the monotonicity of the period function for planar hamiltonian vector fields if the hamiltonian system with hamiltonian h x y 12y2 v x has a center at the origin it is shown that the period function for the family of periodic trajectories surrounding the center is monotone when the function v v 2 is convex this theorem is applied to determine existence and uniqueness of solutions for the neumann boundary value problem x a x 2 bx c x 0 x 1 0

identifying weak foci and centers in the maxwell bloch system abstract in this paper we identify weak foci and centers in the maxwell bloch system a three dimensional quadratic system whose three equilibria are all possible to be of center focus type applying irreducible decomposition and the isolation of real roots in computation of algebraic varieties of lyapunov quantities on an approximated center manifold we prove that at most 6 limit cycles arise from hopf bifurcations and give conditions for exact number of limit cycles near each weak focus further applying algorithms of computational commutative algebra we find darboux polynomials and give some center manifolds in closed form globally on which we identify equilibria to be centers or singular centers by integrability and time reversibility on a center manifold we prove that those centers are of at most second order

unique normal forms for planar vector fields soit x un germe d un champ vectoriel analytique ou formel en equilibre a o r 2 avec des valeurs propres purement imaginaires a 0 i a 0 0 alors x est formellement equivalent a precisement un des champs vectoriels polynomiaux suivant ou x 1 2 cos y 1 2 sin pour des coordonnees appropriees x y 7b a 0 a 0 7b a k a 0 k 7b b j a 0 a 1 a j j j 1 b j j 0

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

classification of the centers their cyclicity and isochronicity for the generalized quadratic polynomial differential systems in this paper we classify the centers the cyclicity of their hopf bifurcation and the isochronicity of the polynomial differential systems in r2 of degree d that in complex notation z x iy can be written as z i z zz d 22 az2 bzz cz 2 where d 2 is an arbitrary even positive integer r and a b c c note that if d 2 we obtain the well known class of quadratic polynomial differential systems which can have a center at the origin

general framework for phase synchronization through localized sets we present an approach which enables one to identify phase synchronization in coupled chaotic oscillators without having to explicitly measure the phase we show that if one defines a typical event in one oscillator and then observes another one whenever this event occurs these observations give rise to a localized set our result provides a general and easy way to identify ps which can also be used to oscillators that possess multiple time scales we illustrate our approach in networks of chemically coupled neurons we show that clusters of phase synchronous neurons may emerge before the onset of phase synchronization in the whole network producing a suitable environment for information exchanging furthermore we show the relation between the localized sets and the amount of information that coupled chaotic oscillator can exchange

routes to complete synchronization via phase synchronization in coupled nonidentical chaotic oscillators we study the transition route to complete synchronization through phase synchonization in generic coupled nonidentical chaotic oscillators through numerical studies two routes are found i e one via lag synchronization the other via the intermittent chaotic burst state without lag synchronization we claim that these two routes are universal as evidence we analyze several examples on the basis of the conventional theory of intermittency in the presence of noise

characterization in bi parameter space of a non ideal oscillator abstract we investigate the dynamical behavior of a non ideal duffing oscillator a system composed of a mass spring pendulum driven by a dc motor with limited power supply to identify new features on duffing oscillator parameter space due to the limited power supply we provide an extensive numerical characterization in the bi parameter space by using lyapunov exponents following this procedure we identify remarkable new organized distribution of periodic windows the ones known as arnold tongues and also shrimp shaped structures in addition we also identify intertwined basins of attraction for coexisting multiple attractors connected with tongues

on the period function in a class of generalized lotka volterra systems in this note motivated by the recent results of wang et al wang et al local bifurcations of critical periods in a generalized 2d lv system appl math comput 214 2009 17 25 we study the behaviour of the period function of the center at the point 1 1 of the planar differential systemu u p 1 v q v email protected q u p 1 where p q email protected r with pq 0 and m 0 our aim is twofold firstly we determine regions in the parameter space for which the corresponding system has a center with a monotonic period function secondly by taking advantage of the results of wang et al we show some properties of the bifurcation diagram of the period function and we make some comments for further research the differential system under consideration is a generalization proposed by farkas and noszticzius of the lotka volterra model

