positive radial solutions and non radial bifurcation for semilinear elliptic equations in annular domains abstract we discuss the existence and multiplicity of positive radial solutions and the non radial bifurcation of u f u 0 in and u 0 on where is an annular domain of r n n 2 we prove that if f u 0 for u 0 and lim u f u u then there exists 0 such that there are at least two positive radial solutions for each 0 at least one for and none for if f 0 0 lim u 0 f u u 1 and uf u 1 e f u for u 0 e 0 then there exists a variational solution for 0 1 where 1 is the least eigenvalue of if f 0 0 lim u 0 f u u 0 and lim u f u u then there exists at least one positive radial solution for any 0 we obtain some precise multiplicity results for narrow annulus and show that the non radial bifurcation occurs if the growth of f u is rapid enough as u

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

applications of equivariant degree for gradient maps to symmetric newtonian systems abstract we consider g s 1 with being a finite group for which the complete euler ring structure in u g is described the multiplication tables for d 6 s 4 and a 5 are provided in the appendix the equivariant degree for g orthogonal maps is constructed using the primary equivariant degree with one free parameter we show that the g orthogonal degree extends the degree for g gradient maps in the case of g s 1 introduced by g ba in k g ba w krawcewicz j wu an equivariant degree with applications to symmetric bifurcation problems i construction of the degree bull london math soc 69 1994 377 398 the computational results obtained are applied to a symmetric autonomous newtonian system for which we study the existence of 2 periodic solutions for some concrete cases we present the symmetric classification of the solution set for the systems considered

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

bifurcation from higher eigenvalues in nonlinear elliptic equations continua that meet infinity on considere des problemes aux valeurs propres non lineaires pour des equations aux derivees partielles elliptiques i j 1 a ij x u xd i x j q x u f x u sur u 0 sur r n borne

multi parameter bifurcation and asymptotics for the singular lane emden fowler equation with a convection term to the second variable and g is unbounded around the origin the asymptotic behaviour of the solution around the bifurcation point is also established provided g u behaves like u around the origin for some 0 1 our approach relies on finding explicit sub and supersolutions combined with various techniques related to the maximum principle for elliptic equations the analysis we develop in this paper shows the key role played by the convection term u p

unbounded sets of solutions of non cooperative elliptic systems on spheres abstract the aim of this paper is to show that any continuum of nontrivial solutions of a non cooperative system of elliptic equations on the sphere s n 1 bifurcating from the set of trivial solutions is unbounded moreover we characterize bifurcation points of this system at which the global symmetry breaking phenomenon occurs as the main tool we use the degree for so 2 invariant strongly indefinite functionals defined in 13

counterexamples to some conjectures on the number of solutions of nonlinear equations on construit trois contre exemples resolvant par la negative certains problemes d analyse non lineaire

periodic solutions of autonomous newtonian systems partially asymptotically linear at infinity abstract the goal of this article is to study the existence continuation and global bifurcation of autonomous newtonian systems partially asymptotically linear at infinity the main idea is to prove a kind of splitting lemma and apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 16

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

global bifurcations of solutions of emden fowler type equation δu x λf u x on an annulus in rn n 3 abstract in this article we study global and symmetry breaking bifurcations of solutions of emden fowler type elliptic differential equations as the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals namely we will apply degree for s 1 equivariant orthogonal maps defined in rybicki 1994 nonlinear anal tma 23 83 102

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

symmetry breaking for a class of semilinear elliptic problems abstract this document considers a nonlinear elliptic problem suppose this problem has a family of positive radial solutions parameterized by r i e ur x this paper studies the possibility of the existence of nonradial solutions of bifurcating from the radial solutions family answering a question posed by smoller and wasserman it is shown this happens if f satisfies suitable assumptions therefore the global structure of the nonradial solution set is investigated

periodic solutions of symmetric autonomous newtonian systems abstract in this paper we show the existence and bifurcation of t periodic solutions of a special form for an autonomous newtonian system with symmetry if the phase space r 2 n is equipped with the structure of an orthogonal representation w w and the potential v w r is invariant then for every such a solution the set of indices of nonvanishing fourier coefficients is finite and depends on w only if the potential v depends on the squares of complex coordinates then for every such a solution t is the minimal period

