synchronization in weighted scale free networks with degree degree correlation abstract we study synchronization phenomena in scale free networks of asymmetrically coupled dynamical systems featuring degree degree correlation in the topology of the connection wiring we show that when the interaction is dominant from the high degree to the low degree or from the low degree to the high degree nodes synchronization is enhanced for assortative or disassortative degree degree correlations

non isochronicity of the center at the origin in polynomial hamiltonian systems with even degree nonlinearities abstract in 2002 x jarque and j villadelprat proved that no center in a planar polynomial hamiltonian system of degree 4 is isochronous and raised a question is there a planar polynomial hamiltonian system of even degree which has an isochronous center in this paper we give a criterion for non isochronicity of the center at the origin of planar polynomial hamiltonian systems moreover the orders of weak centers are determined our results answer a weak version of the question proving that there is no planar polynomial hamiltonian system with only even degree nonlinearities having an isochronous center at the origin

brief paper l2 norm performance index of synchronization and lqr control synthesis of complex networks in this paper the synchronizability problem of dynamical networks is addressed where better synchronizability means faster network synchronization with lower vibration the l 2 norm of the error vector e is taken as a performance index to measure this kind of synchronizability for the equilibrium synchronization case it is shown that there is a close relationship between the l 2 norm of the error vector e and the h 2 norm of the transfer function g of the linearized network about the equilibrium point consequently the effect of the network coupling topology on the h 2 norm of the transfer function g is analyzed finally an lqr controller is designed according to the lqr problem in modern control theory which can drive the whole network to its equilibrium point and meanwhile minimize the l 2 norm of the output of the linearized network

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

synchronization in complex dynamical networks with nonsymmetric coupling based on the work of nishikawa and motter who have extended the well known master stability framework to include non diagonalizable cases we develop another extension of the master stability framework to obtain criteria for global synchronization several criteria for global synchronization are provided which generalize some previous results the jordan canonical transformation method is used in stead of the matrix diagonalization method especially we show clearly that the synchronizability of a dynamical network with nonsymmetric coupling is not always characterized by its second largest eigenvalue even though all the eigenvalues of the nonsymmetric coupling matrix are real furthermore the effects of the asymmetry of coupling on synchronizability of networks with different structures are analyzed numerical simulations are also done to illustrate and verify the theoretical results on networks in which each node is a dynamical limit cycle oscillator consisting of a two cell cellular neural network

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

on the period function of x f x x 2 g x 0 abstract we study the period function t of a center o of the title s equation a sufficient condition for the monotonicity of t or for the isochronicity of o is given such a condition is also necessary when f and g are odd and analytic in this case a characterization of isochronous centers is given some classes of plane systems equivalent to such equation are considered including some kukles systems

period function for a class of hamiltonian systems abstract this paper studies the period function of the class of hamiltonian systems x h y y h x where h x y has the special form h x y f x g y and the origin is a non degenerate center more concretely if t h denotes the period of the periodic orbit contained in h x y h we solve the inverse problem of characterizing all systems with a given function t h we also characterize the limiting behaviour of t at infinity when the origin is a global center and apply this result to prove among other results that there are no nonlinear polynomial isochronous centers in this family

the number and degree distribution of spanning trees in the tower of hanoi graph the number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph such as its reliability communication aspects synchronization and so on however the practical enumeration of spanning trees and the study of their properties remain a challenge particularly for large networks in this paper we study the number and degree distribution of the spanning trees in the hanoi graph we first establish recursion relations between the number of spanning trees and other spanning subgraphs of the hanoi graph from which we find an exact analytical expression for the number of spanning trees of the n disc hanoi graph this result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree then we introduce a vertex labeling which allows to find for each vertex of the graph its degree distribution among all possible spanning trees

an algebraic approach to the classification of centers in polynomial lienard systems abstract we consider the second order lienard system x f x x g x 0 where f x and g x are polynomials which we rewrite in the equivalent two dimensional form x y y g x yf x we show that the local question of whether a critical point of this system is a center can be expressed in terms of global conditions on f and g using these results we give a simple classification of all such centers we also address the problem of the coexistence of centers foci and limit cycles for these systems systems with degenerate centers are also considered

