the hyperelliptic limit cycles of the lienard systems abstract for lienard systems x y y f m x y g n x with f m and g n real polynomials of degree m and n respectively in h zoladek algebraic invariant curves for the lienard equation trans amer math soc 350 1998 1681 1701 the author showed that if m 3 and m 1 n 2 m there always exist lienard systems which have a hyperelliptic limit cycle llibre and zhang j llibre xiang zhang on the algebraic limit cycles of lienard systems nonlinearity 21 2008 2011 2022 proved that the lienard systems with m 3 and n 5 have no hyperelliptic limit cycles and that there exist lienard systems with m 4 and 5 n 8 which do have hyperelliptic limit cycles so it is still an open problem to characterize the lienard systems which have an algebraic limit cycle in cases m 4 and m 1 n 2 m in this paper we will prove that there exist lienard systems with m 5 and m 1 n 2 m which have hyperelliptic limit cycles

uniqueness of dirichlet forms associated with systems of infinitely many brownian balls in ℝd dirichlet forms associated with systems of infinitely many brownian balls in d are studied introducing a linear operator l0 defined on a space of smooth local functions we show the uniqueness of dirichlet forms associated with self adjoint markovian extensions of l0 we also discuss the ergodicity of the reversible process associated with the dirichlet form

a note on existence and uniqueness of limit cycles for lienard systems we consider the lienard equation and we give a sufficient condition to ensure existence and uniqueness of limit cycles we compare our result with some other existing ones and we give some applications

interacting brownian motions in infinite dimensions with logarithmic interaction potentials we investigate the construction of diffusions consisting of infinitely numerous brownian particles moving in mathbb r d and interacting via logarithmic functions two dimensional coulomb potentials these potentials are very strong and act over a long range in nature the associated equilibrium states are no longer gibbs measures we present general results for the construction of such diffusions and as applications thereof construct two typical interacting brownian motions with logarithmic interaction potentials namely the dyson model in infinite dimensions and ginibre interacting brownian motions the former is a particle system in mathbb r while the latter is in mathbb r 2 both models are translation and rotation invariant in space and as such are prototypes of dimensions d 1 2 respectively the equilibrium states of the former diffusion model are determinantal or pfaffian random point fields with sine kernels they appear in the thermodynamical limits of the spectrum of the ensembles of gaussian random matrices such as goe gue and gse the equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non hermitian gaussian random matrices known as the ginibre ensemble

on the existence and uniqueness of limit cycles for generalized lienard systems abstract in this paper we consider a generalized lienard system d x d t y f x 0 1 d y d t g x where f is continuous and differentiable on an open interval b 1 a 1 with b 1 0 a 1 assume that there exist a and b with b 1 b 0 a a 1 such that x f x 0 as b x a and x f x 0 as a x a 1 or b 1 x b a new uniqueness theorem of limit cycles for the lienard system 0 1 is obtained an example is given to show the application of the theorem

alien limit cycles in lienard equations this paper aims at providing an example of a family of polynomial lienard equations exhibiting an alien limit cycle this limit cycle is perturbed from a 2 saddle cycle in the boundary of an annulus of periodic orbits given by a hamiltonian vector field the hamiltonian represents a truncated pendulum of degree 4 in comparison to a former polynomial example not only the equations are simpler but a lot of tedious calculations can be avoided making the example also interesting with respect to simplicity in treatment

limit cycle bifurcations of some lienard systems in this paper we first give some general theorems on the limit cycle bifurcation for near hamiltonian systems near a double homoclinic loop or a center as a preliminary then we use these theorems to study some polynomial lienard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems

limit cycle bifurcations by perturbing a cuspidal loop in a hamiltonian system abstract in this paper we first study the analytical property of the first melnikov function for general hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the hamiltonian value corresponding to the loop then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4 5 or 6 limit cycles in a neighborhood of loop as an application of our main results we consider some polynomial lienard systems and find 4 5 or 6 limit cycles

abelian integrals and limit cycles abstract the paper deals with generic perturbations from a hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles in this paper we show that near a 2 saddle cycle the number of limit cycles produced in unfoldings with one unbroken connection can exceed the number of zeros of the related abelian integral even if the latter represents a stable elementary catastrophe we however also show that in general finite codimension of the abelian integral leads to a finite upper bound on the local cyclicity in the treatment we introduce the notion of simple asymptotic scale deformation

on the period function of lienard systems abstract we study the period functiontof a centeroof a lienard system a sufficient condition for the monotonicity oft or for the isochronicity ofo is given such a condition is also necessary whenfandgare analytic andgis odd in this case a characterization of isochronous centers of lienard systems is given strict monotonicity and global monotonicity oftare also investigated