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

linear series with cusps and n fold points a linear series v s on a curve x has an n fold point along a divisor d of degree n if dim vr h x d dim v 1 the linear series has a cuspofordere atapoint p if dim frch0 xs e l p dim z l linear series with cusps and n fold points are shown to exist if certain inequalities are satisfied the dimensions of the families of linear series with cusps are determined for general curves introduction throughout this paper we work over the complex numbers c let x be a smooth projective curve of genus g a gd on x is a linear series of degree d and dimension r on x i e a pair v s consisting of a line bundle s of degree d on x and an r l dimensional subspace vof h x s when p g r d g r 1 g r d o then x has gd s and the family of gd s on x forms a projective scheme gd x of dimension p g r d furthermore if x is general in moduli then gd x is smooth of dimension p g r d acgh definitic n if d is an effective divisor of degree n 2 on x we say that a gd v s has an n fold point along d if dim v n h x 2 d r we say that a gd has a cusp of order e at a point p if it has an e l fold point along the divisor e 1 p the aim of this paper is to prove the following i if g n p g r d n 1 r n o and p g r 1 d n o then there exists a gd on x with an n fold point along some divisor d of degree n ii if p g r d n 1 r o p g r 1 d n 0 and d is any divisor of degree n on x then there exists a gd on x with an n fold point along d iii if p g r d er 1 0 and p g r 1 d e 1 o then there exists a gd on x with a cusp of order e at some point p iv if x is general in moduli then the family of gd s on x with a cusp of order e has dimension p g r d er 1 if it is nonempty assertion iv has been proved independently by marc coppens c it is easy to see that the hypotheses p g r 1 d n 0 in assertions i and ii and p g r 1 d e 1 0 in assertion iii are necessary if x is general in moduli received by the editors january 20 1987 1980 mclthemcltlcs subject clclsslficcltion 1985 revision primary 14h10 1987 american mathematical society 0002 9947 87 1 00 25 per page

bifurcation analysis on a class of three dimensional quadratic systems with twelve limit cycles abstract this paper concerns bifurcation of limit cycles in a class of 3 dimensional quadratic systems with a special type of symmetry normal form theory is applied to prove that at least 12 limit cycles exist with 6 6 distribution in the vicinity of two singular points yielding a new lower bound on the number of limit cycles in 3 dimensional quadratic systems a set of center conditions and isochronous center conditions are obtained for such systems moreover some simulations are performed to support the theoretical results

the center conditions and local bifurcation of critical periods for a lienard system in this paper we study the problems of centers and isochronous centers and the local bifurcation of critical periods for a lienard system with forth damping calculating the singular point values and period constants we find all center conditions and isochronous center conditions moreover the numbers of local critical periods bifurcating from centers and isochronous centers is obtained by computing the orders of weak centers

an approach to solving systems of polynomials via modular arithmetics with applications the objective of this paper is twofold first we describe a method to solve large systems of polynomial equations using modular arithmetics then we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations

the period function of generalized loudʼs centers abstract in this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied this family is an extension to higher degree of loud s systems the origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function we study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period giving also its character maximum or minimum finally we propose a complete conjectural bifurcation diagram of the period function of these generalized loud s centers

a criticality result for polycycles in a family of quadratic reversible centers abstract we consider the family of dehomogenized loud s centers x y x 1 x x d x 2 f y 2 y where d f r 2 and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus this number is defined exactly the same way as the well known notion of cyclicity of a limit periodic set and we call it criticality the previous results on the issue for the family x r 2 distinguish between parameters with criticality equal to zero regular parameters and those with criticality greater than zero bifurcation parameters a challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters which form a set b of codimension 1 in r 2 in the present paper we succeed in proving that a subset of b has criticality equal to one

isochronous centers of lienard type equations and applications abstract in this work we study eq e x f x x 2 g x 0 with a center at 0 and investigate conditions of its isochronicity when f and g are analytic not necessary odd a necessary and sufficient condition for the isochronicity of 0 is given this approach allows us to present an algorithm for obtained conditions for a point of e to be an isochronous center in particular we find again by another way the isochrones of the quadratic loud systems l d f we also classify a 5 parameters family of reversible cubic systems with isochronous centers

kukles revisited advances in computing techniques we revisit the kukles system to show how advances in computer hardware and software have improved symbolic calculations we confirm directly that there are no non persistent centres for the kukles system we also prove an earlier conjecture regarding the isochronous centres for the extended kukles system we introduce a technique whereby modular resultants can be used to investigate the properties of resultants that cannot be readily calculated using the existing software

metamorphoses of phase portraits of vector field in the case of symmetry of order 4 to investigate the topological structure of hopf bifurcations in a periodically time dependent system one usually makes use of its normal form which can be reduced into two dimensional space by using center manifold method 1 2 the truncated normal form is usually time independent and the vector field induced by such normal form is invariant with respect to some group of symmetry it is well known that the nondegenerate bifurcations of phase portraits with the symmetry of order other than 4 have been well investigated 3 4 but the obtained results about the bifurcations in the case of order 4 are still partial and far from complete

