on the hopf index and the conley index the following generalization of the poincare hopf index theorem is proved if s is an isolated invariant set of a flow on a manifold m then the sum of the hopf indices on s is equal up to a sign to the euler characteristic of the homology conley index of 5 one of the classical results in dynamical systems relating local and global information is the poincare hopf index theorem if sf is a vectorfield on a compact manifold m with isolated zeros then the intersection number of with the zero section of tm is defined to be the sum of the hopf indices of the zeros i ist f yhlx s if is without boundary or if af is outward on dm the poincare hopf theorem asserts that the intersection number of sf is equal to the homological euler characteristic of m i see 4 7 this has the restriction that the local information the hopf indices is only related to the global information the betti numbers when the vectorfield is outward on the boundary pugh 8 removes this restriction in the case of generic contact of with m by relating the intersection number to a sum of euler characteristics dm j where r is the codimensional exit region and is its boundary in this paper we further develop this result by relating the hopf index and intersection numbers to the conley index for dynamical systems suppose that x is a smooth vectorfield on m with s c m an isolated invariant set of the induced flow then the intersection number of a on a neighborhood of s is up to a sign the euler characteristic of the homology conley index of s in m the conley euler number 5 in particular if all zeros on af on s are nondegenerate then the conley index and the hopf index are related by the formula nx m s 2ix there are two aspects to this result first it gives the hopf index and intersection numbers a homological interpretation in a broad setting second it shows that the properties of the hopf index such as additivity stability etc are also properties of conley euler numbers to develop the theorem we first received by the editors february 8 1988 1980 mathematics subject classification 1985 revision primary 34c35 55m20 secondary 54h20 58f25 work supported in part by oarpa applied and computational mathematics program 1989 american mathematical society 0002 9947 89 1 00 5 25 per page

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

hopf bifurcation for equivariant conservative and time reversible systems we study the bifurcation of small periodic solutions at a non semi simple 1 1 resonance in equivariant conservative or equivariant time reversible systems by using an equivariant liapunov schmidt method and restricting to solutions with an appropriate isotropy we reduce the problem to a scalar bifurcation equation the analysis of this equation shows a bifurcation behaviour similar to that found for the hamiltonian hopf bifurcation

isolating segments fixed point index and symbolic dynamics ii homoclinic solutions we present a topological method for detecting homoclinic solutions in nonautonomous ode s as an application we show that there exist infinitely many geometrically distinct homoclinic solutions to the trivial 0 solution in the planar systemz z 1 ei t z 2 for 0 0 495

a cohomological conley index for maps on metric spaces we define a cohomological conley index of an isolated invariant set of a time discrete semidynamical system on a metric space we prove the homotopy invariance and the additivity properties of the index for applications to time periodic parabolic partial differential equations pdes the reader is referred to the authors paper nontrivial full bounded solutions of time periodic semilinear parabolic pdes proc roy soc edinburgh to appear

reference motion in deformable bodies under rigid body motion and vibration part ii evaluation of the coefficient of restitution for impacts abstract the dynamic analysis of rigid body impacts is usually performed by using the coefficient of restitution as a measure of the mechanical energy lost in the process the coefficient of restitution provides an algebraic equation that allows impulse momentum balance equations to be solved this paper reports a method for calculating the kinematic coefficient of restitution for the impact of deformable bodies by using a numerical simulation procedure calculations were done within the framework of the floating reference frame approach as shown here for the first time discriminating between reference velocities and rigid body equivalent velocities is essential with a view to accurately calculating the kinematic coefficient of restitution thus the velocities at which the contact points approach and depart must be calculated as rigid body equivalent velocities if references velocities are used instead the resulting coefficient of restitution lacks physical significance the proposed method is illustrated with two applications namely an axial impact of a rigid body on a deformable rod and a transverse impact of a beam on a fixed stop

periodic solutions in a given set of differential systems the existence of a periodic solution in a given set of caratheodory differential systems is studied the proof of the main statement is based on the wazewski type approach jointly with the degree arguments an illustrating example is supplied and a comparison with the analogous results of other authors is indicated

