quasilinear asymptotically periodic schrodinger equations with subcritical growth abstract in this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established the basic tools employed here are the mountain pass theorem and the concentration compactness principle by using a change of variable the quasilinear equation is reduced to a semilinear equation whose respective associated functional is well defined in h 1 r n and satisfies the geometric hypotheses of the mountain pass theorem

numerical semi global analysis of a 1 2 resonant hopf hopf bifurcation abstract in this paper a numerical semi global analysis of the dynamics near a 1 2 resonant hopf hopf bifurcation on a four dimensional mathematical model of a simple nonlinear oscillator is performed the 1 2 resonant hopf hopf bifurcation is a codimension three singularity denoted by two pairs of purely imaginary eigenvalues with frequency ratio 1 2 a structure involving 1 1 and 1 2 resonant neimark sacker bifurcations is clearly identified both resonances are coupled by lower codimension singularities such as generalized hopf and period doubling cusp points and cyclic fold curves a three parameter semi global analysis is performed and some of the codimension two singularities unfolded by the 1 2 resonant hopf hopf bifurcation are identified several codimension three points are also detected the obtained results can be useful for further theoretical analysis of the corresponding normal form

on a fractional schrodinger equation with periodic potential abstract in this paper we first consider the spectrum of the fractional schrodinger operator s v x on r n where s 0 1 and v x be continuous periodic in x using a new nonlocal normal derivative we prove that the operator has purely continuous spectrum which is bounded below and consists of closed disjoint intervals then when 0 belongs to a spectral gap of s v x we establish an existence result for the fractional schrodinger equation via a new linking theorem it is worth to mention that the nonlinear term in our problem does not satisfy the periodicity and the existence result is even new for the case s 1

double hopf bifurcation for van der pol duffing oscillator with parametric delay feedback control the stability and bifurcation of a van der pol duffing oscillator with the delay feedback are investigated in which the strength of feedback control is a nonlinear function of delay a geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and hopf bifurcations the hopf bifurcation curves and multi stable regions are obtained as two parameters vary some weak resonant and non resonant double hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the death island regions simultaneously by applying the center manifold theory the normal forms near the double hopf bifurcation points as well as classifications of local dynamics are analyzed furthermore some quasi periodic and chaotic motions are verified in both theoretical and numerical ways

restrictions and unfolding of double hopf bifurcation in functional differential equations abstract the normal form of a vector field generated by scalar delay differential equations at nonresonant double hopf bifurcation points is investigated using the methods developed by faria and magalhaes j differential equations 122 1995 181 we show that 1 there exists linearly independent unfolding parameters of classes of delay differential equations for a double hopf point which generically map to linearly independent unfolding parameters of the normal form equations ordinary differential equations 2 there are generically no restrictions on the possible flows near a double hopf point for both general and z 2 symmetric first order scalar equations with two delays in the nonlinearity and 3 there always are restrictions on the possible flows near a double hopf point for first order scalar delay differential equations with one delay in the nonlinearity and in nth order scalar delay differential equations n 2 with one delay feedback

semiclassical ground states for quasilinear schrodinger equations with three times growth abstract in this paper we study the quasilinear schrodinger equation 2 u v x u 2 u u 2 q x u 3 u h 1 r 3 where 0 is a parameter v and q are positive bounded functions for the equation with three times growth we establish the existence of ground states for small using the method of nehari manifold we also describe the concentration phenomena of ground states as 0

a positive ground state solution for a class of asymptotically periodic schrodinger equations with critical exponent abstract in the paper by using a new condition a class of asymptotically periodic schrodinger equations with critical exponent are studied via the variational method a positive ground state solution is obtained

existence and concentration of ground state solutions for critical schrodinger equation with steep potential well abstract in this paper we study the following schrodinger equation u 1 a x u f u u 2 2 u in r n where n 3 and 0 we obtain a ground state solution for the above equation moreover the concentration behavior of the ground state solution is also described as

on perturbations of a class of a periodic m laplacian equation with critical growth 1 after the well known results of brezis nirenberg 10 problems involving ellipticequationswithcriticalgrowthinboundeddomainshavebeenstudiedbyseveralauthors see for example 7 15 11 13 17 and the references therein the special features of this class of problems considered in this paper is that it isdened in r

