correlation dimension for paired discrete time series we present a method for detecting the dimension of a dynamical system encompassing simultaneously two distinct discrete time series this method is an extension of the technique introduced by grassberger and procaccia for single time series and allows to evaluate the common correlation dimension of the chaotic attractor the method is applied to some mathematical models and to multiple single neuron spike trains

a family of three dimensional virtual elements with applications to magnetostatics we consider as a simple model problem the application of virtual element methods vems to the linear magnetostatic three dimensional problem in the formulation of kikuchi in doing so we also introduce new serendipity vem spaces where the serendipity reduction is made only on the faces of a general polyhedral decomposition assuming that internal degrees of freedom could be more easily eliminated by static condensation these new spaces are meant more generally for the combined approximation of h 1 conforming 0 forms h curl conforming 1 forms and h div conforming 2 forms functional spaces in three dimensions and they could surely be useful for other problems and in more general contexts

positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

weak singularities may help periodic solutions to exist abstract in a periodically forced semilinear differential equation with a singular nonlinearity a weak force condition enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem the originality of the arguments relies in that contrary to the customary situation in the available references a weak singularity facilitates the arguments of the proofs

positive periodic solutions for planar differential systems with repulsive singularities on the axes abstract in this paper we provide an existence result of periodic solutions to planar systems having a repulsive singularity on both the axes we require a nonresonance assumption related to the spectrum of the periodic problem associated to the steen s planar system y c x 3 x x d y 3 y x 0 y 0 which is isochronous see 15 as a particular case we obtain the existence of periodic solutions to a periodically perturbed steen s planar system

positive periodic solutions of second order differential equations with weak singularities abstract we establish the existence of positive periodic solutions of the second order differential equation u a t u f t u c t via schauder s fixed point theorem where a l 1 r t z r c l 1 r t z r f is a caratheodory function and is singular at u 0 our main results generalize some recent results by p j torres

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

multiplicity of positive periodic solutions to superlinear repulsive singular equations abstract in this paper we study positive periodic solutions to the repulsive singular perturbations of the hill equations it is proved that such a perturbation problem has at least two positive periodic solutions when the anti maximum principle holds for the hill operator and the perturbation is superlinear at infinity the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

periodic solutions of the second order differential equations with singularity abstract in this paper we deal with the existence of periodic solutions of the second order differential equations x g x p t with singularity near origin by using the phase plane analysis methods we prove that the given equation has at least one periodic solution when g x exhibits semilinear condition for positive large x and singularity condition at origin and the primitive function g x 1 x g s d s tends to positive infinity as x 0 moreover we construct a counter example to show that our result is no longer valid when g x tends to some finite constant as x 0

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

periodic solutions for damped differential equations with a weak repulsive singularity abstract this paper deals with the existence of positive t periodic solutions for the following damped differential equation x p t x q t x f t x c t where p q c l 1 r are t periodic functions and f car r r r is t periodic in the first variable we will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem

multiplicity of positive solutions for singular three point boundary value problems at resonance abstract in this paper we study the existence and multiplicity of positive solutions for the three point boundary value problems with singular nonlinear perturbations at resonance the existence results are applicable to both the case of a strong singularity and the case of a weak singularity the proof is based on a nonlinear alternative leray schauder principle and a fixed point theorem in cones for completely continuous operators

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

parallel solvers for virtual element discretizations of elliptic equations in mixed form abstract the aim of this paper is twofold on the one hand we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three dimensional elliptic equations on polyhedral meshes on the other hand we focus on the parallel solution of the linear system arising from such discretization considering both direct and iterative parallel solvers in the latter case we develop two block preconditioners one based on the approximate schur complement and one on a regularization technique both these topics are numerically validated by several parallel tests performed on a linux cluster more specifically we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

singularly perturbed higher order periodic boundary value problems abstract the existence of positive periodic solutions is proved for the higher order singular differential equation 0 1 x n a n 1 x n 1 a 1 x a 0 t x e t and for some generalizations including a class of differential systems with singular cyclic feedback