random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

a critical study of the use of the generalized impulse momentum balance equations in flexible multibody systems abstract this paper analyses some aspects of the application of the generalized impulse momentum balance equations to flexible multibody systems these equations are the result of extending impulsive dynamics to the collision of flexible and constrained bodies application of the generalized impulse momentum balance provides a system of algebraic equations that is solved for the velocity jumps and the impulses associated with the contact force and the reactions involved as shown in this paper the physical significance of the coefficient of restitution which is included in the formulation to account for energy losses can be misinterpreted the floating frame of reference approach was used in combination with the finite element technique to describe flexibility and component mode synthesis to reduce the number of flexible co ordinates in the collision problem studied the number of elastic co ordinates used to describe flexibility and the time step used in the numerical integration essentially influence the dynamic simulation of the impact process the underlying message of this research is that the collision simulation entails solving many different balances however as shown in this paper this is not the result of the real phenomenon of succession of impacts these facts are illustrated with some numerical simulations of impacts of rigid bodies on flexible elements

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

isolating segments fixed point index and symbolic dynamics iii applications abstract we prove the existence of complicated dynamics in some class of periodic systems on the plane the proof is based on the continuation of the considered system to the model one

pullback attractors for the non autonomous fitzhugh nagumo system on unbounded domains abstract the existence of a pullback attractor is established for the singularly perturbed fitzhugh nagumo system defined on the entire space r n when external terms are unbounded in a phase space the pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far field values of solutions although the limiting system has no global attractor we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded and hence do not blow up as a small parameter approaches zero

wazewski type theorem for non autonomous systems of equations with a disconnected set of egress points in this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form y t f t y t where f t r n r n and t is a time scale for a given set t r n we formulate conditions for function f which guarantee that at least one solution y of the above system stays in unlike previous papers the set is considered in more general form i e the time section t is an arbitrary closed bounded set homeomorphic to the disk for every t t and the boundary t does not contain only egress points thanks to this we can investigate a substantially wider range of equations with various types of bounded solutions a relevant example is considered the results are new also for non autonomous systems of difference equations and the systems of impulsive differential equations

deployment of a multi link flexible structure abstract deployment of a multi link beam structure undergoing locking is analyzed in the timoshenko beam theory in the modeling of the system dynamic forces are assumed to be torques and restoring forces due to the torsion spring at each joint hamilton s principle is used to determine the equations of motion and the finite element method is adopted to analyze the system newmark time integration and newton raphson iteration methods are used to solve for the non linear equations of motion at each time step the locking at the joints of the multi link flexible structure is analyzed by the momentum balance method numerical results are compared with the previous experimental data the angles and angular velocities of each joint tip displacement and velocity of each link are investigated to study the motions of the links at each time step to analyze the effect of thickness on the motion of the link the angle and the tip displacement of each link are compared according to the various slenderness ratios additionally in order to investigate the effect of shear the tip displacements of a timoshenko beam are compared with those of an euler bernoulli beam

existence of global attractors for two dimensional newton boussinesq equation abstract in this paper we study global attractors of two dimensional newton boussinesq equation we obtain the existence of h 0 1 h 0 1 global attractors and h 2 h 2 global attractors in two dimensional bounded domains meanwhile we get the existence of h 0 1 h 0 1 global attractors in two dimensional channels

time reversible and equivariant pitchfork bifurcation abstract in this note we describe a new phenomenon in steady state bifurcations of time reversible and equivariant vector fields when the eigenspace associated with a pair of imaginary eigenvalues that passes through zero onto the real line is separable from the nullspace associated with the fixed zeros of the spectrum we prove that a pitchfork bifurcation occurs

remark on complicated dynamics of some planar system abstract in this paper we study the complicated dynamics generated by the planar periodic system z z 1 e i t z 2 for 0 0 495 we prove the existence of infinitely many geometrically distinct periodic solutions with some special properties

a geometric method for detecting chaotic dynamics a new method of detection of chaos in dynamical systems generated by time periodic nonautonomous differential equations is presented it is based on the existence of some sets called periodic isolating segments in the extended phase space satisfying some topological conditions by chaos we mean the existence of a compact invariant set such that the poincare map is semiconjugated to the shift on two symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the poincare map as an application we prove that the planar equationz 1 ei t z 2 zgenerates chaotic dynamics provided 0 1 288