quasilinear elliptic equations with critical growth via perturbation method abstract we consider a class of quasilinear schrodinger equations which include the modified nonlinear schrodinger equations a new perturbation approach is used to treat the critical exponent case giving new existence results

infinitely many solutions of a symmetric dirichlet problem here q is a bounded domain in r with smooth boundary and f i r is c and satisfies certain growth conditions if m 1 and f is an even function hence f r r is an odd function then ambrosetti and rabinowitz l 21 proved the existence of infinitely many weak solutions of d using a symmetric version of the mountain pass theorem a similar result is due to michalek 3 who considered the case where m 21 rm s c and f is invariant under a free action of the cyclic group z p on c here z p is considered as a subgroup of s c c acting essentially via scalar multiplication on c in this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of d let g be a compact lie group and p g o m an orthogonal representation of g on lrm then g acts on irm via the map g x ir ir g u y gv p g v we write v for the space rm considered as a g space do there exist infinitely many solutions of d if f is invariant under this symmetry i e f gu f u for all g e g and u e v without further assumptions on p the answer to this question is no in general for example if p g id for all g e g then there is no symmetry at all and one cannot expect infinitely many solutions more generally one should exclude the existence of fixed points vg u e vi gu u for all g e g 0 our first result says that there exist infinitely many solutions of d if the following condition is satisfied

nonresonant hopf hopf bifurcation and a chaotic attractor in neutral functional differential equations abstract an algorithm for calculating the third order normal form of a nonresonant hopf hopf singularity in a neutral functional differential equation nfde is established the van der pol equation with extended delay feedback is investigated as an nfde of second order the existence of hopf hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm periodic solutions and quasi periodic solutions are found with the aid of the bifurcation diagram and corresponding numerical illustrations are presented with the breaking down of the 3 torus a chaotic attractor appears in this nfde of second order following the ruelle takens newhouse scenario which usually arises for an ordinary differential equation of order at least 4 this transition is shown via both theoretical and numerical approaches

quasilinear asymptotically periodic elliptic equations with critical growth abstract the existence of a nontrivial solution for a class of asymptotically periodic equations involving the p laplacian operator in the euclidean space is established the solutions are obtained by an application of the mountain pass theorem and the concentration compactness principle the lack of compactness due to the fact that the domain is unbounded is compensated by comparing the minimax level associated with the original problem and the minimax level associated with the periodic limit problem

hopf pitchfork bifurcation in van der pol s oscillator with nonlinear delayed feedback abstract first we identify the critical values for hopf pitchfork bifurcation second we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system furthermore we find some interesting phenomena such as the coexistence of two asymptotically stable states two stable periodic orbits and two attractive quasi periodic motions which are verified both theoretically and numerically

nonlinear perturbations of a periodic elliptic problem with critical growth abstract using the well known mountain pass theorem due to ambrosetti and rabinowitz we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in r n with critical growth

non resonant double hopf bifurcation of a class b laser system we investigate the double hopf bifurcation at positive equilibrium point of a class b laser system firstly we give the critical values of hopf and double hopf bifurcations secondly we implement the normal form method and the center manifold theory for a class b laser system and derive the universal unfolding of the double hopf bifurcation thirdly some dynamic behaviors such as periodic and quasi periodic solutions are observed using numerical simulation

dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks abstract we study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single hopf bifurcation oscillator our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips frequency suppression multiple periodic states and chaos such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator we construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms we find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity phase reversals radial trapping phase jumps and spiraling patterns in the amplitude space our results may find useful applications in physical chemical or biological systems

concentration compactness principle for nonlocal scalar field equations with critical growth abstract the aim of this paper is to study a concentration compactness principle for homogeneous fractional sobolev space d s 2 r n for 0 s min 1 n 2 as an application we establish palais smale compactness for the lagrangian associated to the fractional scalar field equation s u f x u for 0 s 1 moreover using an analytic framework based on d s 2 r n we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range

schrodinger equations with asymptotically periodic terms we study the existence of non trivial solutions for a class of asymptotically periodic semilinear schrodinger equations in n by combining variational methods and the concentration compactness principle we obtain a non trivial solution for the asymptotically periodic case and a ground state solution for the periodic one in the proofs we apply the mountain pass theorem and its local version