higher order limit cycle bifurcations from non degenerate centers abstract the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number has led in recent works to consider only the lowest terms of these constants in this work we improve the results obtained in this direction for polynomials systems of the form x y p n x y y x q n x y where p n and q n are a homogeneous polynomial of degree n we use center bifurcation to estimate the cyclicity of a unique singular point of focus center type for different values of n and compare with the results given by the conjecture presented in 15

linearizability and local bifurcation of critical periods in a cubic kolmogorov system since chicone and jacobs investigated local bifurcation of critical periods for quadratic systems and newtonian systems in 1989 great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation this paper is devoted to the linearizability and local bifurcation of critical periods for a cubic kolmogorov system we use the darboux method to give explicit linearizing transformations for isochronous centers investigating the finite generation for the ideal of all period constants which are of the polynomial form in six parameters we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center moreover we prove that the maximum number of critical periods is reachable

the successive derivatives of the period function of a plane vector field previously we provided an expression which generalized the classical melnikov function to any order for the first nonzero derivative of a return mapping our method relied on the decomposition of a 1 form associated to the relative cohomology of the perturbed hamiltonian with the same techniques we give a formula for the first nonzero derivative of a period function we focus on the particular example ofh 1 2 x2 y2 and then we define a class of hamiltonians for which the same computation remains valid finally we investigate relations with birkhoff normal form

bifurcation of limit cycles and integrability of planar dynamical systems in complex form we describe a new algorithm for the calculation of liapunov quantities for two dimensional systems in terms of the complex form of the system it is more transparent conceptually and more efficient computationally we demonstrate the advantages of this approach by deriving the integrability conditions and information about the bifurcation of limit cycles for some cubic systems

algebraic families of analytic functions i the aim of this paper is to consider analytic functions arising in connection with algebraic differential equations from the following point of view taylor coefficients of such functions usually turn out to be polynomials or rational functions in the parameters of the problem following pioneering work of bautin 1 we relate analytic properties of the functions considered with algebraic properties of various ideals generated by their taylor coefficients bautin s approach has been successfully applied to the study of the poincare mapping of the planar polynomial systems in 4 9 27 and to the study of the period function of polynomial hamiltonian systems in 5 6 12 13 however the study of the poincare mapping of the polynomial differential equations closely related to the second part of hilbert 16th problem is probably the most difficult among a wide variety of problems to which bautin s approach can be applied in particular counting zeroes of solutions of algebraic differential equations is an important and mostly open problem to our best knowledge the first application of bautin s method to this problem was given in 26 in the present paper we systematically apply this method in the following two situations to algebraic functions and to solutions of the first order polynomial differential equations higher order linear equations are also considered in section 5 article no de963250

local bifurcations of critical periods for cubic lienard equations with cubic damping continuing chicone and jacobs work for planar hamiltonian systems of newton s type in this paper we study the local bifurcation of critical periods near a nondegenerate center of the cubic lienard equation with cubic damping and prove that at most 2 local critical periods can be produced from either a weak center of finite order or the linear isochronous center and that at most 1 local critical period can be produced from nonlinear isochronous centers

restricted independence in displacement function for better estimation of cyclicity abstract since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a center focus equilibrium in this paper we consider a restricted independence to a parametric curve which gives a method not only to increase the lower bound for the cyclicity of the center focus equilibrium but also to be available when those focal values are not independent we apply this method to a nondegenerate cubic center focus variety and prove that the cyclicity reaches its an upper bound

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

local bifurcations of critical periods in a generalized 2d lv system abstract in this paper we investigate a generalized two dimensional lotka volterra system which has a center we give an inductive algorithm to compute polynomials of periodic coefficients find structures of solutions for systems of algebraic equation corresponding to isochronous centers and weak centers of finite order and derive conditions on parameters under which the considered equilibrium is an isochronous center or a weak center of finite order we show that with appropriate perturbations at most two critical periods bifurcate from the center

limit cycle bifurcations from a non degenerate center in this work we discuss the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number moreover we calculate the minimum number of bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators in particular we consider polynomial systems of the form x y pn x y y x qn x y where pn and qn are a homogeneous polynomial of degree n we use center bifurcation rather than multiple hopf bifurcations used a previous work 19 to estimate the cyclicity of a unique singular point of focus center type for n 4 5 6 7 and compare with the results given by the conjecture presented in 18

isochronicity into a family of time reversible cubic vector fields in this work we study necessary and sufficient conditions for the existence of isochronous centers into a family of cubic time reversible systems this class of reversible systems is characterized by the existence of an inverse integrating factor which is a certain power of an invariant straight line