a topological method for bounded solutions of nonautonomous ordinary differential equations the existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied this is done by associating the given equation to nonlinear autonomous ones by means of a family of skew product flows related by homotopy the existence of a bounded solution to the original differential equation is then related to the nontriviality of a certain conley index associated with the autonomous differential equations the existence of nontrivial bounded solutions is also considered the differential equations studied are perturbations of homogeneous ones

the conley index and the method of averaging in particular one useful theorem see e g hale s for details states roughly that if f satisfies certain uniformity hypotheses and the averaged equation has x 0 as a hyperbolic equilibrium point then the timedependent equation has a unique almost periodic solution in a neighborhood of zero for sufficiently small e 0 this solution is uniformly asymptotically stable if in addition x 0 is asymptotically stable for the averaged equation the hyperbolicity of x 0 is clearly crucial even for autonomous equations for example consider f x e x3 x2 we obtain fo x x3

attractors for reaction diffusion equations in unbounded domains abstract in this paper we study the asymptotic behaviour of solutions for parabolic non linear evolution equations in r n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 r n

isolating segments fixed point index and symbolic dynamics abstract an extension of the recently introduced srzednicki wojcik method for detecting chaotic dynamics in periodically forced ordinary differential equations is presented as an application of the method we construct a topological model for the planar equation z 1 e i t z 2 z z c 1 and we show by a continuation argument that the symbolic dynamics on three symbols for the topological model continues to eq 1 for 0

dynamics of a globally coupled oscillator array abstract we study a set of n globally coupled ordinary differential equations of the form encountered in circuit analysis of superconducting josephson junction arrays particular attention is paid to two kinds of simple time periodic behavior known as in phase and splay phase states some results valid for general n as well as further results for n 2 and n are presented a recurring theme is the appearance of very weak dynamics near the periodic states the implications for josephson junction arrays are discussed

existence of global attractors for the benjamin bona mahony equation in unbounded domains in this paper we investigate the asymptotic behavior of the benjamin bona mahony equation in unbounded domains we prove the existence of a global attractor when the equation is defined in a three dimensional channel the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

conley index of poincare maps in isolating segments abstract we calculate the discrete time conley index of the poincare map of a time periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment as an application we present results on the existence of bounded solutions of some planar equations

conditions for local reversing symmetries in dynamical systems abstract dynamical systems may possess symmetries and reversing symmetries local reversing symmetries around fixed points of dynamical systems are characterized as a property of their normal form expansions in this paper an inductive method is introduced to derive conditions for a fixed point to possess a local reversing symmetry of finite order the method is introduced for dynamical systems with continuous time flows the analogous problem for dynamical systems with discrete time mappings is discussed afterwards conditions for local reversing symmetries are given explicitly for fixed points of planar flows these conditions may be used as a negative criterion to show that a given dynamical system does not have a reversing symmetry

reversible equivariant linear systems in this paper we classify the structure of linear reversible systems vector fields on rn that are equivariant with respect to a linear representation of a compact lie group h we assume the time reversal symmetry r also acts linearly and is such that the group g that is generated by h and r is again a compact lie group the main tool for the classification is the representation theory of compact lie groups the results are applied to some generic eigenvalue movements of linear reversible equivariant systems

on periodic solutions of certain nth order differential equations we present two theorems on periodic solutions of nth order ordinary differential equations with the right hand side t periodic with respect to the time variable we apply a topological method based on the lefschetz fixed point theorem and the wazewski principle

dynamics and stability of a class of low reynolds number swimmers near a wall we study the dynamic stability of low reynolds number swimming near a plane wall from a control theoretic viewpoint we consider a special class of swimmers having a constant shape focus on steady motion parallel to the wall and derive conditions under which it is passively stable without sensing or feedback we study the geometric structure of the swimming equation and highlight the relation between stability and reversing symmetry of the dynamical system finally our numerical simulations reveal the existence of stable periodic motion the results have implications for design of miniature robotic swimmers as well as for explaining the attraction of micro organisms to surfaces

