stochastic bifurcation in duffing van der pol oscillators global analysis of bifurcation in the duffing van der pol oscillators under both additive and multiplicative random excitations is explored in detail by the generalized cell mapping method using digraph system parameters are chosen in the range that two co exist attractors and a saddles as an alternative definition stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value it is found that under certain conditions stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation by this global analysis the mechanism of development occurrence and evolution of stochastic bifurcation can be explored clearly and vividly

products and compositions with the dirac delta function the need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation junction conditions and curved shock waves earlier definitions are briefly reviewed and new definitions are proposed using non standard analysis basic properties are established and some products and compositions with the delta distribution are explicitly evaluated with these definitions the domain of validity of the nonlinear differential equations of classical field theory can be extended to include rankine hugoniot equations are derived from the euler equations an immediate application to quantum field theory is pointed out

the serre spectral sequence of a multiplicative fibration in a fibration omegaf omegaj under right arrow omegax omega pi under right arrow omegab we show that finiteness conditions on f force the homology serre spectral sequence with f p coefficients to collapse at some finite term this in particular implies that as graded vector spaces h omegax is almost isomorphic to h omegab circle times h omegaf one consequence is the conclusion that x is elliptic if and only if b and f are

the equivalence of darmois israel and distributional method for thin shells in general relativity a distributional method to solve the einstein s field equations for thin shells is formulated the familiar field equations and jump conditions of darmois israel formalism are derived a careful analysis of the bianchi identities shows that for cases under consideration they make sense as distributions and lead to jump conditions of darmois israel formalism

intermittency in the presence of noise we analyse iterations of maps on an interval with an added noise term in the neighbourhood of an intermittency threshold we rigorously derive a universal scaling function for the laminar time expressed as a function of the distance from the threshold and the variance of the noise

comment on asymptotic properties of coupled langevin equations the asymptotic behavior of coupled langevin equations in the limit of weak noise is studied by general normal form techniques in the vicinity of a pitchfork bifurcation the non gaussian behavior of the critical variable is established the conditional probability of the noncritical variable around the center manifold is determined it is shown that in certain cases the distribution of this later variable may be non gaussian

on a generic property of geodesic flows in this paper we are concerned with a sufficient condition for a riemannian metric on a compact simply connected manifold to have infinitely many geometrically distinct closed geodesics 1969 gromoll meyer proved in gm that for every riemannian metric on a compact simply connected manifold m there are infinitely many geometrically distinct closed geodesics if the sequence bi m f i of betti numbers of the free loop space m is unbounded for some field f using the theory of minimal models vigue poirrier sullivan proved that the rational cohomology algebra h m i q of m is generated by a single element if and only if the sequence bi m i q i of betti numbers of the free loop space m of m is bounded it is a conjecture that the same statement holds for all fields of prime characteristic partial results are due to mccleary ziller mz and halperin vigue poirrier hv now we turn to the manifolds for which the hypothesis of gromoll meyer s theorem does not hold for example spheres and projective spaces then stability properties of the closed geodesics become important a closed geodesic is hyperbolic if the linearized poincare map has no eigenvalue of norm 1 from the bumpy metrics theorem due to abraham ab and anosov an2 and from a pertubation result due to klingenberg takens kt one can conclude a c4 generic metric on a compact manifold has either a non hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic in the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by moser mo in the second case there are infinitely many closed geodesics if m is simply connected due to results by hingston hi and the author ra1 hence it follows from these results that a c4 generic metric on a compact simply connected manifold has infinitely many geometrically distinct closed geodesics we remark that it is an open question whether there is a riemannian metric on a simply connected compact manifold all of whose closed geodesics are hyperbolic in ra2 we show that the examples of metrics on the 2

connected graded hopf algebras we study algebraic and homological properties of two classes of infinite dimensional hopf algebras over an algebraically closed field k of characteristic zero the first class consists of those hopf k algebras that are connected graded as algebras and the second class are those hopf k algebras that are connected as coalgebras for many but not all of the results presented here the hopf algebras are assumed to have finite gel fand kirillov dimension it is shown that if the hopf algebra h is a connected graded hopf algebra of finite gel fand kirillov dimension n then h is a noetherian domain which is cohen macaulay artin schelter regular and auslander regular of global dimension n it has s2 idh and is calabi yau detailed information is also provided about the hilbert series of h our results leave open the possibility that the first class of algebras is properly contained in the second for this second class the hopf k algebras of finite gel fand kirillov dimension n with connected coalgebra the underlying coalgebra is shown to be artin schelter regular of global dimension n both these classes of hopf algebra share many features in common with enveloping algebras of finite dimensional lie algebras for example an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra nevertheless we construct as one of our main results an example of a hopf k algebra h of gel fand kirillov dimension 5 which is connected graded as an algebra and connected as a coalgebra but is not isomorphic as an algebra to u g for any lie algebra g

asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise i cusp bifurcation we show how a singular perturbation technique based on the introduction of properly scaled variables enables us to derive the asymptotic properties of coupled langevin equations in the limit of weak noise this technique can be applied when the macroscopic steady state is asymptotically or marginally stable in the close vicinity of a cusp bifurcation point a simple prescription for the adiabatic elimination of the fast variable is established the critical variable exhibits amplified non gaussian fluctuations on a slow time scale the properties of the fast variable depend on the nonlinearity of the system under consideration because of its coupling to the critical variable it may exhibit amplified fluctuations of non gaussian nature

noise and bifurcations the influence of while noise on bifurcating dynamical systems is investigated using both fokker planck and functional integral methods noise leads to fuzzy bifurcations where physically relevant quantities become smooth functions of the bifurcation parameters we study dynamical and probabilistic quantities such as invariant measures liapunov exponents correlation functions and exit times the behavior of these quantities near the deterministic bifurcation point changes for distinct values of the control parameter therefore the very concept of bifurcation point becomes meaningless and must be replaced by the notion of bifurcation region

an algebraic model for chains on ωbg p we provide an interpretation of the homology of the loop space on the p completion of the classifying space of a finite group in terms of representation theory and demonstrate how to compute it we then give the following reformulation if f is an idempotent in kg such that f kg is the projective cover of the trivial module k and e 1 f then we exhibit isomorphisms for n 2 h n bg p k tor e kg e n 1 kg e e kg h n bg p k ext n 1 e kg e e kg e kg further algebraic structure is examined such as products and coproducts restriction and steenrod operations

the cumulant approach for investigating the noise influence on mode locking bifurcations the influence of noise on mode locking bifurcations is investigated for the circle map and for coupled logistic maps the technique based on the cumulant expansion is used for the bifurcation analysis of these systems it is shown that this cumulant analysis in gaussian approximation provides a suitable description of the influence of weak noise we find that the universal scaling properties for the circle map in the critical point for the golden mean sequence are the same as those obtained analytically using the path integral technique we find the same scaling behaviour in the case of weak multiplicative noise

bifurcations in fluctuating systems the center manifold approach bifurcations in fluctuating dynamical systems are studied using the ideas of center manifold reduction the method provides not only a systematic procedure for the reduction of the system to a small number of variables but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points the joint probability density factorizes into a stationary gaussian densityp v u in the fast variablesv and a time dependent densityp u f in the slow variablesu describing the dynamics on the center manifoldv v0 u p u t obeys a reduced fokker planck equation that can be written in a normal form by means of local nonlinear transformations both additive and multiplicative white noise are considered as is colored noise the results extend and formalize haken s concept of adiabatic elimination of fast variables

on spiking models for synaptic activity and impulsive differential equations we illustrate the problems that can arise in writing differential equations that include dirac delta functions to model equations with state dependent impulsive forcing specifically difficulties arise in the interpretation of the products of distributions with discontinuous functions we suggest several methods to resolve these ambiguities such as using limiting sequences and asymptotic analysis with applications of the results given for discrete maps these suggestions are applied to a popular model describing synaptic connections in the brain

obstructions to nonnegative curvature and rational homotopy theory we establish a link between rational homotopy theory and the problem which vector bundles admit a complete riemannian metric of nonnegative sectional curvature as an application we show for a large class of simply connected nonnegatively curved manifolds that if c lies in the class and t is a torus of positive dimension then most vector bundles over c x t admit no complete nonnegatively curved metrics

temporal localization of limit cycles in a noise driven chemical oscillator abstract we study the influence of multiplicative random external perturbation on the selkov model exhibiting limit cycles the dynamics are described by stochastic differential equations the numerically computed ensemble averages of the concentrations exhibit temporal localization of the limit cycles for small amplitudes of the noise the localization time is smaller when the amplitude of the noise is higher to understand this behavior a stochastic complex ginzburg landau equation is derived using normal form transformations on the stochastic evolution equations in the case of weak noise the equations for the radial and the phase components of the amplitude separate and can be solved exactly the solution is used to compute the damping factors which were found to be gaussian we also show that asymptotically the radial part fluctuates near the average value and the phase is found to be a gaussian random variable with time dependent mean and variance and may be responsible for the temporal localization of the limit cycle

stochastic calculus application to dynamic bifurcations and threshold crossings for the dynamic pitchfork bifurcation in the presence of white noise the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter the threshold crossing problem used for example to model the firing of a single cortical neuron is considered concentrating on quantities that may be experimentally measurable but have so far received little attention expressions for the statistics of pre threshold excursions occupation density and last crossing time of zero are compared with results from numerical generation of paths