reachability of maximal number of critical periods without independence abstract the reachability of the maximal number of critical periods bifurcated from a weak center is judged with independence of period quantities since the independence is a sufficient condition in this paper we give a method to judge the reachability without the independence our main idea is to consider how the number of zeros for derivative of the period function changes as a curve in the parameter space scans near a specific point and choose an appropriate curve to find the largest number of zeros we apply this method to two polynomial differential systems having zero dimensional isochronicity variety and nonzero dimensional one separately to show that the maximal number is reached and obtain greater numbers than those obtained by classical independence separately with our method the two examples give the possibility that more critical periods arise from an isochronous center than a finite order weak center an opposite fact to many known results

phase portraits of separable hamiltonian systems abstract we study a generalization of potential hamiltonian systems h x y y 2 f x with one degree of freedom namely those with hamiltonian functions of type h x y f x g y which will be denoted by x h we present an algorithm to obtain the phase portrait including the behaviour at infinity of x h when f and g are arbitrary polynomials indeed from the graphs of the one variable functions f and g we are able to give the full description on the poincare disk therefore extending the well known method to obtain the phase portrait of potential systems in the finite plane the fact that the phase portraits can be fully described in terms of the two one variable real functions f and g allows as well a complete study of the bifurcation diagrams in complete families of vector fields the algorithm can be applied to study separable hamiltonian systems with one degree of freedom which include a vast amount of examples in physical applications

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

on the bifurcation of double homoclinic loops of a cubic system abstract a cubic symmetric system x y e a 1 x a 2 a 3 1 x 3 b 1 3 x 2 y a 2 3 a 3 1 x y 2 b 1 1 y 3 y x x 3 e a 1 y b 1 1 x 3 a 2 3 a 3 1 x 2 y b 1 3 x y 2 a 2 a 3 1 y 3 is considered by computing the focus quantities and saddle quantities we get the quantities which determine the stability of the double homoclinic loops appearing then combining hopf bifurcation and double homoclinic loop bifurcation we prove that seven limit cycles can bifurcated from the double homoclinic loops in the above cubic system as far as we know this result on the number of limit cycles bifurcated from double homoclinic loops of the cubic system is new

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

bifurcation of limit cycles and integrability conditions for 6 parameter families of polynomial vector fields of arbitrary degree abstract we characterize the center variety and the maximum order of any focus of the 6 parameter family of real planar polynomial vectors given in complex notation by z i z a z n 1 z j 1 b z n 2 z j 2 c z n 3 z j 3 where a b c c 0 n 1 j 1 n 2 j 2 n 3 j 3 n k j k 1 for k 1 2 3 n 1 j 1 n 2 j 2 n 3 j 3 1 n 3 j 3 1 n 2 j 2 1 n 1 j 1 and 1 n 1 j 1 1 n 2 j 2 0

bi center problem for some classes of z2 equivariant systems we investigate the simultaneous existence of two centers and their isochronicity for two families of planar z2 equivariant differential systems first for a family studied by liu and li 2011 we present the necessary and sufficient conditions for the existence of an isochronous bi center next we give conditions for the existence of a bi center and study its isochronicity for a planar z2 equivariant quintic system having two weak foci or centers we also give two examples of non hamiltonian cubic systems with three isochronous centers

the period function of classical lienard equations in this paper we study the number of critical points that the period function of a center of a classical lienard equation can have centers of classical lienard equations are related to scalar differential equations x x f x x 0 with f an odd polynomial let us say of degree 2 1 we show that the existence of a finite upperbound on the number of critical periods only depending on the value of can be reduced to the study of slow fast lienard equations close to their limiting layer equations we show that near the central system of degree 2 1 the number of critical periods is at most 2 2 we show the occurrence of slow fast lienard systems exhibiting 2 2 critical periods elucidating a qualitative process behind the occurrence of critical periods it all provides evidence for conjecturing that 2 2 is a sharp upperbound on the number of critical periods we also show that the number of critical periods multiplicity taken into account is always even

conditions for the existence of a center for the kukles homogeneous systems abstract in this work we study necessary and sufficient conditions for the existence of centers in two families of linear centers with homogeneous quartic and quintic nonlinearities systems of this class are called kukles homogeneous systems systems of this type were studied for the first time by kukles who studied a linear center with cubic nonhomogeneous nonlinearities

the period in the lotka volterra system is monotonic abstract periodic solutions in a class of hamiltonian systems with one degree of freedom containing the lotka volterra system are considered their period is represented as an integral appropriate for numerical evaluation as well as for theoretical investigations the monotonicity of the period in the lotka volterra system follows as a simple consequence