on non radially symmetric bifurcation in the annulus we discuss the radially symmetric solutions and the non radially symmetric bifurcation of the semilinear elliptic equation u 2 eu 0 in and u 0 on where x r2 a2 0 a the upper branch of radial solutions has a non radially symmetric bifurcation symmetry breaking at each k a k 1 as a 0 the radial solutions will tend to the radial solutions on the disk and 0 a 1 the critical number on the disk

on algebraic problems behind the brouwer degree of equivariant maps abstract given a finite group g and two unitary g representations v and w possible restrictions on topological degrees of equivariant maps between representation spheres s v and s w are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in s v denoted by v effective applications of these congruences is limited by answers to the following questions i under which conditions is v 1 and ii does there exist an equivariant map with the degree easy to calculate in the present paper we address both questions we show that v 1 for each irreducible non trivial c g module if and only if g is solvable this provides a new solvability criterion for finite groups for non solvable groups we use 2 transitive actions to construct complex representations with non trivial characteristic regarding the second question we suggest a class of norton algebras without 2 nilpotents giving rise to equivariant quadratic maps which admit an explicit formula for the degree

global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n body problem abstract given n point masses turning in a plane at a constant speed this paper deals with the global bifurcation of periodic solutions for the masses in that plane and in space as a special case one has a complete study of n identical masses on a regular polygon and a central mass the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of the orthogonal degree theory all the symmetries of the bifurcating branches

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

computation of the equivariant 1 stem abstract let g be a compact lie group in this paper combining a short exact sequence obtained by balanov and krawcewicz with some additional topological techniques we complete the computation of the first equivariant stem 1 g st using the exact sequence and a property of nonabelian connected compact lie groups whose proof was suggested to us by r oliver we show that this group is finite if and only if g is finite

degenerate branching points of autonomous hamiltonian systems abstract this paper deals with non constant 2 periodic solutions of x t j h x t where 0 and h c 2 r n r n r with 2 h x 0 i 0 0 2 2 h x 0 for x 0 h 1 0 sufficient conditions for the existence of connected branches of such solutions bifurcating from x 0 0 have been formulated the corresponding theorem concerning connected branches of arbitrary periodic nonstationary trajectories of the hamiltonian system x t j h x t emanating from x 0 has been proved minimal periods of trajectories near x 0 have been described

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

existence and continuation of solutions for a nonlinear neumann problem abstract in this article we study the existence continuation and bifurcation from infinity of nonconstant solutions for a nonlinear neumann problem we apply the leray schauder degree and the degree for s o 2 equivariant gradient operators defined by the second author in s rybicki s o 2 degree for orthogonal maps and its applications to bifurcation theory nonlinear anal tma 23 1 1994 83 102

separation of global solution branches of elliptic systems with symmetry via nodal properties the idea that nodal properties are invariant along global solution branches of bifurcation problems goes back to the work of kolodner l and wolkowsky 2 31 in the context of nonlinear sturm liouville eigenvalue problems in that setting one can count the number of zeros of the solution a real valued function of a single real variable for the same class of problems it was shown later by crandall and rabinowitz 4 that this node count is inherited from the associated eigenfunction of the linearized problem about the trivial solution and that the invariance of nodal properties implies separation and unboundedness of global bifurcating solution branches it is natural to seek generalizations of these ideas to higher dimensions e g to problems involving more than one independent variable second order elliptic partial differential equations and or more than one dependent variable systems of second order differential equations these lead to immediate difficulties for elliptic partial differential equations we are confronted with the task of characterizing the set of zeros of functions of two or more variables in recent papers 5 6 we have shown how this problem simplifies considerably for equations with symmetries specifically in that setting we have demonstrated that the entire nodal set comprising onlyfixed hyper planes is precisely inherited from the eigenfunction of the linearization and is invariant along global branches of solutions possessing certain prescribed symmetries the proof requires groupinvariance arguments combined with subtle maximum principles in this work we take up the more challenging problem of systems of second order elliptic equations in the presence of symmetry to illustrate the difficulties involved we first discuss the case of a system of second order sturm liouville problems of the form

the g invariant implicit function theorem in infinite dimensions ii our basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries the result is sufficiently general to cover a great many applications it generalizes some earlier work of the author and corrects and improves some work of vanderbauwhede we also consider the breaking of symmetries problem and the variational case finally we apply our results to study the periodic solutions of an ordinary differential equation

