on the computation of the coefficients associated with high order normal forms a new procedure for obtaining high order normal forms and the associated coefficients is presented it is assumed that the jacobian of the system considered is in a diagonal form in comparison with existing normal form approaches this procedure lends itself more readily to symbolic calculations like maple and the calculations of high order normal forms together with the associated coefficients are carried out much more conveniently to illustrate the approach five examples are presented examples 1 and 3 also contain a comparison of the results obtained by the methods of normal forms and averaging

almost planar homoclinic loops in r3 in this paper we study homoclinic loops of vector fields in 3 dimensional space when the two principal eigenvalues are real of opposite sign which we call almost planar we are interested to have a theory for higher codimension bifurcations almost planar homoclinic loop bifurcations generically occur in two versions non twisted and twisted loops we consider high codimension homoclinic loop bifurcations under generic conditions the generic condition forces the existence of a 2 dimensional topological invariant ring non necessarily unique which is a topological cylinder in the non twisted case and a topological mobius band in the twisted case if the third eigenvalue is negative resp positive the limit set resp limit set of trajectories starting in a neighborhood of the loop is contained in the invariant ring the dynamics is then given by fixed points or periodic points of period 2 of a 1 dimensional map which admits a nice asymptotic expansion allowing to treat homoclinic loop bifurcations of arbitrarily high codimension in the non twisted case

limit cycles bifurcated from a class of quadratic reversible center of genus one abstract we investigate the bifurcation of limit cycles in a class of planar quadratic reversible non hamiltonian systems this systems has a center of genus one the exact upper bound of the number of limit cycles is given

analysis on limit cycles of zq equivariant polynomial vector fields with degree 3 or 4 this paper presents a study on the limit cycles of zq equivariant polynomial vector fields with degree 3 or 4 previous studies have shown that when q 2 cubic order systems can have 12 small amplitude limit cycles in this paper particular attention is focused on the cases of q 3 it is shown that for cubic order systems when q 3 there exist 3 small limit cycles and 1 big limit cycle while for q 4 it has 4 small limit cycles and 1 big limit cycle and when q 5 there is only 1 small limit cycle for fourth order systems the cases for even q are the same as the cubic order systems when q 5 it can have 10 small limit cycles while for q 7 there exists only 1 small limit cycle the case q 3 is not considered in this paper numerical simulations are presented to illustrate the theoretical results

bifurcation of small limit cycles in z5 equivariant planar vector fields of order 5 abstract in this paper we consider bifurcation of small limit cycles from hopf type singular points in z 5 equivariant planar vector fields of order 5 we apply normal form theory and the technique of solving coupled multivariate polynomial equations to prove that the maximal number of small limit cycles that such vector fields can have is 25 in addition we show that no large limit cycles exist thus h 5 25 where h n denotes the hilbert number of the n th degree polynomial vector fields this improves the best result of h 5 24 existing in the current literature

bifurcation of small limit cycles in cubic integrable systems using higher order analysis abstract in this paper we present a method of higher order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations this method is equivalent to higher order melinikov function approach used for studying bifurcation of limit cycles around a center but simpler attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by oladek 1995 24 can indeed have eleven limit cycles under perturbations at least up to 7th order moreover the pattern of numbers of limit cycles produced near the center is discussed up to 39th order perturbations and no more than eleven limit cycles are found

computation of normal forms via a perturbation technique considering qualitative behavior of a non linear dynamical system often leads to first simplifying the differential equations or finding their normal forms a perturbation technique for computing normal forms is presented this technique associated with the method of multiple scales can be used to systematically find a unique form for a given set of differential equations the technique is discussed in detail through the analysis of hopf bifurcation it is shown that for hopf bifurcation the method only requires solving two dimensional matrix systems for any higher order normal forms of a generaln dimensional system with the aid of the symbolic language maple this approach is straightforward and is computationally efficient and fast furthermore a simple verification scheme is given for verifying the normal forms and associated non linear transformations obtained using any methodology examples are presented to demonstrate the applicability of the perturbation technique and the computation efficiency of the maple program

a case study for homoclinic chaos in an autonomous electronic circuit a trip taken from takens bogdanov to hopf sbil nikov abstract the dynamics of an autonomous electronic oscillator is analysed both theoretical study with methods of local bifurcation theory and numerical simulation by using continuation methods to detect global behaviours are carried out comment is made on the presence of homoclinic connections which organise the periodic and nonperiodic behaviour the richness and complexity of the periodic oscillations are described and the presence of isolated branches of odd period orbits are emphasized further attention is paid to the chaotic attractor coexisting with sil nikov homoclinicity even in the presence of a hopf degeneracy the hopf sil nikov singularity

a cubic kolmogorov system with six limit cycles abstract we consider a class of cubic kolmogorov systems we show in particular that a maximum of six small amplitude limit cycles can bifurcate from a critical point in the first quadrant and we discuss the number of invariant lines

one point extensions and derived equivalence abstract work of the first author with de la pena m barot j a de la pena proc amer math soc 127 1999 647 655 concerned with the class of algebras derived equivalent to a tubular algebra raised the question whether a derived equivalence between two algebras can be extended to one point extensions the present paper yields a positive answer