type ii intermittency in the driven double scroll circuit abstract in this work we show experimental evidences confirmed by numerical results from type ii intermittency in the driven double scroll circuit numerically we found a new scaling power law dependence on the critical parameter this result is a consequence of the new global bifurcation scenario for the t 2 torus breakdown observed in this system a homoclinic saddle connection is the nonlinear mechanism responsible for the reinjection of the trajectory around a repelling focus in fact in this global scenario the total laminar phase is the spiraling laminar period usually considered plus the time the trajectory spends in the vicinity of the saddle points

existence of solitary waves and periodic waves for a perturbed generalized bbm equation abstract the existence of solitary waves and periodic waves for a perturbed generalized bbm equation is established by using geometric singular perturbation theory attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree four exhibiting a cuspidal loop it is proven that the wave speed c 0 h is decreasing on h 0 1 12 by analyzing the ratio of abelian integrals the upper and lower bounds of the limit wave speed are given moreover the relation between the wave speed and the wavelength of traveling waves is obtained

upper bounds in phase synchronous weak coherent chaotic attractors an approach is presented for coupled chaotic systems with weak coherent motion from which we estimate the upper bound value for the absolute phase difference in phase synchronous states this approach shows that synchronicity in phase implies synchronicity in the time of events a characteristic explored to derive an equation to detect phase synchronization based on the absolute difference between the time of these events we demonstrate the potential use of this approach for the phase coherent and the funnel attractor of the rossler system as well as for the spiking bursting rulkov map

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

bifurcation of local critical periods in the generalized loud s system abstract we study the bifurcation of local critical periods in the differential system x y bx n 1 y y x dx n fx n 2 y 2 where b d f r and n 3 is a fixed natural number here by local we mean in a neighbourhood of the center at the origin for n even we show that at most two local critical periods bifurcate from a weak center of finite order or from the linear isochrone and at most one local critical period from a nonlinear isochrone for n odd we prove that at most one local critical period bifurcates from the weak centers of finite or infinite order in addition we show that the upper bound is sharp in all the cases for n 2 this was proved by chicone and jacobs in c chicone m jacobs bifurcation of critical periods for plane vector fields trans am math soc 312 1989 433 486 and our proof strongly relies on their general results about the issue

the period function of reversible quadratic centers abstract in this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers namely the dehomogenized loud s systems the local bifurcation diagram of the period function at the center is fully understood using the results of chicone and jacobs c chicone m jacobs bifurcation of critical periods for plane vector fields trans amer math soc 312 1989 433 486 most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center the techniques that we use here are different from the ones in c chicone m jacobs bifurcation of critical periods for plane vector fields trans amer math soc 312 1989 433 486 because while the period function extends analytically at the center it has no smooth extension to the polycycle at best one can hope that it has some asymptotic expansion another major difficulty is that the asymptotic development has to be uniform with respect to the parameters in order to prove that a parameter is not a bifurcation value we study also the bifurcations in the interior of the period annulus and we show that there exist three germs of curves in the parameter space that correspond to this type of bifurcation moreover we determine some regions in the parameter space for which the corresponding period function has at least one or two critical periods finally we propose a complete conjectural bifurcation diagram of the period function of the dehomogenized loud s systems our results can also be viewed as a contribution to the proof of chicone s conjecture c chicone review in mathscinet ref 94h 58072

the period function for hamiltonian systems with homogeneous nonlinearities abstract the paper deals with hamiltonian systems with homogeneous nonlinearities we prove that such systems have no isochronous centers that the period annulus of any of its centres is either bounded or the whole plane and that the period function associated to the origin has at most one critical point

from isochronous potentials to isochronous systems abstract there is a wide literature involving the study of isochronous equations of the type x t v x t 0 where v is a c 2 function in this paper we show how the kinetic energy t y 1 2 y 2 can be modified still preserving the isochronicity property of the corresponding system more generally we provide estimates for the periods and show an application to the steen s equation and other systems related to the anharmonic potential v x a x 2 b x 2

isochronicity for a class of reversible systems abstract this paper deals with the relation between isochronicity and first integral for a class of reversible systems x u x y y f x y which associates to the first integral of the form h x y f x y 2 g x two necessary and sufficient conditions are given to characterize isochronicity for these systems moreover we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n

limit cycles in 4 star symmetric planar piecewise linear systems abstract our interest is centered in the study of the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is x y 0 we consider the symmetric case that is one vector field defined in the odd quadrants and the other in the even ones we deal with equilibrium points of center focus type with matrices in real jordan form in each vector field when the infinity is monodromic in this case we provide the center classification at infinity we prove that the maximum order of a weak focus is five moreover we show the existence of systems exhibiting five limit cycles bifurcating from infinity