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

technical communique asymptotic rejection of unknown sinusoidal disturbances in nonlinear systems this paper deals with global disturbance rejection of nonlinear systems the disturbance is assumed to be sinusoidal with completely unknown phases amplitude and frequencies but the number of distinct frequencies or the order of the corresponding unknown linear exosystem is known different from the common structural assumptions of nonlinear systems needed in literature for disturbance rejection of nonlinear systems the proposed method only requires the information of control design with a known lyapunov function when the system is disturbance free and a mild assumption needed for internal model design the proposed disturbance rejection algorithm extends complete global rejection of unknown sinusoidal disturbances for nonlinear dynamic systems beyond the common nonlinear models such as the strict feedback forms and the output feedback forms

multiple periodic solutions for γ symmetric newtonian systems abstract the existence of periodic solutions in symmetric newtonian systems x f x can be effectively studied by means of the o 2 equivariant gradient degree with values in the euler ring u o 2 in this paper we show that in the case of being a finite group the euler ring u o 2 and the related basic degrees are effectively computable using euler ring homomorphisms the burnside ring a o 2 and the reduced o 2 degree with no free parameters we present several examples of newtonian systems with various symmetries for which we show existence of multiple periodic solutions we also provide exact value of the equivariant topological invariant for those problems

periodic solutions of asymptotically linear autonomous newtonian systems with resonance abstract in this paper we study the existence continuation and bifurcation from infinity of 2 periodic solutions of autonomous newtonian systems we underline that the resonant case is considered to prove the results we apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 15 and the angle condition introduced by bartsch and li 1997 in 16

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

new proof and generalizations of the demidowitsch schneider criterion a new proof of the demidowitsch schneider criterion on the absence of closed trajectories of r3 is given the new proof is generalized and applied in several directions rn vector fields n 3

global bifurcations of solutions of elliptic differential equations abstract the aim of this article is to prove global bifurcation theorems for s 1 equivariant potential operators of the form compact perturbation of identity as an application we prove that components of the set of nontrivial solutions of system formula which bifurcate from the set of trivial solutions are unbounded in suitably chosen space the important point to note here is that our results apply only to systems with an even number of equations as the main tool we use degree theory for s 1 equivariant potential operators constructed by the author in 16 and developed in 17

existence uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations we study positive solutions of the dirichlet problem a u x f u x 0 x d u x o x e ad where d is an n ball we find necessary and sufficient conditions for solutions to be nondegenerate we also give some new existence and uniqueness theorems in this paper we study positive solutions of the dirichlet problem au x f u x o x f2

hopf bifurcations of functional differential equations with dihedral symmetries abstract we discuss the joint impact of temporal delay and spatial dihedral symmetries on the occurence and multiplicity of hopf bifurcations for a system of fdes by applying the equivariant degree theory we establish a result on the existence of multiple branches of nonconstant periodic solutions and classify their symmetries general results are illustrated by a ring of identical oscillators with identical coupling between adjacent cells

a larger family of nonlinear systems for the repetitive learning control a generalization of the pid 1 control has been recently presented in marino et al 2012 which guarantees the output tracking of smooth periodic reference signals with known period for uncertain minimum phase nonlinear systems in output feedback form with any relative degree 1 in this note we show that the repetitive learning control proposed in marino et al 2012 also solves the same output tracking problem for a larger class of uncertain nonlinear systems in normal form provided that the inverse system dynamics satisfy the demidovich condition on suitable compact sets

global bifurcations of critical orbits of g invariant strongly indefinite functionals abstract let h be a separable hilbert space which is an orthogonal representation of a compact lie group g and let h r be a g invariant strongly indefinite functional of the class c 1 to study critical orbits of we have defined the degree for g invariant strongly indefinite functionals which is an element of the euler ring u g using this degree we have formulated the rabinowitz alternative for g invariant strongly indefinite functionals the abstract results are applied to the study of global bifurcations of weak solutions of non cooperative systems of elliptic differential equations