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

zeroes of complete elliptic integrals for 1 2 resonance abstract we give an estimate for the number of zeroes of the elliptic integral i h a x y dx b x y dy along the ovals h x y h h x y y 2 2 x 2 x 4 in terms of the degrees of the symmetric polynomials a and b

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

derived tame blowing up of tree algebras abstract let k be an algebraically closed field and a be a tree algebra we consider here a class obtained by the blowing up of a tree algebra a at a set of vertices d of a such an algebra is denoted by a d the objective of this paper is to prove the equivalence between the derived tameness and the non negativity of the euler form for algebras of this form we also show that in this case if d is a non empty set then a d must be derived equivalent to a special incidence algebra called semichain algebra

modeling and analysis of recurrent autoimmune disease many autoimmune diseases are characterized by a pattern of recurrence and remission in which periods of apparent self tolerance are punctuated by intervals of recurring autoimmunity we introduce a newly discovered class of terminally differentiated regulatory t cells hla dr t reg cells into an existing autoimmune disease model our newly developed 4 dimensional model exhibits recurrent dynamics which are preserved in a reduced and rescaled 3 dimensional model as well applying dynamical systems theory we analyze the dynamics underlying this behavior in both the 4 dimensional and 3 dimensional models and further prove that the recurrent behavior or oscillation arises due to a hopf bifurcation or a persistent oscillation rather than from homoclinic orbits numerical simulations are conducted to verify the analytical results and identify the recurrent parameter region

the cyclicity of triangles and segments in quadratic systems abstract the paper deals with quadratic systems having separatrix contour in the form of a triangle or a conic segment the limit cycles appearing near this configuration after quadratic perturbation is studied the maximal number of limit cycles is 3 for the bounded triangle and 2 for the unbounded triangle and for the bounded segment

non linear stability analysis of a complex rotor stator contact system in this paper a non linear strategy based on the centre manifold the rational approximants and the alternating frequency time domain method has been developed in order to study the non linear dynamical behaviour of a system in the neighbourhood of a critical steady state equilibrium point the stability analysis and the non linear dynamics of a complex braking system with a non linear rotor stator contact are presented moreover one of the most important steps of this paper is the determination of the non linear behaviour and the limit cycle amplitudes of this complex system in order to conduct this study the dynamic response is evaluated by using applying the centre manifold the rational approximants and the alternating frequency time domain method that permit to obtain rapidly and efficiently the non linear behaviour of the system the dynamic response obtained by applying this method is compared with that evaluated through numerical integration

linear recursion formulas of quantities of singular point and applications in this paper we study the algorithm of focus quantities and saddle quantities of a real planar autonomous differential system two linear recursion formulas for computation of quantities of singular point are obtained using the recursion formulas to compute focus quantities or saddle quantities of a real planar autonomous differential system we need only make use of four fundamental operation of arithmetic in the coefficients of the system then avoid complex integrating recursion operations which is necessary in more usual approaches therefore the calculation can be readily done with using computer symbol operation system such as mathematica they are far more practical than more usual approaches from the algorithmic point of view and by which the focus quantities and saddle quantities of a polynomial system can be computed and simplified in a much simpler way as a result of straightforward application we obtain with relative ease the first 8 focus quantities at origin of a cubic system and the sufficient and necessary conditions of integrability in the neighborhood of the origin of this system

on hopf cyclicity of planar systems we investigate the maximal number of limit cycles which appear under perturbations in hopf bifurcations by using the first order melnikov function with multiple parameters

the displacement map associated to polynomial unfoldings of planar hamiltonian vector fields we study the displacement map associated to small one parameter polynomial unfoldings of polynomial hamiltonian vector fields on the plane its leading term the generating function m t has an analytic continuation in the complex plane and the real zeroes of m t correspond to the limit cycles bifurcating from the periodic orbits of the hamiltonian flow we give a geometric description of the monodromy group of m t and use it to formulate sufficient conditions for m t to satisfy a differential equation of fuchs or picard fuchs type as examples we consider in more detail the hamiltonian vector fields inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i and inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i possessing a rotational symmetry of order two and three respectively in both cases m t satisfies a fuchs type equation but in the first example m t is always an abelian integral that is to say the corresponding equation is of picard fuchs type while in the second one this is not necessarily true we derive an explicit formula of m t and estimate the number of its real zeroes