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

decomposition of algebraic sets and applications to weak centers of cubic systems there are many methods such as grobner basis characteristic set and resultant in computing an algebraic set of a system of multivariate polynomials the common difficulties come from the complexity of computation singularity of the corresponding matrices and some unnecessary factors in successive computation in this paper we decompose algebraic sets stratum by stratum into a union of constructible sets with sylvester resultants so as to simplify the procedure of elimination applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center we determine the order of weak centers and discuss the bifurcation of critical periods

isochronicity for several classes of hamiltonian systems abstract in this paper we study isochronous centers of analytic hamiltonian systems giving special attention to the polynomial case we first revisit the potential systems and we show the connection between isochronicity and involutions we then study a more general system namely the ones associated to hamiltonians of the form h x y a x b x y c x y 2 as an application we classify the cubic polynomial hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial hamiltonian isochronous centers

computing center conditions for vector fields with constant angular speed we investigate the planar analytic systems which have a center focus equilibrium at the origin and whose angular speed is constant the conditions for the origin to be a center in fact an isochronous center are obtained concretely we find conditions for the existence of a cw commutator of the field we cite several subfamilies of centers and obtain the centers of the cuartic polynomial systems and of the families y x h1 hm x y h1 hm t and y x h2 h2n x y h2 h2n t with hi homogeneous polynomial in x y of degree i in these cases the maximum number of limit cycles which can bifurcate from a fine focus is determined

nondegenerate centers for abel polynomial differential equations of second kind in this paper we study the center problem for abel polynomial differential equations of second kind computing the focal values and using modular arithmetics and grbner bases we find the center conditions for such systems for lower degrees

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

weak centers and bifurcation of critical periods in reversible cubic systems abstract in this paper we investigate nonhomogeneous cubic differential systems with reversibility which guarantees that the systems have a center at the origin we apply the ritt wu method to process algebraic equations and inequalities using maple v 3 on a computer we give an inductive algorithm for computing the period coefficient polynomials we find the structure of solutions of systems of algebraic equations corresponding to isochronous centers and weak centers of every finite order and we derive conditions on the parameters under which the origin is an isochronous center or a weak center of finite order we show that with appropriate perturbations local bifurcation of critical periods will occur from weak centers of finite order and isochronous centers

on urabe s criteria of isochronicity we give a short proof of urabe s criteria for the isochronicity of periodical solutions of the equation we show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must all be non polynomial and not symmetric an even function of the coordinate x

a class of reversible cubic systems with an isochronous center abstract we study cubic polynomial differential systems having an isochronous center and an inverse integrating factor formed by two different parallel invariant straight lines such systems are time reversible we find nine subclasses of such cubic systems see theorem 8 we also prove that time reversible polynomial differential systems with a nondegenerate center have half of the isochronous constants equal to zero see theorem 3 we present two open problems

limit cycles of a predator prey model with intratrophic predation abstract we present some properties of a differential system that can be used to model intratrophic predation in simple predator prey models in particular for the model we determine the maximum number of limit cycles that can exist around the only fine focus in the first quadrant and show that this critical point cannot be a centre

limit cycles in a disease transmission model we show that certain disease transmission systems have stable limit cycles through hopf bifurcation two parameter families of systems with stable limit cycles are described

non transitive maps in phase synchronization abstract concepts from ergodic theory are used to describe the existence of special non transitive maps in attractors of phase synchronous chaotic oscillators in particular it is shown that for a class of phase coherent oscillators these special maps imply phase synchronization we illustrate these ideas in the sinusoidally forced chua s circuit and two coupled rossler oscillators furthermore these results are extended to other coupled chaotic systems in addition a phase for a chaotic attractor is defined from the tangent vector of the flow finally it is discussed how these maps can be used for the real time detection of phase synchronization in experimental systems

phase synchronization in the perturbed chua circuit we show experimental and numerical results of phase synchronization between the chaotic chua circuit and a small sinusoidal perturbation experimental real time phase synchronized states can be detected with oscilloscope visualization of the attractor using specific sampling rates arnold tongues demonstrate robust phase synchronized states for perturbation frequencies close to the characteristic frequency of the unperturbed chua

global bifurcation destroying the experimental torus t 2 we show experimentally the scenario of a two frequency torus t 2 breakdown in which a global bifurcation occurs due to the collision of a torus with an unstable periodic orbit creating a heteroclinic saddle connection followed by an intermittent behavior

critical period bifurcations of a cubic system we study bifurcations of the period function of a linear centre perturbed by third degree homogeneous polynomials the approach is based on making use of algorithms of computational algebra

