subharmonic and homoclinic bifurcations in a parametrically forced pendulum abstract depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the melnikov method and of averaging methods it is shown that as a parameter is varied repeated resonances of successively higher periods occur culminating in homoclinic orbits according to the theorem of smale homoclinic bifurcation is the source of the unstable chaotic motions observed for some selected parameter sets the theoretical predictions are tested by numerical calculations very good agreement is found between analytical and numerical results

resonance bands in a two degree of freedom hamiltonian system abstract in perturbations of integrable two degree of freedom hamiltonian systems the invariant kam tori are typically separated by zones of instability or resonance bands inhabited by elliptic and hyperbolic periodic orbits and homoclinic orbits we indicate how the melnikov method or the method of averaging can asymptotically predict the widths of these bands in specific cases and we compare these predictions with numerical computations for a pair of linearly coupled simple pendula we conclude that even for low order resonances the first order asymptotic results are generally useful only for very small coupling 10 4

on optimal performance of nonlinear energy sinks in multiple degree of freedom systems abstract we study the problem of optimizing the performance of a nonlinear spring mass damper attached to a class of multiple degree of freedom systems we aim to maximize the rate of one way energy transfer from primary system to the attachment and focus on impulsive excitation of a two degree of freedom primary system with an essentially nonlinear attachment the nonlinear attachment is shown to be able to perform as a nonlinear energy sink nes by taking away energy from the primary system irreversibly for some types of impulsive excitations using perturbation analysis and exploiting separation of time scales we perform dimensionality reduction of this strongly nonlinear system our analysis shows that efficient energy transfer to nonlinear attachment in this system occurs for initial conditions close to homoclinic orbit of the slow time scale undamped system a phenomenon that has been previously observed for the case of single degree of freedom primary systems analytical formulae for optimal parameters for given impulsive excitation input are derived generalization of this framework to systems with arbitrary number of degrees of freedom of the primary system is also discussed the performance of both linear and nonlinear optimally tuned attachments is compared while nes performance is sensitive to magnitude of the initial impulse our results show that nes performance is more robust than linear tuned mass damper to several parametric perturbations hence our work provides evidence that homoclinic orbits of the underlying hamiltonian system play a crucial role in efficient nonlinear energy transfers even in high dimensional systems and gives new insight into robustness of systems with essential nonlinearity

study of a class of subharmonic motions using a non smooth temporal transformation nstt abstract a nonlinear boundary value problem nbvp formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented the formulation is based on a non smooth temporal transformation nstt to replace the independent temporal variable of the problem with two new piecewise linear periodic temporal variables as a result the problem of computing the subharmonic motions is reduced to a set of nbvps with homogeneous boundary conditions some numerical and analytical solutions of the reduced nbvps are given and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed in particular the formulation shows a definite connection between vibro impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit an interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system

relations and functions in multiset context a multiset is a collection of objects in which repetition of elements is significant in this paper an attempt is made to generalize the concepts of relation function composition and equivalence in the multiset context as a pre requisite a brief survey of the axiomatic approach to the multiset theory is also presented

analytical study of vibrating systems with strong non linearities by employing saw tooth time transformations abstract a special saw tooth temporal transformation technique is formulated it is simple enough to allow analytical computation of strongly non linear free and forced dynamic responses but at the same time can be applied to the analysis of general classes of non linear problems

the set of fuzzy rational numbers and flexible querying a fuzzy bag is a bag in which each occurrence of an element is associated with a grade of membership this notion can be viewed as a generalization of the concepts of set fuzzy set and bag the set of fuzzy integers n f provides a general characterization in which all these different concepts are treated in a uniform way and can then be composed in the field of databases the use of fuzzy bags is motivated by their ability to manage both quantities and preferences however n f becomes too restricted a framework when dealing with queries based on difference or division operations so a more general structure based on the set of fuzzy relative integers z f in which exact differences can be performed has been first developed in this paper we carry on with this approach and we extend z f to the set of fuzzy rational numbers q f this context leads to define a closed system of multiplicative operations and allows to perform exact divisions applied to flexible querying of databases q f and the notion of division on fuzzy numbers allow to generalize the relational division they define a sound basis for queries involving ratios between quantities

on the theory of bags and lists abstract in this paper the concept of bag space has been introduced the concept of bag complement is redefined suitably and some theorems involving bag operations have been established the concept of the theory of lists which is a generalization of the theory of bags has been redefined and many existing and new results have been established based upon this new definition

the existence of arbitrarily many distinct periodic orbits in a two degree of freedom hamiltonian system abstract melnikov s method is used to prove the existence of arbitrarily many elliptic and hyperbolic periodic orbits in the neighborhood of an elliptic orbit of a two degree of freedom hamiltonian system which is almost integrable the existence of such orbits precludes the existence of analytic second integrals of a certain type the methods used permit a detailed analysis of the way in which resonant tori break up between the kam irrational tori which are preserved for weak coupling of two independent nonlinear oscillators

dedekind multisets and function shells abstract a multiset is a set containing repeated elements we define the unique multiset determined by a function based on an idea formulated by dedekind in 1888 the class of all functions which determine the same multiset is called a function shell the formal theory of multisets is revised to allow for infinite repetitions of elements the revised theory of multisets is then interpreted as a theory of function shells with this interpretation a new algebra of functions is defined and interesting properties of function shells are investigated