predicting the synchronization time in coupled map networks an analytical expression for the synchronization time in coupled map networks is given by means of the expression the synchronization time for any given network can be predicted accurately furthermore for networks in which the distributions of nontrivial eigenvalues of coupling matrices have some unique characteristics analytical results for the minimal synchronization time are given

onset of synchronization in large networks of coupled oscillators we study the transition from incoherence to coherence in large networks of coupled phase oscillators we present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength we find that under appropriate conditions the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix we show how with an additional assumption a mean field approximation recently proposed is recovered from our results we test our theory with numerical simulations and find that it describes the transition when our assumptions are satisfied we find that our theory describes the transition well in situations in which the mean field approximation fails we study the finite size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase

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

the number of spanning trees of an infinite family of outerplanar small world and self similar graphs abstract in this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar small world and self similar graphs this number is an important graph invariant related to different topological and dynamic properties of the graph such as its reliability synchronization capability and diffusion properties the calculation of the number of spanning trees is a demanding and difficult task in particular for large graphs and thus there is much interest in obtaining closed expressions for relevant infinite graph families we have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree

enumeration of spanning trees in planar unclustered networks abstract among a variety of subgraphs spanning trees are one of the most important and fundamental categories they are relevant to diverse aspects of networks including reliability transport self organized criticality loop erased random walks and so on in this paper we introduce a family of modular self similar planar networks with zero clustering relevant properties of this family are comparable to those networks associated with technological systems having low clustering like power grids some electronic circuits the internet and some biological systems so it is very significant to research on spanning trees of planar networks however for a large network evaluating the relevant determinant is intractable in this paper we propose a fairly generic linear algorithm for counting the number of spanning trees of a planar network using the algorithm we derive analytically the exact numbers of spanning trees in planar networks our result shows that the computational complexity is o t which is better than that of the matrix tree theorem with o m 2 t 2 where t is the number of steps and m is the girth of the planar network we also obtain the entropy for the spanning trees of a given planar network we find that the entropy of spanning trees in the studied network is small which is in sharp contrast to the previous result for planar networks with the same average degree we also determine an upper bound and a lower bound for the numbers of spanning trees in the family of planar networks by the algorithm as another application of the algorithm we give a formula for the number of spanning trees in an outerplanar network with small world features

non isochronicity of the center in polynomial hamiltonian systems abstract in 2002 jarque and villadelprat proved that planar polynomial hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial hamiltonian systems of even degree recently it was proved that a planar polynomial hamiltonian system is non isochronous if a quantity denoted by m 2 m 2 can be computed such that m 2 m 2 0 as a corollary of this criterion the open question was answered for those systems with only even degree nonlinearities in this paper we consider the case of m 2 m 2 0 and give a new criterion for non isochronicity applying the new criterion we also answer the open question for some cases in which some terms of odd degree are included

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

on rayleigh ritz ratios of a generalized laplacian matrix of directed graphs abstract in c w wu algebraic connectivity of directed graphs linear and multilinear algebra in press a generalization of fiedler s notion of algebraic connectivity to directed graphs was presented that inherits many properties of the algebraic connectivity of undirected graphs and has applications to the synchronization of coupled dynamical systems with both constant and time varying coupling however it did not inherit nonnegativity and some of its bounds for combinatorial properties such as maximum directed cut and isoperimetric number are less strict that their undirected counterparts in particular these bounds do not have the corresponding bounds for undirected graphs as limiting cases the purpose of this paper is to present a refinement of this algebraic connectivity which preserve nonnegativity for strongly connected graphs and have bounds which contain the undirected graphs as special cases in particular we study quantities related to a generalized laplacian matrix of directed graphs and obtain bounds on combinatorial properties such as diameter bandwidth and bisection width for general directed graphs finally we give an application to the synchronization of coupled dynamical systems with constant coupling in particular we show that strong enough cooperative coupling will synchronize a network of coupled systems if the underlying directed graph is strongly connected

isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials in this work we study isochronous centers of two dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree a complete classification of the necessary conditions for the time reversible systems of this class is given in order to have an isochronous center at the origin an open problem is stated for the sufficient conditions moreover we find two nonreversible isochronous families from the center cases known all the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods

counting spanning trees of a type of generalized farey graphs abstract the farey graph f n is derived from the famous farey sequence and it is a small world network with a connectivity distribution decaying exponentially by using the matrix tree theorem zhang et al 2012 obtained the exact formula of the number of spanning trees of f n in this paper by using the electrical network method we consider a type of generalized farey graphs and give the exact solution for the number of spanning trees of these generalized farey graphs which generalizes some previous results about the farey graphs

a class of vertex edge growth small world network models having scale free self similar and hierarchical characters abstract the problem of uncovering the internal operating function of network models is intriguing demanded and attractive in researches of complex networks notice that in the past two decades a great number of artificial models are built to try to answer the above mentioned task based on the different growth ways these previous models can be divided into two categories one type possessing the preferential attachment follows a power law p k k 2 3 the other has exponential scaling feature p k k however there are no models containing above two kinds of growth ways to be presented even the study of interconnection between these two growth manners in the same model is lacking hence in this paper we construct a class of planar and self similar graphs motivated from a new attachment way vertex edge growth network operation more precisely the couple of both them we report that this model is sparse small world and hierarchical and then not only is scale free feature in our model but also lies the degree parameter 3 242 out the typical range note that we suggest that the coexistence of multiple vertex growth ways will have a prominent effect on the power law parameter and the preferential attachment plays a dominate role on the development of networks over time at the end of this paper we obtain an exact analytical expression for the total number of spanning trees of models and also capture spanning trees entropy which we have compared with those of their corresponding component elements

clustering and the synchronization of oscillator networks by manipulating the clustering coefficient of a network without changing its degree distribution we examine the effect of clustering on the synchronization of phase oscillators on networks with poisson and scale free degree distributions for both types of networks increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies surprisingly in scale free networks clustering promotes the synchronization of the most connected nodes hubs even though it inhibits global synchronization as a result they show an additional advanced transition instead of a single synchronization threshold this cluster enhanced synchronization of hubs may be relevant to the brain that is scale free and highly clustered

spectra of sparse regular graphs with loops we derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary lengths the implications of our results for the structural and dynamical properties of network models are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization analytical formulas for the spectrum are obtained for specific lengths of the loops

enumeration of spanning trees on contact graphs of disk packings abstract obtaining the number of spanning trees of complex networks is an outstanding challenge since traditional approaches such as calculating the eigenvalues of the matrix and decomposing of spanning subgraphs are awkward or even infeasible for a large scale network the foundation and importance of this quantity relating to some topological and dynamic properties prompt us to explore the role of determinant identities for laplace matrices we introduce the basic electrically equivalent technique to determine an exact analytical expression for the quantity on the contact graph of disk packings which is proposed by zhang et al 2009 our theoretical results shed light on the relationship between the microscopic change of the quantity and topological iteration of the network in particular we compare the entropy of spanning trees on the network with the other two dimensional and three dimensional lattices we show that the new model is a small world scale free network with the maximum entropy so far found in addition our method for employing the electrically equivalent technique to enumerate spanning trees is general and can be easily extended to other complex networks

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

linearizability conditions of time reversible cubic systems abstract in this paper we present the necessary and sufficient conditions for linearizability of the planar time reversible cubic complex system x x p x y y y q x y from these conditions the necessary and sufficient conditions for the origin to be an isochronous center of the time reversible cubic real system u v f u v v u g u v can be obtained thus the isochronous center problem of time reversible cubic systems is solved completely