differential elimination with dixon resultants in this paper we apply the idea of dixon resultant to algebraic differential equations and introduce the dixon differential resultant we prove that a necessary condition for the existence of a common solution of two algebraic differential equations is that the differential resultant is equal to zero which actually provides a method of elimination and reduces a system of multi variate differential equations to a system of single variate differential equations this result is also generalized to the system of n differential polynomials we give algorithms to realize our method of elimination for systems of differential equations our results and algorithms are demonstrated by some examples

keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

a class of hamiltonian systems with increasing periods on considere un systeme hamiltonien de la forme x g y y f x ou h f et h g satisfont certaines hypotheses on montre qu il y a un intervalle maximal 0 a a 0 tel que chaque solution x t y t avec x 0 a 0 a y 0 0 est periodique avec son orbite enfermant 0 0 soit p a la plus petite periode de cette solution alors p 0 a r est differentiable avec p a 0

global bifurcations of limit and separatrix cycles in a generalized lienard system abstract we construct a canonical cubic dynamical system of kukles type and carry out the global qualitative analysis of its special case corresponding to a generalized lienard equation in particular we prove that the foci of such a lienard system can be at most of second order and that this particular system can have at least three limit cycles in the whole phase plane moreover unlike all previous works on the kukles type systems we study global bifurcations of limit and separatrix cycles using arbitrary including as large as possible field rotation parameters of our canonical system as a result we have obtained a classification of all possible types of separatrix cycles for this generalized lienard system and also all possible distributions of its limit cycles

computing centre conditions for certain cubic systems abstract we present necessary and sufficient conditions for a critical point of certain two dimensional cubic differential systems to be a centre extensive use of the computer algebra system reduce is involved the search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations it has proved to be a difficult problem and full conditions are known for very few classes of systems such conditions are also required in the investigation of hilbert s sixteenth problem concerning the number of limit cycles of polynomial systems

noise scaling of phase synchronization of chaos we investigate the effect of noise on phase synchronization of coupled chaotic oscillators it is found that additive white noise can induce phase slips in integer multiples of 2p s in parameter regimes where phase synchronization is observed in the absence of noise the average time duration of the temporal phase synchronization scales with the noise amplitude in a way that can be described as superpersistent transient we give two independent heuristic derivations that yield the same numerically observed scaling law pacs number s 05 45 xt the phenomenon of synchronous chaos has attracted much attention since the work of pecora and carroll in 1990 1 typically when two chaotic oscillators are coupled together synchronization between them can occur when the coupling strength is large enough recently a more delicate type of synchronization phenomenon was discovered by rosenblum pikovsky and kurths 2 this is the phase synchronization of chaotic oscillators which occurs at smaller coupling strength than that required for complete synchronization briefly if trajectories in each chaotic oscillator can be regarded as a rotation then the phase angle of the rotation increases steadily with time u t 5vt1f t where v is the average rotation frequency and f t is a term characterizing chaotic fluctuations as such the rate of increase of phase can be modeled as a drift v plus a zero mean chaotic process in the absence of coupling the phase angles of the two oscillators u 1 t and u 2 t are uncorrelated that is if one measures the difference du t uu 1 t 2u 2 t u one finds that du t increases steadily with time however when a small amount of coupling is present du t can be confined within 2p while the amplitudes of the rotations are still completely uncorrelated the bifurcation that leads to this phase synchronization was subsequently investigated 3 5 the ability of chaotic systems to have phase synchronization has implications on digital communication with chaos using the natural chaotic symbolic dynamics 6 in such a case it is highly desirable to suppress phase diffusions between chaotic communication channels to ensure proper timing for decoding in this brief report we address to what extent phase synchronization can be observed in laboratory experiments by investigating the effect of noise on phase synchronization our principal results are 1 additive white noise a type of noise encountered commonly in experimental situations can induce phase slips in units of 2p between the coupled oscillators which would otherwise be synchronized in phase in the absence of noise and 2 the average time duration between successive phase slips appears to obey a scaling law with the noise amplitude e