normal form computation without central manifold reduction an efficient method for calculating the normal form and the associated non linear transformations for the semi simple case without central manifold reduction is given in this paper the one step transformation concept is adopted for easy programming this method can be used to calculate high order normal forms of high dimensional ordinary differential equations of non linear oscillators a program in mathematica language is designed to perform the calculation three examples are given in order to verify the method and to show the efficiency of the program

bifurcation and stability analyses for a coupled brusselator model abstract this paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two brusselators of particular interest is the study of the associated hopf bifurcation and double hopf bifurcations the motion of the oscillator may either be periodic bifurcating from a hopf type critical point or quasi periodic bifurcating from a compound critical point furthermore bifurcation analysis reveals that the limit cycles associated with the first brusselator are always stable while that generated by the second brusselator may be unstable if the parameter values are chosen far from the stability boundary it is interesting to note that in the vicinity of the double hopf compound critical point there exist periodic as well as quasi periodic solutions the quasi periodic motion is stable for a small parameter region a robust gauss seidel like implicit finite difference method gs1 has been developed and used for the solution of the resulting initial value problem ivp in addition to being of comparable accuracy judging by the similarity of the profiles generated with the fourth order runge kutta method rk4 the gs1 method will be seen to have better numerical stability property than rk4 unlike the rk4 which fails when large time steps are used to integrate the ivp extensive numerical simulations with appropriate initial data suggest that the gs1 method is unconditionally convergent moreover it is more economical computationally

four small limit cycles around a hopf singular point in 3 dimensional competitive lotka volterra systems abstract the 3 dimensional competitive lotka volterra lv systems have been studied for more than two decades and particular attention has been focused on bifurcation of limit cycles for such a system zeeman 1993 identified 33 stable equivalence classes on a carrying simplex among which only classes 26 31 may have limit cycles it has been shown that all these 6 classes may possess two limit cycles and the existence of three limit cycles was claimed in some of these classes recently gyllenberg and yan 2009 studied the existence of four limit cycles three of them are small amplitude limit cycles due to hopf bifurcation and one additional limit cycle enclosing all the three small amplitude limit cycles is due to the existence of a heteroclinic cycle and proposed a new conjecture including i there exists a 3 d competitive lv system with at least 5 limit cycles ii in the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3 d competitive lv system the vanishing of the first four focus values the vanishing of the zero order focus value means that there is a pair of purely imaginary eigenvalues at the positive equilibrium does not imply that the heteroclinic cycle is neutrally stable and hence it does not imply that the positive equilibrium is a center iii in the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3 d competitive lv system the vanishing of the first three focus values and that the heteroclinic cycle is neutrally stable do not imply the vanishing of the third order focus value and hence they do not imply that the positive equilibrium is a center in this paper we will present two examples belonging to class 27 and another two examples belonging to class 26 which exhibit at least four small amplitude limit cycles in the vicinity of the positive equilibrium due to hopf bifurcations and prove that the items ii and iii in the conjecture are true moreover showing the existence of four small amplitude limit cycles is a necessary step towards proving item i of the conjecture

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

superconvergence of discontinuous galerkin and local discontinuous galerkin methods eigen structure analysis based on fourier approach various superconvergence properties of discontinuous galerkin dg and local dg ldg methods for linear hyperbolic and parabolic equations have been investigated in the past due to these superconvergence properties dg and ldg methods have been known to provide good wave resolution properties especially for long time integrations zhong and shu 2011 26 in this paper under the assumption of uniform mesh and via fourier approach we observe that the error of the dg or ldg solution can be decomposed into three parts 1 dissipation and dispersion errors of the physically relevant eigenvalue this part of error will grow linearly in time and is of order 2 k 1 for dg method and 2 k 2 for ldg method 2 projection error there exists a special projection of the exact solution such that the numerical solution is much closer to this special projection than the exact solution itself this part of error will not grow in time 3 the dissipation of non physically relevant eigenvectors this part of error will be damped exponentially fast with respect to the spatial mesh size x along this line we analyze the error for a fully discrete runge kutta rk dg scheme a collection of numerical examples for linear equations are presented to verify our observations above we also provide numerical examples based on non uniform mesh nonlinear burgers equation and high dimensional maxwell equations to explore superconvergence properties of dg methods in a more general setting