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

normal forms of non resonance and weak resonance double hopf bifurcation in the retarded functional differential equations and applications the norm forms of the double hopf bifurcation for a general dde is established the van der pol duffing oscillator with delayed position and velocity feedback is investigated the dynamical classification near the double hopf bifurcation point is studied the stable periodic doubling solutions and unstable three dimensional torus are found in this paper we firstly present the general framework of calculation of normal forms of non resonance and weak resonance double hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to faria and magalh a es then the dynamical behavior of van der pol duffing oscillator with delayed position and velocity feedback is considered specifically the dynamical classification near the double hopf bifurcation point is investigated by analyzing the obtained normal form finally the numerical simulations support the theoretical results and present some interesting phenomena

nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents in this paper we will discuss the existence of a nontrivial solution of the following class of semilinear biharmonic problem involving critical exponents 6u a x u h x u q 1u k x u p 1u in r u h r n 5 p where 1iqip 2 1 n 4 n 4 and a h k rn r are bounded nonnegative and continuous functions we recall that semilinear and quasilinear problems for second order equations have been extensively studied in the last years for laplace operator and subcritical case they have been studied for instance by rabinowitz 30 coti zelati and rabinowitz 14 kryszewski and szulkin 21 alama and li 1 when a h and k are 1 periodic while in the paper 25 montecchiari has dropped the periodicity condition on a h and

asymptotically periodic superquadratic hamiltonian systems abstract we obtain existence of solutions for the system u v x u f v x u v x r n v v x v f u x u v x r n where n 3 v c r n 0 is periodic and f c 1 r n r 2 r is superquadratic at infinity we consider the case that f is periodic and asymptotically periodic in the proofs we apply variational techniques

a nonautonomous predator prey system with stage structure and double time delays in the present paper we study a nonautonomous predator prey model with stage structure and double time delays due to maturation time for both prey and predator we assume that the immature and mature individuals of each species are divided by a fixed age and the mature predator only attacks the immature prey based on some comparison arguments we discuss the permanence of the species by virtue of the continuation theorem of coincidence degree theory we prove the existence of positive periodic solution by means of constructing an appropriate lyapunov functional we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution two examples are given to illustrate the feasibility of our main results

normal forms for retarded functional differential equations and applications to bogdanov takens singularity abstract the paper addresses for retarded functional differential equations fdes the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity a phase space appropriate to the computation of these normal forms is introduced and adequate nonresonance conditions for the computation of the normal forms are derived as an application the general situation of bogdanov takens singularity and its versal unfolding for scalar retarded fdes with nondegeneracy at second order is considered both in the general case and in the case of differential delay equations of the form t x t x t 1

stability with respect to the delay on a paper of k l cooke and p van den driessche a proposal for the correction of the main result of the title s paper is made some accompanying comments on other points are given so a simpler central result is derived an explicit representation of the set of stable delays is achieved

double hopf bifurcation and quasi periodic attractors in delay coupled limit cycle oscillators abstract we investigate the double hopf bifurcation at zero equilibrium point firstly we give the critical values of hopf and double hopf bifurcations secondly we implement the normal form method and the center manifold theory for delay coupled limit cycle oscillators and derive the universal unfolding and a complete bifurcation diagram of the system thirdly many interesting phenomena such as attractive periodic motion and three dimensional invariant torus are observed using numerical simulation finally the normal forms of several strong resonant cases are listed

ground states for the nonlinear kirchhoff type problems abstract in this paper we study the existence of ground states for the asymptotically periodic kirchhoff type problems with critical growth and three times growth the proof is based on the method of nehari manifold and concentration compactness principle in particular we improve the method of nehari manifold for kirchhoff type equations with three times growth