reversible relative periodic orbits abstract we study the bundle structure near reversible relative periodic orbits in reversible equivariant systems in particular we show that the vector field on the bundle forms a skew product system by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves we also discuss possibilities for drifts along group orbits our results extend those recently obtained in the equivariant context by b sandstede et al 1999 j nonlinear sci 9 439 478 and c wulff et al 2001 ergodic theory dynam systems 21 605 635

nontrivial full bounded solutions of time periodic semilinear parabolic pdes consider the semilinear evolution equation p u a u f t u where a is a sectorial operator on a banach space and f is periodic in t using a time discrete conley index developed in a previous paper 6 we prove a few existence results on bounded solutions of p defined for all t r more specific results are given for time periodic scalar parabolic equations

periodic points of the planar area preserving poincare map inside isolating segment abstract the aim of this note is to study the structure of the set of periodic points of the area preserving poincare map p associated to t periodic local planar process we will use the approach based on the method of periodic isolating segments isolating chains and the fixed point index theory the number of the applications to the processes generated by periodic in time non autonomous planar ode s are given

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

using solution sets for solving boundary value problems for ordinary differential equations where f e c i x r r s c c z r consists of finding fixed points of a solution map e for a suitable linearized problem associated to bv see e g 2 5 6 71 in these problems the map x is either continuous and single valued or upper semicontinuous set valued with convex values these are rather strong restrictions because the set of solutions of a general boundary value problem may well be neither a singleton nor a convex set the case in which the solution map is upper semicontinuous but with nonnecessarily convex values is considered in this paper more precisely we want to consider the case when the values of x form an acyclic set and the case when the values of c are sets with a finite number of elements an application to the existence of periodic solutions for a licnard type equation is also given moreover an existence theorem for a boundary value problem of the following form f l x x x teicr x e r

conley index for random dynamical systems conley index theory is a very powerful tool in the study of dynamical systems in this paper we generalize conley index theory to discrete random dynamical systems our constructions are basically the random version of franks and richeson in j franks d richeson shift equivalence and the conley index trans amer math soc 352 2000 3305 3322 for maps and the relations of isolated invariant sets between time continuous random dynamical systems and corresponding time h maps are discussed two examples are presented to illustrate results in this paper

on periodic solutions of planar polynomial differential equations with periodic coefficients abstract we consider the planar equation z ak l t zk z l where ak l is a t periodic complex valued continuous function equal to 0 for almost all k l n we present sufficient conditions imposed on ak l which guarantee the existence of its t periodic solutions and in the case a0 0 0 the conditions for the existence of nonzero ones we use a method which computes the fixed point index of the poincare andronov operator in isolated sets of fixed points generated by so called periodic blocks the method is based on the lefschetz fixed point theorem and the topological principle of wazewski

time reversibility and particle sedimentation this paper studies an ordinary differential equation ode model called the stokeslet model and describes sedimentation of small clusters of particles in a highly viscous fluid this model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n sided polygon when n 3 hocking j fluid mech 20 1964 pp 129 139 and caflisch et al phys fluids 31 1988 pp 3175 3179 prove the existence of periodic motion in the frame moving with the center of gravity in the cluster in which the particles form an isosceles triangle the study of periodic and quasiperiodic solutions of the stokeslet model is continued with emphasis on the spatial and time reversal symmetry of the model time reversibility is due to infinite viscosity and spatial dihedral symmetry is due to the assumption of identical particles and the symmetry of the trivial solution for three particles the existence of a second family of periodic solutions and a family of quasiperiodic solu

remark on krasnosielski s guiding functions and srzednicki s periodic segments abstract we show that the existence of the complete set of guiding functions guarantees the existence of periodic isolating segment that carries the same information concerning periodic solutions of the non autonomous periodic equations x f x t

hyperbolicity and averaging for the srzednicki wojcik equation for the srzednicki wojcik equation the planar nonautonomous ode parameterized by r z z 1 z 2exp i t z t c using averaging we show how the region of hyperbolicity grows with based on this we give bounds on the sizes of bounded orbits

averaging conley index continuation and recurrent dynamics in almost periodic parabolic equations abstract we study a non autonomous parabolic equation with almost periodic rapidly oscillating principal part and nonlinear interactions we associate to the equation a skew product semiflow and for a special class of nonlinearities we define the conley index of isolated compact invariant sets as the frequency of the oscillations tends to infinity we prove that every isolated compact invariant set of the averaged autonomous equation can be continued to an isolated compact invariant set of the skew product semiflow associated to the non autonomous equation finally we illustrate some examples in which the conley index can be explicitly computed and can be exploited to detect the existence of recurrent dynamics in the equation

asymptotic behavior of the newton boussinesq equation in a two dimensional channel abstract we prove the existence of a global attractor for the newton boussinesq equation defined in a two dimensional channel the asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions we also establish the regularity of the global attractor