generalized hopf bifurcation and its dual generalized homoclinic bifurcation we show the duality between the generalized hopf bifurcation ghb and the generalized homoclinic bifurcation ghb this duality is twofold 1 the poincare normal forms at a weak focus and at a weak saddle 2 the bifurcation diagrams of the ghb and the ghb since the ghb is well known the ghb may be considered as the heart of the article

hopf and homoclinic bifurcations for near hamiltonian systems abstract we study homoclinic bifurcation of limit cycles in perturbed planar hamiltonian systems suppose that a homoclinic loop is defined by h h s our main result is that a new method is established for computing the coefficients of the expansion of melnikov functions at h h s then by using those coefficients more limit cycles would be found around homoclinic loops an example is also provided to illustrate our method

a perturbation method for computing the simplest normal forms of dynamical systems abstract a previously developed perturbation method is generalized for computing the simplest normal form at each level of computation the minimum number of terms are retained of general n dimensional differential equations this direct approach combining the normal form theory with center manifold theory in one unified procedure can be used to systematically compute the simplest or unique normal form two particular singularities of the jacobian of the system are considered in this paper the first one is associated with one pair of purely imaginary eigenvalues hopf type singularity and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues hopf zero type singularity the approach can be easily formulated and implemented using a computer algebra system maple programs have been developed in this paper which can be automatically executed by a user without the knowledge of computer algebra a physical oscillator model is studied in detail to show the computational efficiency of the direct method and the advantage of using the simplest normal form which greatly simplifies the analysis on dynamical systems in particular for bifurcations and stability

on the number of zeros of abelian integrals for a polynomial hamiltonian irregular at infinity abstract up to now most of the results on the tangential hilbert 16th problem have been concerned with the hamiltonian regular at infinity i e its principal homogeneous part is a product of the pairwise different linear forms in this paper we study a polynomial hamiltonian which is not regular at infinity it is shown that the space of abelian integral for this hamiltonian is finitely generated as a r h module by several basic integrals which satisfy the picard fuchs system of linear differential equations applying the bound meandering principle an upper bound for the number of complex isolated zeros of abelian integrals is obtained on a positive distance from critical locus this result is a partial solution of tangential hilbert 16th problem for this hamiltonian as a consequence we get an upper bound of the number of limit cycles produced by the period annulus of the non hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k n under polynomial perturbation of arbitrary degree

perturbation from an asymmetric cubic hamiltonian this paper concerns with the number of limit cycles from an asymmetric hamiltonian of degree three under cubic perturbation eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis two of which are new

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

normal form of double hopf bifurcation in forced oscillators a general four dimensional normal form of a double hopf bifurcation is considered as a particular case the normal form of a forced non autonomous non linear oscillator having two frequencies namely the linear natural frequency and the excitation frequency is studied in detail when these two frequencies form two purely imaginary sub blocks of order two in the real jordan block the system constitutes a double hopf bifurcation in this paper the normal form of the double hopf bifurcation is reduced when the two frequencies are not in resonance in order to use the method of normal form the non autonomous problem is transformed into an autonomous one by a generalized co ordinate transformation the method of undetermined coefficients is used to find the double hopf bifurcation normal form the coefficients of similar monomials rather than similar powers of e are compared to get the normal form to various orders the steady state periodic solutions and the bifurcation equations of the forced non linear vibration system in the case of non resonant are studied a mathematica program is designed to find the normal form three examples are given to use the mathematica program and to compare them with the existing results

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

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

extended canonical algebras and fuchsian singularities the authors introduce a new class of finite dimensional algebras called extended canonical and determine the shape of their derived categories extended canonical algebras arise from a canonical algebra by onepoint extension or coextension by an indecomposable projective module our main results concern the case of negative euler characteristic of the corresponding weighted projective line mathbb x more specifically we establish for a base field of arbitrary characteristic a link to the fuchsian singularity r of mathbb x which for the base field of complex numbers is isomorphic to an algebra of automorphic forms by means of a recent result of orlov we show that the triangulated category of the graded singularities of r in the sense of buchweitz and orlov admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra of particular interest are those cases where the attached coxeter transformation has spectral radius one a k theoretic analysis then shows that this happens exactly for 38 cases including arnold s 14 exceptional unimodal singularities the paper is related to recent independent work by kajiura saito and takahashi