existence and continuation of periodic solutions of autonomous newtonian systems abstract in this article we study the existence and the continuation of periodic solutions of autonomous newtonian systems to prove the results we apply the infinite dimensional version of the degree for so 2 equivariant gradient operators defined by the third author in nonlinear anal theory methods appl 23 1 1994 83 102 and developed in topol meth nonlinear anal 9 2 1997 383 417 using the results due to rabier symmetries topological degree and a theorem of z q wang j math 24 3 1994 1087 1115 and wang symmetries and calculation of the degree chinese ann math 10 1989 520 536 we show that the leray schauder degree is not applicable in the proofs of our theorems because it vanishes

periodic solutions of asymptotically linear autonomous hamiltonian systems abstract in this paper we study the existence and the continuation of periodic solutions of autonomous hamiltonian systems we apply the degree for g invariant strongly indefinite functionals defined in gol biewska and rybicki 2011 2 we consider resonant as well as nonresonant case at stationary solutions

existence of periodic solutions for semilinear reaction diffusion systems abstract in this paper we consider the existence of periodic solutions of reaction diffusion systems by using s 1 degree theory due to dylawerski et al see jodel et al ann pol math 41 1991 243

equivariant bifurcation index abstract we consider a bifurcation index bif g k 0 1 u g defined in terms of the degree for g equivariant gradient maps see g ba 1997 21 rybicki 1994 22 rybicki 2005 23 where g is a real compact connected lie group and u g is the euler ring of g see tom dieck 1977 29 tom dieck 1987 30 the main result of this article is the following bif g k 0 1 u g iff bif t k 0 1 u t where t g is a maximal torus of g it is also shown that all the bifurcation points of weak solutions of the following problem u f u in b n u 0 on s n 1 are global bifurcation points additionally the global symmetry breaking bifurcation points are characterised

global continua of periodic solutions of singular first order hamiltonian systems of n vortex type the paper deals with singular first order hamiltonian systems of the form begin aligned gamma k dot z k t j nabla z k h z t quad z k t in omega subset mathbb r 2 k 1 dots n end aligned where j in mathbb r 2 times 2 defines the standard symplectic structure in mathbb r 2 gamma 1 dots gamma n in mathbb r setminus 0 are given and the hamiltonian h is of n vortex type begin aligned h z 1 dots z n frac 1 2 pi sum genfrac 0 0pt j k 1 j ne k n gamma j gamma k log left z j z k right f z end aligned this is defined on the configuration space z 1 ldots z n in omega n z j ne z k text for j ne k of n different points in the domain omega subset mathbb r 2 the function f omega n rightarrow mathbb r may have additional singularities near the boundary of omega n we prove the existence of a global continuum of periodic solutions z t z 1 t dots z n t in omega n that emanates after introducing a suitable singular limit scaling from a relative equilibrium z t in mathbb r 2n of the n vortex problem in the whole plane where f 0 examples for z include thomson s vortex configurations or equilateral triangle solutions the domain omega need not be simply connected a special feature is that the associated action integral is not defined on an open subset of the space of 2 pi periodic h 1 2 functions the natural form domain for first order hamiltonian systems this is a consequence of the singular character of the hamiltonian our main tool in the proof is a degree for s 1 equivariant gradient maps that we adapt to this class of potential operators

bifurcation and symmetry breaking of solutions of systems of elliptic differential equations in this article we study global bifurcations of weak solutions of the following variational system of elliptic differential equations u au u u in u 0on we prove sufficient conditions for the existence of unbounded continua of nontrivial solutions emanating from the trivial ones and necessary conditions for the existence of bounded continua moreover we describe a symmetry breaking phenomenon that occurs on that continua

the index change and global bifurcation for flows with a first integral the existence continuation and bifurcation of solutions of fixed period r of one parameter families of ordinary differential equations with a first integral may be studied using the topological degrees defined in 6 and 7 and the change of index formula given in 8 the purpose here is threefold first suppose that f x v x 0 so that v is a first integral for x f x we examine how the behaviour of f restricts that of v and vice versa and how the periodic spectra of linear ordinary differential equations are affected by the presence of a linear or non linear which occurs surprisingly naturally first integral second we give classes of one parameter families of ordinary differential equations where the change of index in 8 is non zero