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

a three dimensional hellinger reissner virtual element method for linear elasticity problems abstract we present a virtual element method for the 3d linear elasticity problems based on hellinger reissner variational principle in the small strain theory we propose a low order scheme with a priori symmetric stresses and continuous tractions across element interfaces a convergence and stability analysis is developed and we confirm the theoretical predictions via some numerical tests

existence of a pair of periodic solutions of an o d e generalizing a problem in nonlinear elasticity via variational methods on considere l existence des solutions t periodiques de l equation differentielle x g t x ou g r r r est continue et t periodique en t sous certaines conditions sur g on demontre l existence d au moins deux solutions t periodiques on utilise une methode variationnelle

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

periodic solutions for second order differential equations with a singular nonlinearity abstract this paper deals with the existence of positive t periodic solutions for the following differential equation x a t x f t x c t where a c l 1 0 t and f car 0 t r r the existence results are obtained by using a fixed point theorem in cones

quasi periodic solutions of forced isochronous oscillators at resonance abstract we deal with the existence of quasi periodic solutions of forced isochronous oscillators with a repulsive singularity the nonlinearity is a bounded perturbation using a variant of moser s twist theorem of invariant curves due to ortega r ortega boundedness in a piecewise linear oscillator and a variant of the small twist theorem proc london math soc 79 1999 381 413 we show that there are many quasi periodic solutions and the boundedness of all solutions

a minimax method for a class of hamiltonian systems with singular potentials abstract this paper presents a minimax method which gives existence and multiplicity results for time periodic solutions of a class of hamiltonian systems when a singular potential is present the singularity satisfies the strong force condition of gordon when milder singularities are permitted a notion of generalized t periodic solution is introduced and we get existence and multiplicity results for such solutions

existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity it is proved that a periodically forced second order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a t periodic stable solution for high values of the mean value of the forcing term the method of proof combines a rescaling argument with the analysis of the first twist coefficient of the

dynamical systems with effective like potentials on etudie l existence des solutions t periodiques pour des systemes hamiltoniens d ordre 2 de la forme x v x ou v r n 0 r est de classe c 2 et v x quand x 0

radially symmetric systems with a singularity and asymptotically linear growth abstract we prove the existence of infinitely many periodic solutions for radially symmetric systems with a singularity of repulsive type the nonlinearity is assumed to have a linear growth at infinity being controlled by two constants which have a precise interpretation in terms of the dancer fucik spectrum our result generalizes an existence theorem by del pino et al 1992 4 obtained in the case of a scalar second order differential equation

a multiplicity result for periodic solutions of second order differential equations with a singularity abstract by the use of the poincare birkhoff fixed point theorem we prove a multiplicity result for periodic solutions of a second order differential equation where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity our main theorem is related to previous results by rebelo 1996 1997 4 5 and rebelo and zanolin 1996 6 7 in connection with a problem raised by del pino et al 1992 1

on the 2 dimensional dual minkowski problem abstract we study the 2 dimensional dual minkowski problem which is the following nonlinear problem on unit circle u u g u 1 u 2 u 2 2 k 2 s for any given positive continuous function g with 2 m periodic we prove that it is solvable for all k 1 and m 3 4 5

periodic solutions of duffing equations with semi quadratic potential and singularity abstract in this paper we deal with the existence of periodic solutions of the second order differential equations x g x p t with singularity we prove that the given equation has at least one periodic solution when g x has singularity at origin g x 1 x g s d s satisfies n 2 4 lim inf x 2 g x x 2 lim sup x 2 g x x 2 n 1 2 4 for large positive x where n 0 is an integer and g x tends to positive infinity as x 0

periodic orbits of radially symmetric keplerian like systems a topological degree approach abstract we are concerned with non autonomous radially symmetric systems with a singularity which are t periodic in time by the use of topological degree theory we prove the existence of large amplitude periodic solutions whose minimal period is an integer multiple of t precise estimates are then given in the case of keplerian like systems showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system

twist periodic solutions of second order singular differential equations abstract we study the twist periodic solutions of second order singular differential equations such twist periodic solutions are stable in the sense of lyapunov and present much interesting dynamical features around them the proof is based on the third order approximation method the estimates of periodic solutions of ermakov pinney equations and the estimates on rotation numbers of hill equations play an important role in the analysis