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

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

collective fixed points and maximal elements with applications to abstract economies abstract in this paper we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is condensing as an application of our collective fixed points theorem we derive the coincidence theorem for two families of multivalued maps defined on product spaces then we give some existence results for maximal elements for a family of ls majorized multivalued maps whose product is condensing we also prove some existence results for maximal elements for a family of multivalued maps which are not ls majorized but their product is condensing as applications of our results some existence results for equilibria of abstract economies are also derived the results of this paper are more general than those given in the literature

fixed point and maximal element theorems with applications to abstract economies and minimax inequalities in this paper we establish some fixed point theorems for a family of multivalued maps under mild conditions by using our fixed point theorems we derive some maximal element theorems for a particular family of multivalued maps namely the condensing multivalued maps as applications of our results we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences further applications of our results are also given to minimax inequalities for a family of functions

nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation abstract the transient and steady state response of an oscillator with hysteretic restoring force and sinusoidal excitation are investigated hysteresis is modeled by using the bilinear model of caughey with a hybrid system formulation a novel method for obtaining the exact transient and steady state response of the system is discussed stability and bifurcations of periodic orbits are studied using poincare maps results are compared with asymptotic expansions obtained by caughey the bilinear hysteretic element is found to act like a soft spring several sub harmonic resonances are found in the system however no chaotic behavior is observed away from the sub harmonic resonance the asymptotic expansions and the exact steady state response of the system are seen to match with good accuracy

representing lattices using many valued relations this paper provides a representation theorem of lattices using many valued relations we show that any many valued relation can be associated to a unique lattice which is a meet sublattice of a product of chains conversely to any lattice we can associate a many valued relation such that its associated lattice is isomorphic to the initial one thereby we obtain a representation theorem of lattices using many valued relations moreover since several many valued relations might have the same associated lattice we give a characterization of the minimal many valued relation that can be associated to a lattice we then sketch a polynomial time algorithm which computes such a minimal relation from either a lattice or an arbitrary relation this representation presents several advantages it is smaller than the usual binary representation all known reconstruction algorithms working on binary relation can be used without loss of efficiency it can be used by existing data mining processes

higher level constructs for families and multisets abstract in an earlier paper we argued that the concept of multisets should be based on families that is on sets and functions between sets rather than as usual on sets and cardinal numbers we showed how certain fundamental problems regarding the distinguishablity of objects as well as unexpected anomalies of the basic operations of union intersection and complement can be avoided elegantly using families rather than multisets on the other hand there is a trivial mapping of families to multisets hence using families does not introduce any significant formal complications the difficulties with the usual definition of multisets by multiplicities reach beyond the philosophical or metamathematical foundations for the basic operations one can find acceptable compromises for higher level set constructs like cartesian products projections relations functions or the power set the multiset counterparts are rather contrived and many of the constructions leave the strict definition of multisets actually introducing family like constructs through a back door in the earlier paper we proposed to use families instead of multisets to resolve the basic problems in this paper we show that families instead of multisets support the higher level constructions even when no appropriate multiset based constructions exist we also show that multisets form a category and that the natural mapping from families to multisets is a functor this emphasizes our claim that for a definition of multisets one should start with families and only introduce multiplicities as a secondary concept

non smooth approximations of the limiting phase trajectories for the duffing oscillator near 1 1 resonance this paper demonstrates an analytical description of the non stationary response of the forced duffing oscillator near the 1 1 resonance our approach exploits the concept of the limiting phase trajectories lpts the lpt is understood as a trajectory of the system being initially at rest and motion over the lpt corresponds to a maximum possible energy transfer from an external source of harmonic excitation to the oscillator it is shown that for a wide range of parameters the behavior of the duffing oscillator moving over the lpt is similar to the dynamics of a free particle between two motion limiters the vibro impact hypothesis suggests a perturbation technique based on non smooth generating functions which yields an explicit asymptotic expression for the lpt different types of the dynamical behavior of the oscillator subjected to harmonic or narrow band perturbations are studied

complicated dynamics of a linear oscillator with a light essentially nonlinear attachment abstract we study the dynamics of a two degree of freedom dof nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear nonlinearizable stiffness we consider first the undamped system and perform a numerical study based on non smooth transformations to determine its periodic solutions in a frequency energy plot it is found that there is a sequence of periodic solutions bifurcating or emanating from the main backbone curve of the plot we then study analytically the periodic orbits of the undamped system using a complexification averaging technique in order to determine the frequency contents of the various branches of solutions and to understand the types of oscillation performed by the system at the different regimes of the motion the transient responses of the weakly damped system are then examined and numerical wavelet transforms are used to study the time evolutions of their harmonic components we show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics as it causes complicated transitions between modes in the damped transient motion in addition there is the possibility of strong passive energy transfer energy pumping from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions

multiset topologies induced by multiset relations multiset topology is a collection of multisets which satisfies the axioms of topology in this paper multiset topologies are obtained using binary multiset relations the relationship between multiset topologies is investigated and some of the properties of these multiset topologies are proved the quasi discrete multiset topology is obtained from a symmetric multiset relation instead of an equivalence multiset relation and several examples are given to indicate counter connections finally the rough multiset is introduced and the multiset topology induced by a multiset relation is used to generalize the rough multiset concept

selecting families and their applications deguire and lassonde p deguire m lassonde familles selectantes topol methods nonlinear anal 5 1995 261 269 extend the concept of continuous selection and introduce the notion of selecting family for a family of set valued mappings in this paper we first establish a new existence theorem of selecting families the existence of the selecting families will be then used in order to obtain several fixed component theorems of fan browder type an intersection result a maximal element theorem for a family of set valued mappings and a minimax inequality

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