limit cycle bifurcations near a 2 polycycle or double 2 polycycle of planar systems abstract in this paper we study the bifurcation problem of limit cycles for a near hamiltonian system near a 2 polycycle or double 2 polycycle we obtain new results on the number of limit cycles by using the method of stability changing of a homoclinic loop this method also provides a new way to find alien limit cycles especially we found some alien limit cycles of new type

coexistence of stable limit cycles in a generalized curie weiss model with dissipation in this paper we modify the langevin dynamics associated to the generalized curie weiss model by introducing noisy and dissipative evolution in the interaction potential we show that when a zero mean gaussian is taken as single site distribution the dynamics in the thermodynamic limit can be described by a finite set of odes depending on the form of the interaction function the system can have several phase transitions at different critical temperatures because of the dissipation effect not only the magnetization of the systems displays a self sustained periodic behavior at sufficiently low temperature but in certain regimes any finite number of stable limit cycles can exist we explore some of these peculiarities with explicit examples

the center conditions for a lienard system we obtain necessary and sufficient coefficient conditions for a center at the origin for a lienard system with nonlinearities of degree six the necessity of the conditions is derived from the first seven focus quantities and their sufficiency is proved by cherkas s method

benford s law and first letter of words a universal first letter law fll is derived and described it predicts the percentages of first letters for words in novels the fll is akin to benford s law bl of first digits which predicts

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

infinite dimensional stochastic differential equations related to random matrices we solve infinite dimensional stochastic differential equations isdes describing an infinite number of brownian particles interacting via two dimensional coulomb potentials the equilibrium states of the associated unlabeled stochastic dynamics are the ginibre random point field and dyson s measures which appear in random matrix theory to solve the isdes we establish an integration by parts formula for these measures because the long range effect of two dimensional coulomb potentials is quite strong the properties of brownian particles interacting with two dimensional coulomb potentials are remarkably different from those of brownian particles interacting with ruelle s class interaction potentials as an example we prove that the interacting brownian particles associated with the ginibre random point field satisfy plural isdes

hopf bifurcation for two types of lienard systems abstract in this paper we study the maximal number of limit cycles in hopf bifurcations for two types of lienard systems and obtain an upper bound of the number in some cases the upper bound is the least called the hopf cyclicity

limit cycles of a class of lienard systems with restoring forces of seventh degree abstract the study of limit cycles for lienard system is very important not only in theoretical studies but also in applications in this paper we study the number of limit cycles for a class of lienard systems with restoring forces of seventh degree let h n m denote the maximum number of limit cycles bifurcated from the generalized lienard system x y y g x f x y where f x and g x are polynomials in x and deg f n det g m we greatly improve the existing results of h n m for m 7 n 4 and m 7 n 2 n with 4 n 20

a cubic system with thirteen limit cycles abstract we construct a planar cubic system and demonstrate that it has at least 13 limit cycles the construction is essentially based on counting the number of zeros of some abelian integrals

benford s law first significant digit and distribution distances for testing the reliability of financial reports in developing countries abstract we discuss a common suspicion about reported financial data in 10 industrial sectors of the 6 so called main developing countries over the time interval 2000 2014 these data are examined through benford s law first significant digit and through distribution distances tests it is shown that several visually anomalous data have to be a priori removed thereafter the distributions much better follow the first digit significant law indicating the usefulness of a benford s law test from the research starting line the same holds true for distance tests a few outliers are pointed out

linear estimate of the number of zeros of abelian integrals for a kind of quartic hamiltonians abstract an upper bound b n 7n 5 is derived for the number of zeros of abelian integrals i h h g x y dy f x y dx on the open interval where h is an oval lying on the algebraic curve h x y 1 2 y2 u x h deg u x 4 and is the maximal interval of existence of h f x y g x y are polynomials of x and y and n max deg f x y deg g x y

covid 19 flattening the curve and benford s law for many countries attempting to control the fast rising number of coronavirus cases and deaths the race is on to flatten the curve since the spread of coronavirus disease 2019 covid 19 has taken on pandemic proportions in the absence of significant control interventions the curve could be steep with the number of covid 19 cases growing exponentially in fact this level of proliferation may already be happening since the number of patients infected in italy closely follows an exponential trend thus we propose a test when the numbers are taken from an exponential distribution it has been demonstrated that they automatically follow benford s law bl as a result if the current control interventions are successful and we flatten the curve i e we slow the rate below an exponential growth rate then the number of infections or deaths will not obey bl for this reason bl may be useful for assessing the effects of the current control interventions and may be able to answer the question how flat is flat enough in this study we used an epidemic growth model in the presence of interventions to describe the potential for a flattened curve and then investigated whether the epidemic growth model followed bl for ten selected countries with a relatively high mortality rate among these countries south korea showed a particularly high degree of control intervention although all of the countries have aggressively fought the epidemic our analysis shows that all countries except for japan satisfied bl indicating the growth rates of covid 19 were close to an exponential trend based on the simulation table in this study bl test shows that the data from japan is incorrect