stabilizing weighted complex networks real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper the v stability problem is investigated for a class of asymmetric weighted coupled networks with nonidentical node dynamics which includes the unweighted network as a special case pinning control is suggested to stabilize such a coupled network the complicated stabilization problem is reduced to measuring the semi negative property of the characteristic matrix which embodies not only the network topology but also the node self dynamics and the control gains it is found that network stabilizability depends critically on the second largest eigenvalue of the characteristic matrix the smaller the second largest eigenvalue is the more the network is pinning controllable numerical simulations of two representative networks composed of non chaotic systems and chaotic systems respectively are shown for illustration and verification

tracing the evolution of physics on the backbone of citation networks many innovations are inspired by past ideas in a nontrivial way tracing these origins and identifying scientific branches is crucial for research inspirations in this paper we use citation relations to identify the descendant chart i e the family tree of research papers unlike other spanning trees that focus on cost or distance minimization we make use of the nature of citations and identify the most important parent for each publication leading to a treelike backbone of the citation network measures are introduced to validate the backbone as the descendant chart we show that citation backbones can well characterize the hierarchical and fractal structure of scientific development and lead to an accurate classification of fields and subfields

synchronization analysis of delayed complex networks with time varying couplings abstract in this paper a new method is presented to analyze the linear stability of the synchronized state in arbitrarily coupled complex dynamical systems with time delays the coupling configurations are not restricted to the symmetric and irreducible connections or the non negative off diagonal links the stability criteria are obtained by using lyapunov krasovskii functional method and subspace projection method these criteria reveal the relationship between coupling matrices and stability of the dynamical networks

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

on local synchronisability of nonlinear networked systems with a unit inner coupling matrix and switching topology the local synchronisation of the nonlinear networked systems with switching topology is studied in this article for the special case in which the inner coupling matrix is a unit matrix the critical idea is to convert synchronisation problem to consensus problem such that the switching topology can be dealt with effectively without any extra assumption on the relationship among the involved topologies a novel local synchronous criterion consisting of the maximum lyapunov exponent of nodal self dynamics and the consensus convergent rate of the switching topology is given to illustrate the effectiveness of the analytical results two numerical examples on chaos synchronisation are simulated respectively for the periodic switching case and randomly switching case

synchronization is optimal in nondiagonalizable networks we consider maximization of the synchronizability of oscillator networks by assigning weights and directions to the links of a given interaction topology by extending the master stability formalism to all possible network structures we show that unless some oscillator is linked to all the others maximally synchronizable networks are necessarily nondiagonalizable and can always be obtained by imposing unidirectional information flow with normalized input strengths the results provide insights into hierarchical structures observed in complex networks in which synchronization is important

optimization of synchronization in gradient clustered networks we consider complex clustered networks with a gradient structure where the sizes of the clusters are distributed unevenly such networks describe actual networks in biophysical systems and in technological applications more closely than the previous models theoretical analysis predicts that the network synchronizability can be optimized by the strength of the gradient field but only when the gradient field points from large to small clusters a remarkable finding is that if the gradient field is sufficiently strong synchronizability of the network is mainly determined by the properties of the subnetworks in the two largest clusters these results are verified by numerical eigenvalue analysis and by direct simulation of synchronization dynamics on coupled oscillator networks

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

detecting link failures in complex network processes using remote monitoring abstract we study whether local structural changes in a complex network can be distinguished from passive remote time course measurements of the network s dynamics specifically the detection of link failures in a network synchronization process from noisy measurements at a single network component is considered by phrasing the detection task as a maximum a posteriori probability hypothesis testing problem we are able to obtain conditions under which the detection is 1 improved over the a priori and 2 asymptotically perfect in terms of the network spectrum and graph we find that in the case where the detector has knowledge of the network s state perfect detection is possible under general connectivity conditions regardless of the measurement location when the detector does not have state knowledge a remote signature permits improved but not perfect detection under the same connectivity conditions at its essence detectability is achieved because of the close connection between a network s topology its eigenvalues and local response characteristics