on some fourth order semilinear elliptic problems in r n the existence of two solutions for a fourth order semilinear elliptic problem involving critical growth from the viewpoint of sobolev embedding was established the basic tools used in the analysis were the mountain pass theorem constrained minimization and concentration compactness principle the existence of positive principle eigenvalues for the corresponding linear elliptic problem was also investigated

existence of solutions for asymptotically periodic fractional schrodinger equation abstract in this paper we study the following nonlinear fractional schrodinger equation s u v x u f x u x r n where s 0 1 s is the fractional laplacian operator of order s and v f are asymptotically periodic functions in x under suitable conditions the existence of nontrivial solution is obtained by using variational methods and concentration compactness argument

a positive ground state solution for a class of asymptotically periodic schrodinger equations in this paper by using the reformative conditions a class of asymptotically periodic schrodinger equations are studied via the variational method a positive ground state solution is obtained

on superlinear problems without the ambrosetti and rabinowitz condition abstract existence and multiplicity results are obtained for superlinear p laplacian equations without the ambrosetti and rabinowitz condition to overcome the difficulty that the palais smale sequences of the euler lagrange functional may be unbounded we consider the cerami sequences our results extend the recent results of miyagaki and souto o miyagaki m souto superlinear problems without ambrosetti and rabinowitz growth condition j differential equations 245 2008 3628 3638

on the analysis of the double hopf bifurcation in machining processes via centre manifold reduction the single degree of freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double hopf bifurcation point centre manifold reduction and normal form calculations are performed to investigate the long term dynamics of the cutting process the normal form of the four dimensional centre subsystem is derived analytically and the possible topologies in the infinite dimensional phase space of the system are revealed it is shown that bistable parameter regions exist where unstable periodic and in certain cases unstable quasi periodic motions coexist with the equilibrium taking into account the non smoothness caused by loss of contact between the tool and the workpiece the boundary of the bistable region is also derived analytically the results are verified by numerical continuation the possibility of transient chaotic motions in the global non smooth dynamics is shown

existence of solutions for quasilinear elliptic equations with hardy potential in this paper we consider the following quasilinear elliptic equation with hardy potential and dirichlet boundary condition i j 1ndj aij x u diu 12 i j 1ndsai j x u diudju x 2u f x u in where n n 3 is a smooth bounded domain di xi dsaij x s saij x s and 0 n 22 2 and x 2 is called the hardy potential by using the perturbation method we prove the existence of infinitely many solutions for the above problem

on fractional schrodinger equation with periodic and asymptotically periodic conditions abstract in this paper we study the following fractional schrodinger equation s u v x u f x u x r n where s s 0 1 denotes the fractional laplacian under some weaker conditions on the nonlinearity we obtain the existence of solutions for the above problem in periodic case and asymptotically periodic case via variational methods respectively

double hopf bifurcation of coupled dissipative stuart landau oscillators with delay in this paper the double hopf bifurcation of delay coupled dissipative stuart landau oscillators models are considered where the time delay is regarded as one of the parameters and the other is the natural frequency it is found that as the values of the two parameters vary a number of double hopf bifurcation points arise consequently it has been found that the system exhibits very rich complex dynamics some resonant and non resonant double hopf bifurcation phenomena can be observed some numerical simulations support our analysis result

on ground state solutions for quasilinear elliptic equations with a general nonlinearity in the critical growth abstract in this paper we consider the following problem p u v u p 2 u f u u w 1 p r n where 1 p n and f has critical growth the purpose of this paper is to study the existence of the ground state solution for a class of quasilinear elliptic equations some recent results are extended

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

on a class of semilinear schrodinger equations with indefinite linear part in this paper we are concerned with the existence of ground state solutions for schrodinger equation u v x u k x u 2 2u f x u u h1 rn where n 4 v is periodic in x and 0 lies in a spectral gap of v k and f are asymptotically periodic in x

on einstein four manifolds with s 1 actions we study closed einstein 4 manifolds which admit s1 actions of a certain type i e warped products in particular we classify them up to isometry when the fixed point of the s1 action satisfies certain natural geometric conditions the proof uses the bochner weitzenbock formula for 1 forms and the theory of minimal surfaces in 3 manifolds