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

comment on noise and bifurcations we calculate in exact form the first correction in a parameter measuring the strength of the noise to the effective potential for one variable diffusion processes the use of this potential to study transitions is discussed

dynamics of the stochastic duffing oscillator in gaussian approximation abstract the previously developed general concept of the gaussian approximation allows the description of the driven stochastic duffing oscillator by ordinary differential equations these equations are analyzed analytically as well as numerically to investigate the modification of simple bifurcations and of the feigenbaum cascade by stochastic forces furthermore the method allows a qualitative discussion of the stochastic system in the deterministic irregular region

codimension two bifurcations of nonlinear systems driven by fuzzy noise abstract by means of a fuzzy generalized cell mapping method a duffing van der pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period one attractors grow and merge as the intensity of the fuzzy noise is increased by introducing a small symmetry breaking parameter to break the symmetry the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations considering both the intensity of the fuzzy noise and the symmetry breaking parameter together as controls a codimension two bifurcation of fuzzy attractors is defined and examples of additive and multiplicative fuzzy noise are given such a codimension two bifurcation is fuzzy noise induced effects which cannot be seen in the deterministic systems

continuity of type i intermittency from a measure theoretical point of view we study a simple dynamical system which displays a so called type i intermittency bifurcation we determine the bowen ruelle measure e and prove that the expectation e g of any continuous functiong and the kolmogoroff sinai entropyh e are continuous functions of the bifurcation parameter therefore the transition is continuous from a measure theoretical point of view those results could be generalized to any similar dynamical system

index parity of closed geodesics and rigidity of hopf fibrations the diameter rigidity theorem of gromoll and grove 1987 states that a riemannian manifold with sectional curvature 1 and diameter 2 is either homeomorphic to a sphere locally isometric to a rank one symmetric space or it has the cohomology ring of the cayley plane ca the reason that they were only able to recognize the cohomology ring of ca is due to an exceptional case in another theorem gromoll and grove 1988 a riemannian submersion m b b with connected fibers that is defined on the euclidean sphere m is metrically congruent to a hopf fibration unless possibly m b 15 8 we will rule out the exceptional cases in both theorems our argument relies on a rather unusal application of morse theory for that purpose we give a general criterion which allows to decide whether the morse index of a closed geodesic is even or odd

on the berger conjecture for manifolds all of whose geodesics are closed we define a besse manifold as a riemannian manifold m g all of whose geodesics are closed a conjecture of berger states that all prime geodesics have the same length for any simply connected besse manifold we firstly show that the energy function on the free loop space of a simply connected besse manifold is a perfect morse bott function with respect to a suitable cohomology secondly we explain when the negative bundles along the critical manifolds are orientable these two general results then lead to a solution of berger s conjecture when the underlying manifold is a sphere of dimension at least four

oscillators with chaotic behavior an illustration of a theorem by shil nikov using an explicit one parameter family of differential equations describing oscillators with feedback effects we prove the existence of values of the parameters such that there exist infinitely many unstable periodic orbits of saddle type the proof relies on a theorem by shil nikov which we propose as an explanation for the origin and structure of the chaotic behavior displayed by many well known third order differential systems

escape statistics for systems driven by dichotomous noise i general theory previous results on first passage time statistics for systems driven by dichotomous noise are extended in order to cover the escape from regions including fixed points of the stochastic flow for such regions a treatment splitting the escape through one or the other boundary is required the obtained escape probabilities and mean exit times are relevant for the complete characterization of stochastic systems undergoing bifurcations

rational l s category and its applications let s be a 1 connected cw complex of finite type and put cat0 s lusternik schnirelmann category of the localization sq this invariant is char acterized in terms of the minimal model of s it is shown that if is the fibre inclusion of a fibration it is also shown that if dim h s q oo then either dim it s q oo or the groups nk s q grow exponentially with k introduction in this paper topological space means a pointed path connected normal space s1 such that the inclusion of the basepoint is a cofibration and the rational singular homology is finite dimensional in each degree all vector spaces and algebras have q as ground field homology and cohomology of spaces is singular with rational coefficients we shall be dealing with category of spaces cat s in the sense of lusternik and schnirelmann 18 a recent survey is given in 15 but our definition differs by one from that of 15 so that for us spheres have category one not two our purpose is to provide a computationally useful characterization of the rational category cat0 5 of space 5 and to apply this in a variety of situations the definition of cat0 5 is sketched below and given precisely in 4 it satisfies cat0 s cauls when 5 is 1 connected cat0 5 coincides with the category of the localization sq which has been studied by toomer in 26 and 27 and lemaire and sigrist in 17 the characterization problem was posed by berstein 26 most of the applications depend on the following fundamental mapping theorem which we derive as a consequence of our characterization