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

averaging principle and hyperbolic evolution equations abstract an averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand side is a continuous perturbation of the time dependent family of linear operators determining a linear evolution system it generalizes classical henry s results for perturbations of sectorial operators on fractional spaces it is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by kato

periodic solutions of complex valued differential equations and systems with periodic coefficients has been considered by srzednicki 11 13 using a combination of generalized conley s isolating blocks the index of singular points of autonomous systems and the lefschetz fixed point theorem recently it has been shown that slight extensions of srzednicki s results could be obtained in a simpler and more elementary way using continuation theorems of the leray schauder type see 6 9 the aim of the present paper is to show that the same methodology can be adapted to prove the existence of periodic solutions for more general classes of equations and systems article no 0054

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

euler poisson equations and existence of quasi periodic motions of heavy tops abstract we study the existence of quasi periodic motions of an asymmetric heavy top by using two different approaches the standard moser weinstein theorem and an extended liapounov center theorem we take advantage of the multiple reversibility of the euler poisson equations and are able to show that the extended liapounov center theorem yields more normal modes than does the moser weinstein theorem

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

a landesman lazer type result for periodic parabolic problems on rn at resonance abstract we are concerned with t periodic solutions of nonautonomous parabolic problem of the form u t u v x u f t x u t 0 x r n with v l r n l p r n where 2 p if n 1 2 and n p for n 3 and t periodic continuous perturbation f r n r r the so called resonant case is considered i e when n ker v 0 and f is bounded we derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the brouwer topological degree of the time averaging of the mapping f n n f being the restriction of f to n by use of the formula and continuation techniques we show that landesman lazer type conditions imply the existence of t periodic solutions

on periodic solutions inside isolating chains abstract we develop a geometric approach to problems concerning the existence of t periodic solutions of a non autonomous time t periodic ordinary differential equation we consider isolating segments subsets of the extended phase space of the equation which in some ways resemble isolating blocks from the theory of isolated invariant sets the union of several contiguous isolating segments is called an isolating chain isolating segments determine some homomorphisms in reduced singular homologies the main theorem asserts that the lefschetz number of the composition of the homomorphisms determined by segments such that their union is a periodic isolating chain is equal to the fixed point index of the poincare map of the equation in the set of initial values of t periodic solutions contained inside the chain we give some applications of the theorem to planar polynomial equations in particular we prove that the equation z z 5 sin 2 t z has four nonzero periodic solutions provided 0 336

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

random attractors for the stochastic fitzhugh nagumo system on unbounded domains abstract the existence of a random attractor for the stochastic fitzhugh nagumo system defined on an unbounded domain is established the pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables these estimates are obtained by a cut off technique

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

non autonomous attractors for singularly perturbed parabolic equations on rn abstract we study the asymptotic behavior of solutions of a class of singularly perturbed non autonomous parabolic equations defined on r n with unbounded external terms we first prove the existence of a pullback attractor for the perturbed equation in l 2 r n and then establish the upper semicontinuity of these attractors as small perturbations approach zero the uniform estimates on the tails of solutions are used to overcome the difficulty caused by the non compactness of sobolev embeddings on unbounded domains

stabilizing effect of large average initial velocity in forced dissipative pdes invariant with respect to galilean transformations abstract we describe a topological method to study the dynamics of dissipative pdes on a torus with rapidly oscillating forcing terms we show that a dissipative pde which is invariant with respect to the galilean transformations with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms we apply the technique to the viscous burgers equation and the incompressible 2d navier stokes equations with a time dependent forcing we prove that for a large initial average speed the equation admits a bounded eternal solution which attracts all other solutions forward in time for the incompressible 3d navier stokes equations we establish the existence of a locally attracting solution