relationship between the vertex centered linearity preserving scheme and the lowest order virtual element method for diffusion problems on star shaped polygons abstract as a generalization of the finite element method the virtual element method vem has made remarkable achievements in the algorithm and analysis of simulating various problems recently vertex centered linearity preserving scheme vlps wu et al 2016 is a linear finite volume scheme which is currently only applied to the numerical simulation of diffusion problems in this paper we study the relationship between vlps and the lowest order vem for diffusion problems on star shaped polygonal meshes from an algebraic point of view the global stiffness matrix of vlps with a special stabilization term coincides with that of the lowest order vem while the load terms are generally different specifically we find that the global stiffness matrices of the two methods can be split as the consistency parts and the stability parts and the consistency parts are the same while the stability parts coincide under some assumptions as a by product a new stability term is obtained for vlps besides a post processing procedure is suggested for the lowest order vem to preserve the positivity of the cell centered unknowns and maintain at the same time the local conservation on the primary meshes the positivity existence and uniqueness of the cell centered unknowns are studied numerical experiments confirm the theoretical findings and demonstrate the efficiency of the post processing procedure on various polygonal meshes

positive solutions to a singular neumann problem abstract we show the existence of positive solution for the following class of singular neumann problem u a x u h x u p in b r with u 0 on b r where r 0 0 is a positive parameter 0 p 0 1 b r b r 0 r n a b r r and h b r r are radially symmetric nonnegative c 1 functions using variational methods and sub and supersolutions we obtain a solution for an approximated problem involving mixed boundary conditions the limit of the approximated solutions is a positive solution

a neumann problem with logarithmic nonlinearity in a ball abstract the nonlinear elliptic problem u log u h x u p in a ball b r r n with neumann boundary conditions u 0 on b r were studied by means of a double perturbation argument we obtain a positive radial solution u c 2 b r 0 c b r

positive bound states of systems of nonlinear schrodinger equations abstract we study the existence of positive bound states of non autonomous systems of nonlinear schrodinger equations both the singular case and the regular case are discussed the proof is based on a nonlinear alternative principle of leray schauder

quantifying the closeness of fractal measures numerically fractal measures are represented by finite sets of discrete points e g those generated by trajectories we introduce a cross correlation integral to evaluate the distinctness and overlap of two such sets and present a criterion to determine up to what precision the two sets are realizations of the same measure

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forces this paper is devoted to study the existence of positive solutions to the second order semipositone periodic boundary value problem x a t x f t x x 0 x 1 x 0 x 1 here f t x may be singular at x 0 and may be superlinear at x our analysis relies on a fixed point theorem in cones

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

periodic motion of a system of two or three charged particles we provide necessary and sufficient conditions for the existence of t periodic solutions of a system of second order ordinary differential equations that models the motion of two or three collinear charged particles of the same sign

periodic solutions for second order singular damped differential equations abstract we study the existence of positive periodic solutions for second order singular damped differential equations by combining the analysis of the sign of green s functions for the linear damped equation together with a nonlinear alternative principle of leray schauder recent results in the literature are generalized and significantly improved

an elliptic singular system with nonlocal boundary conditions abstract we study the existence of solutions for the nonlinear second order elliptic system u g u f x where g c r n s r n with s r n bounded using topological degree methods we prove an existence result under a geometric condition on g moreover we analyze the particular case of an isolated repulsive singularity under a nirenberg type condition we prove the existence of a sequence of solutions of appropriate approximated problems that converges to a generalized solution

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

positive periodic solutions of hill s equations with singular nonlinear perturbations abstract we study the existence and multiplicity of positive periodic solutions of hill s equations with singular nonlinear perturbations the new results are applicable to the case of a strong singularity as well as the case of a weak singularity the proof relies on a nonlinear alternative principle of leray schauder and a fixed point theorem in cones some recent results in the literature are generalized and improved

non collision periodic solutions of second order singular dynamical systems we study the existence of non collision periodic solutions for second order singular dynamical systems the repulsive case and the attractive case are dealt with using a unified topological approach the proof is based on a well known fixed point theorem for completely continuous operators involving a new type of cone we do not need to consider so called strong force conditions moreover for the repulsive case the critical case can be covered recent results in the literature even in the scalar case are complemented generalized and improved

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved