustly in many applications energy momenta bifurcation diagram is constructed as a tool for studying the global structure of 3 d o f near integrable systems a link between the diagram shape pr and the resonance structure is found the differences between the dynamics appearing in 2 and 3 d o f systems exhibiting prs are studied analytically and numerically the numerical study demonstrates that prs are an unavoidable source of large and fast instabilities in typical 3 d o f systems

from the ginzburg landau model to vortex lattice problems we introduce a coulombian renormalized energy w which is a logarithmic type of interaction between points in the plane computed by a renormalization we prove various of its properties such as the existence of minimizers and show in particular using results from number theory that among lattice configurations the triangular lattice is the unique minimizer its minimization in general remains open

bifurcation to chaotic scattering abstract we investigate a novel type of bifurcation to chaos which occurs in the context of chaotic scattering in chaotic scattering the deflection angle versus impact parameter is singular on a set of impact parameters which is fractal this behavior is caused by the presence of a chaotic invariant set of unstable bounded orbits in the bifurcation considered here the chaotic set arises abruptly in a sense to be discussed as the particle energy e decreases from above a critical value em to below we call this transition an abrupt bifurcation to fully developed chaotic scattering numerical computation of the dimension d of the chaotic set shows that in agreement with theoretical prediction d 1 in em e 1 near the abrupt bifurcatio

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

dimension reduction techniques for the minimization of theta functions on lattices we consider the minimization of theta functions theta lambda alpha sum p in lambda e pi alpha p 2 amongst lattices lambda subset mathbb r d by reducing the dimension of the problem following as a motivation the case d 3 where minimizers are supposed to be either the bcc or the fcc lattices a first way to reduce dimension is by considering layered lattices and minimize either among competitors presenting different sequences of repetitions of the layers or among competitors presenting different shifts of the layers with respect to each other the second case presents the problem of minimizing theta functions also on translated lattices namely minimizing l u mapsto theta l u alpha another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers the first direction leads to the question of minimization amongst orthorhombic lattices whereas the second is relevant for asymptotics questions which we study in detail in two dimensions

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series

rigorous chaos verification in discrete dynamical systems abstract in this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems the existence of such points guarantees that a smale horseshoe is embedded in the mapping which therefore is chaotic we take special interest in the 2 dimensional case and describe a method to obtain a computer assisted proof of chaos for these systems numerical results are presented for the standard map