the period function of the generalized lotka volterra centers abstract the present paper deals with the period function of the quadratic centers in the literature different terminologies are used to classify these centers but essentially there are four families hamiltonian reversible q 3 r codimension four q 4 and generalized lotka volterra systems q 3 lv chicone c chicone review in mathscinet ref 94h 58072 conjectured that the reversible centers have at most two critical periods and that the centers of the three other families have a monotonic period function with regard to the second part of this conjecture only the monotonicity of the hamiltonian and q 4 families w a coppel l gavrilov the period function of a hamiltonian quadratic system differential integral equations 6 1993 1357 1365 y zhao the monotonicity of period function for codimension four quadratic system q 4 j differential equations 185 2002 370 387 has been proved concerning the q 3 lv family no substantial progress has been made since the middle 80s when several authors showed independently the monotonicity of the classical lotka volterra centers f rothe the periods of the volterra lokta system j reine angew math 355 1985 129 138 r schaaf global behaviour of solution branches for some neumann problems depending on one or several parameters j reine angew math 346 1984 1 31 j waldvogel the period in the lotka volterra system is monotonic j math anal appl 114 1986 178 184 by means of the first period constant one can easily conclude that the period function of the centers in the q 3 lv family is monotone increasing near the inner boundary of its period annulus i e the center itself thus according to chicone s conjecture it should be also monotone increasing near the outer boundary which in the poincare disc is a polycycle in this paper we show that this is true in addition we prove that except for a zero measure subset of the parameter plane there is no bifurcation of critical periods from the outer boundary finally we show that the period function is globally i e in the whole period annulus monotone increasing in two other cases different from the classical one

critical periods of perturbations of reversible rigidly isochronous centers abstract in this paper we discuss bifurcation of critical periods in an m th degree time reversible system which is a perturbation of an n th degree homogeneous vector field with a rigidly isochronous center at the origin we present period bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function this procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods applying our procedure to the case n m 2 we determine the maximum number of critical periods and their location to the case n m 3 we investigate the bifurcation of critical periods up to the first order in e and obtain the expression of the second period bifurcation function when the first one vanishes

computing integrability conditions for a cubic differential system abstract the integrability conditions for certain systems of differential equations p y y q y where p and q are cubic polynomials are obtained using two distinct computational approaches such conditions are required in the investigation of hilbert s 16 th problem and in the development of algorithms for the automatic solution of differential equations

linear series with an n fold point on a general curve a linear series v s on a curve x has an n fold point along a divisor d of degree n if dim v n h x s d dimv 1 the dimensions of the families of linear series with an n fold point are determined for general curves we work over the field of complex numbers c at x be a smooth projective curve a gd on x is a linear series of dimension r and degree d on x i e a pair v s consisting of a line bundle y of degree d and an r 1 dimensional subspace v c h x s the gd s on x are parameterized by a projective scheme gd x if x is general in moduli then dimgd x p g r d g r l g r d acgh defixlition we say that a gd v s has an n fold point along a divisor d of degree n 2 in x if dim vnh x s d r if z x v is a flat proper irreducible family of smooth curves there exists a scheme gd xlv which is projective over v whose fiber over each v e v is gd xu where xv ot 1 v eh 2 it is easy to see that by considering the construction of gd x xv xn xn as the degeneracy locus of a vector bundle map there is a closed subscheme n x v c gd x xv xn xn such that the fiber over each point p1 pn e xvn consists of the gd s on xv with an n fold point along e pj we will write n x if v is a point let xg denote the moduli space of smooth curves of genus g 3 and let u be the open subset of xg corresponding to curves without nontrivial automorphisms let z u be the universal curve over u we will prove the following theorem theorem with g n 2 g 3 r 2 and the notation as above dimn zu p g r d r n 1 n when u is a sufficiently general point of u

dynamics of the two frequency torus breakdown in the driven double scroll circuit in this work we numerically identify three scenarios for the two frequency torus breakdown to chaos using the driven double scroll circuit with varying driving parameters two of these scenarios follow the curryyorke route to chaos for one scenario we identify the transition to chaos through the onset of a heteroclinic tangle and its heteroclinic points in the other scenario chaos appears via period doubling bifurcations the third scenario is through the type ii intermittency for which a quasiperiodic torus grows in size breaks by touching external saddle points and forms a heteroclinic saddle connection these dynamic scenarios have distinct structure evolutions for the curry yorke route chaos appears softly and alternates with phase locking while for the type ii intermittency chaos appears abruptly and is preserved for a large range of the varying driving parameter s1063 651x 98 05210 6

an explicit expression of the first liapunov and period constants with applications in this paper we study systems in the plane having a critical point with pure imaginary eigenvalues and we search for effective conditions to discern whether this critical point is a focus or a center in the case of it being a center we look for additional conditions in order to be isochronous we stress that the essential differences between the techniques used in this work and the more usual ones are basically two the elimination of the integration constants when we consider primitives of functions see also remark 3 2 and the fact that we maintain the complex notation in the whole study thanks to these aspects we reach with relative ease an expression of the first three liapunov constants v3 v5 andv7 and of the first two period ones p2andp4 for a general system as far as we know this is the first time that a general and compact expression ofv7has been given moreover the use of a computer algebra system is only needed in the computation ofv7andp4 these results are applied to give a classification of centers and isochronous centers for certain families of differential equations