n v limit for stochastic dynamics in continuous particle systems we provide an n v limit for the infinite particle infinite volume stochastic dynamics associated with gibbs states in continuous particle systems on d d 1 starting point is an n particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset d with finite volume lebesgue measure v the aim is to approximate the infinite particle infinite volume stochastic dynamic by the above n particle dynamic in as n and v such that n v where is the particle density first we derive an improved ruelle bound for the canonical correlation functions under an appropriate relation between n and v then tightness is shown by using the lyons zheng decomposition the equilibrium measures of the accumulation points are identified as infinite volume canonical gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle infinite volume stochastic dynamics via the associated martingale problem assuming a property closely related to markov uniqueness and weaker than essential self adjointness via mosco convergence techniques we can identify the accumulation points as markov processes and show uniqueness i e all accumulation corresponding to one invariant canonical gibbs measure coincide the proofs work for general repulsive interaction potentials of ruelle type and all temperatures densities and dimensions d 1 respectively may have a nontrivial negative part and infinite range as e g the lennard jones potential additionally our result provides as a by product an approximation of grand canonical gibbs measures by finite volume canonical gibbs measures with empty boundary condition

scaling limit of stochastic dynamics in classical continuous systems we investigate a scaling limit of gradient stochastic dynamics associated with gibbs states in classical continuous systems on rd d 1 the aim is to derive macroscopic quantities from a given microscopic or mesoscopic system the scaling we consider has been investigated by brox in 1980 rost in 1981 spohn in 1986 and guo and papanicolaou in 1985 under the assumption that the underlying potential is in c30 and positive we prove that the dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the dirichlet form of a generalized ornstein uhlenbeck process the proof is based on the analysis and geometry on the configuration space which was developed by albeverio kondratiev and rockner in 1998 and works for general gibbs measures of ruelle type hence the underlying potential may have a singularity at the origin only has to be bounded from below and may not be compactly supported therefore singular interactions of physical interest are covered as for example the one given by the lennard jones potential which is studied in the theory of fluids furthermore using the lyons zheng decomposition we give a simple proof for the tightness of the scaled processes we also prove that the corresponding generators however do not converge in the l2 sense this settles a conjecture formulated by brox by rost and by spohn

the law of the leading digits and the world religions abstract benford s law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger ones we investigate here numerical data on the country wise adherent distribution of seven major world religions i e christianity islam buddhism hinduism sikhism judaism and baha ism to see if the proportion of the leading digits occurring in the distribution conforms to benford s law we find that the adherent data of all the religions except christianity excellently does conform to benford s law furthermore unlike the adherent data on christianity the significant digit distribution of the three major christian denominations i e catholicism protestantism and orthodoxy obeys the law thus in spite of their complexity general laws can be established for the evolution of religious groups

the perturbations of a class of hyper elliptic hamilton systems with a double homoclinic loop through a nilpotent saddle abstract in this paper we consider the upper bounds of a number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 under the perturbations of 0 1 x 2 x 2 3 x 3 4 x 4 y y where 0 1 and i r i 0 4 the unperturbed system has a double homoclinic loop with a nilpotent saddle of order 1 the sharp upper bounds are obtained for each of the cases of 1 4 0 1 3 0 2 3 0 and 3 4 0 when abelian integrals are defined in the bounded period annuli

new lower bounds for the hilbert number of polynomial systems of lienard type abstract in this paper we study the maximal number of limit cycles of some kinds of polynomial lienard systems with arbitrary degree and obtain some new lower bounds for the hilbert number of the systems which improve truly the certain existing results

multiple limit cycles of some strongly nonlinear lienard van der pol oscillator in this paper we investigate limit cycles of some lienard van der pol oscillator the system has a double homoclinic loop and two cuspidal loops if the damping effect has vanished by the related melnikov function theory and bifurcation theories the limit cycles near the singular circle and center with their distribution are found the number of limit cycles obtained also reveals that some recent results on the lower bounds of the maximal number of limit cycles bifurcated from this kind of systems can be improved

limit cycles near an eye figure loop in some polynomial lienard systems abstract in this paper we study the number of limit cycles in the family d h e 0 where h y 2 2 0 x g u d u y f x d x with g x x x 2 1 x 2 1 4 2 and f x an even polynomial of degree 10 we will consider mainly the bifurcation of limit cycles near the eye figure loop and the center of d h 0 our investigation focuses on the lower bound of the maximal number of limit cycles for these systems in particular we show that the perturbed system can have at least 8 limit cycles when deg f x 10