symmetric liapunov center theorem for minimal orbit abstract using the techniques of equivariant bifurcation theory we prove the existence of non stationary periodic solutions of symmetric systems q t u q t in any neighborhood of an isolated orbit of minima q 0 of the potential u we show the strength of our result by proving the existence of new families of periodic orbits in the lennard jones two and three body problems and in the schwarzschild three body problem

robust nonlinear integral control it is well known from linear systems theory that an integral control law is needed for asymptotic set point regulation under parameter perturbations the note presents a similar result fur a class of nonlinear systems in the presence of an unknown equilibrium due to uncertain nonlinearities and dynamic uncertainties both partial state and output feedback cases are considered sufficient small gain type conditions are identified for existence of linear and nonlinear control laws a procedure for robust nonlinear integral controller design is presented and illustrated via a practical example of fan speed control

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

degenerate bifurcation points of periodic solutions of autonomous hamiltonian systems abstract we study connected branches of nonconstant 2 periodic solutions of the hamilton equation x t j h x t where 0 h c 2 r n r n r and 2 h x 0 a 0 0 b for x0 h 1 0 the hessian 2h x0 can be singular we formulate sufficient conditions for the existence of such branches bifurcating from given x0 0 as a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the hamiltonian system x t j h x t emanating from x0 we describe also minimal periods of trajectories near x0

on silnikov s homoclinic saddle focus theorem abstract consideration is given to the chaotic dynamics near an orbit homoclinic to a saddle focus fixed point of silnikov type a new type of symbolic system is used to describe the dynamics on the unstable manifold and its relation to the silnikov dynamics techniques based on the silnikov method are developed for this unifying treatment these new ideas and techniques should be applicable for a wide variety of homoclinically and heteroclinically related problems

global adaptive output regulation for a class of nonlinear systems with iiss inverse dynamics using output feedback abstract this paper generalizes our recent results on global robust output regulation of output feedback systems with integral input to state stable iiss inverse dynamics in two aspects first we consider the general relative degree case instead of the unity relative degree case for this purpose the one step approach proposed previously has to be generalized to a recursive approach second we allow the exosystem to be uncertain for this purpose we need to introduce an adaptive control technique to estimate the unknown parameter vector in the exosystem a convergence analysis of the estimated parameter vector will also be given

existence of limit cycles for coupled van der pol equations abstract in this paper we consider the existence of limit cycles of coupled van der pol equations by using s1 degree theory due to dylawerski et al see ann polon math 62 1991 243

on the positive solutions of semilinear equations δ ℎ 0 on the compact manifolds in this paper we study the existence nonexistence and uniqueness of positive solutions of semilinear equations u u hu p 0 on compact riemannian manifolds as well as on bounded smooth domains in r n with homogeneous dirichlet or neumann boundary conditions

a unifying framework for global regulation via nonlinear output feedback from iss to iiss this paper presents a unifying framework for the problem of robust global regulation via output feedback for nonlinear systems with integral input to state stable inverse dynamics subject to possibly unknown control direction the contribution of the paper is two fold firstly we consider the problem of global regulation instead of global asymptotic stabilization gas for systems with generalized dynamic uncertainties it is shown by an elementary example that gas is not solvable using conventional smooth output feedback secondly we reduce the stability requirements for the disturbance and demand relaxed assumptions for the system using our framework most of the known classes of output feedback form systems are broadened in several directions unmeasured states and unknown parameters can appear nonlinearly restrictive matching and growth assumptions are removed the dynamic uncertainty satisfies the weaker condition of sontag s integral input to state stability and the sign of high frequency gain may be unknown a constructive strategy is proposed to design a dynamic output feedback control law that drives the state to the origin while keeping all other closed loop signals bounded

output regulation for linear systems via adaptive internal model given a stabilizable and detectable linear system with additive disturbances and output references generated by a linear stable exosystem with unknown parameters and known order the problem of designing a global output feedback regulator which asymptotically achieves output regulation and disturbance rejection is considered the system is assumed to be known while the frequencies of the exosystem are unknown all exosystem oscillatory modes are assumed to be excited by the initial condition a global solution is proposed consisting of a dynamic output feedback controller which includes exponentially convergent estimates of the unknown frequencies

bifurcation and symmetry breaking let e and f be banach spaces e f and an interval in r in this paper we use bifurcation theory to study the set of solutions of an equation m u 0 where m is a smooth operator m e f