abnormal synchronizing path of delay coupled chaotic oscillators on the edge of stability abstract in this paper the transition of synchronizing path of delay coupled chaotic oscillators in a scale free network is highlighted mainly through the critical transmission delay makes chaotic oscillators be coupled on the edge of stability we find that the transition of synchronizing path is abnormal which is characterized by the following evidences a synchronization process starts with low degree rather than high degree ones b the high degree nodes do not undertake the role of hub c the synchronized subnetworks show a poor small world property as a result of hubs absence d the clustering synchronization behavior emerges even community structure is absent in the scale free network this abnormal synchronizing path suggests that the diverse synchronization behaviors occur in the same topology which implies that the relationship between dynamics and structure of network is much more complicated than the common sense that the structure is the foundation of dynamics moreover it also reveals the potential connection from the transition of synchronization behavior to disorder in real complex networks e g alzheimer disease

effects of degree correlation on the controllability of networks abstract in this paper we investigate the effects of degree correlation on the controllability of undirected networks with different degree mixing patterns based on simulation analysis considering random pinning max degree pinning and mix degree pinning some relevance factors on controllability such as the control gain and the number of pinned nodes are discussed in detail it is found that disassortative mixing enhances the network controllability contrast to assortative mixing to which the network controllability is sensible furthermore too large or too low value of the feedback gain can reduce the controllability from the viewpoint of practical application mix degree pinning strategy is suggested in assortative network while in disassortative network max degree pinning scheme is better than random pinning

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

isochronicity and linearizability of planar polynomial hamiltonian systems in this paper we study isochronicity and linearizability of planar polynomial hamiltonian systems first we prove a theorem which supports a negative answer to the following open question stated by jarque and villadelprat in 15 do there exist planar polynomial hamiltonian systems of even degree having an isochronous center additionally we obtain some conditions for linearizability of complex cubic hamiltonian systems

isochronous centers of polynomial hamiltonian systems and a conjecture of jarque and villadelprat abstract we study the conjecture of jarque and villadelprat stating that every center of a planar polynomial hamiltonian system of even degree is nonisochronous this conjecture has already been proved for quadratic and quartic systems using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields we are able to describe a very large class of nonisochronous hamiltonian systems of even arbitrarily large degree

choosing effective controlled nodes for scale free network synchronization abstract previous studies concerning pinning control of complex network synchronization have very often demonstrated that in an unweighted symmetrical scale free network controlling the high degree nodes is more efficient than controlling randomly chosen ones due to the heterogeneity of the node degree or edge connection distribution of the scale free network small degree nodes have relatively high probabilities of being chosen at random but their control has less influence on the other nodes through the network this raises the question of whether or not controlling the high degree nodes is always better than controlling the small ones in scale free networks our answer to this is yes and no in this study we carry out extensive numerical simulations to show that in an unweighted symmetrical barabasi albert scale free network when the portion of controlled nodes is relatively large controlling the small nodes becomes better than controlling the big nodes and controlling randomly chosen nodes has approximately the same effect as controlling the big ones however we also show that for normalized weighted scale free networks controlling the big nodes is in fact always better than controlling the small ones

nonexistence of isochronous centers in planar polynomial hamiltonian systems of degree four abstract in this paper we study centers of planar polynomial hamiltonian systems and we are interested in the isochronous ones we prove that every center of a polynomial hamiltonian system of degree four that is with its homogeneous part of degree four not identically zero is nonisochronous the proof uses the geometric properties of the period annulus and it requires the study of the hamiltonian systems associated to a hamiltonian function of the form h x y a x b x y c x y 2 d x y 3

stochastic synchronization in blinking networks of chaotic maps in this paper we analyze stochastic synchronization of coupled chaotic maps over blinking networks composed of a pristine static network and stochastic on off couplings between any pair of nodes we focus on mean square linear stability of the synchronized state by analyzing the time evolution of the second moment of the variation transverse to the synchronization manifold by projecting the variational equations on the eigenvectors of a higher order state matrix describing this variational dynamics we establish a necessary and sufficient condition for stochastic synchronization based on the largest lyapunov exponent of the map and the spectral radius of such matrix this condition is further simplified by computing closed form results for the spectral properties of the moments of the graph laplacian associated to the intermittent coupling and using classical eigenvalue bounds we illustrate the main results through simulations on synchronization of chaotic henon maps

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