bifurcation of ten small amplitude limit cycles by perturbing a quadratic hamiltonian system with cubic polynomials abstract this paper contains two parts in the first part we shall study the abelian integrals for oladek s example 13 in which the existence of 11 small amplitude limit cycles around a singular point in a particular cubic vector field is claimed we will show that the bases chosen in the proof of 13 are not independent which leads to failure in drawing the conclusion of the existence of 11 limit cycles in this example in the second part we present a good combination of melnikov function method and focus value or normal form computation method to study bifurcation of limit cycles an example by perturbing a quadratic hamiltonian system with cubic polynomials is presented to demonstrate the advantages of both methods and 10 small amplitude limit cycles bifurcating from a center are obtained by using up to 5th order melnikov functions

perturbations of non hamiltonian reversible quadratic systems with cubic orbits abstract this paper is concerned with degree n polynomial perturbations of a class of planar non hamiltonian reversible quadratic integrable system whose almost all orbits are cubics we give an estimate of the number of limit cycles for such a system if the first order melnikov function abelian integral m 1 h is not identically zero then the perturbed system has at most 5 for n 3 and 3 n 7 for n 4 limit cycles on the finite plane if m 1 h is identically zero but the second melnikov function is not then an upper bound for the number of limit cycles on the finite plane is 11 for n 3 and 6 n 13 for n 4 respectively in the case when the perturbation is quadratic n 2 there exists a neighborhood u of the initial non hamiltonian polynomial system in the space of all quadratic vector fields such that any system in u has at most two limit cycles on the finite plane the bound for n 2 is exact

on the structure of poincare lyapunov constants for the weak focus of polynomial vector fields in this paper generalized poincare lyapunov constants vi i 1 2 are defined and an explicit formula is given for the lyapunov functions it is shown that vi is determined modulo v1 vi 1 this is used for the investigation and creation of limit cycles

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

pressure forcing and dispersion analysis for discontinuous galerkin approximations to oceanic fluid flows this paper is part of an effort to examine the application of discontinuous galerkin dg methods to the numerical modeling of the general circulation of the ocean one step performed here is to develop an integral weak formulation of the lateral pressure forcing that is suitable for usage with a dg method and with a generalized vertical coordinate that includes level terrain fitted isopycnic and hybrid coordinates as examples this formulation is then tested in special cases with analyses of dispersion relations and numerical stability and with some computational experiments these results suggest that the advantages of dg methods may significantly outweigh their disadvantages in the settings tested here this paper also outlines some other issues that need to be addressed in future work

a numerical approach to bifurcations from quadratic centers abstract a combination of analytical and numerical work is done to analyze bifurcation of limit cycles from non hamiltonian codimension three quadratic centers the winding curve c 3of cyclicity three separatrix cycles qualitatively located in earlier work is determined numerically evidence is given that the 2 2 3 2 and 3 3 configurations of limit cycles do not bifurcate from this class of quadratic centers

the cyclicity of the period annulus of the quadratic hamiltonian triangle abstract the paper studies quadratic hamiltonian centers surrounded by a separatrix contour having a form of a triangle it is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to three

bifurcation of limit cycles in a cubic order planar system around a nilpotent critical point abstract in this paper bifurcation of limit cycles is considered for planar cubic order systems with an isolated nilpotent critical point normal form theory is applied to compute the generalized lyapunov constants and to prove the existence of at least 9 small amplitude limit cycles in the neighborhood of the nilpotent critical point in addition the method of double bifurcation of nilpotent focus is used to show that such systems can have 10 small amplitude limit cycles near the nilpotent critical point these are new lower bounds on the number of limit cycles in planar cubic order systems near an isolated nilpotent critical point moreover a set of center conditions is obtained for such cubic systems

multiple time scales and pressure forcing in discontinuous galerkin approximations to layered ocean models this paper addresses some issues involving the application of discontinuous galerkin dg methods to ocean circulation models having a generalized vertical coordinate these issues include the following 1 determine the pressure forcing at cell edges where the dependent variables can be discontinuous in principle this could be accomplished by solving a riemann problem for the full system but some ideas related to barotropic baroclinic time splitting can be used to reduce the riemann problem to a much simpler system of lower dimension such splittings were originally developed in order to address the multiple time scales that are present in the system 2 adapt the general idea of barotropic baroclinic splitting to a dg implementation a significant step is enforcing consistency between the numerical solution of the layer equations and the numerical solution of the vertically integrated barotropic equations the method used here has the effect of introducing a type of time filtering into the forcing for the layer equations which are solved with a long time step 3 test these ideas in a model problem involving geostrophic adjustment in a multi layer fluid in certain situations the dg formulation can give significantly better results than those obtained with a standard finite difference formulation

thirteen limit cycles for a class of cubic hamiltonian system with higher order perturbed terms abstract the number and distribution of limit cycles of a cubic hamiltonian system under higher order perturbed terms is investigated by using the bifurcation theory and the method of detection function we obtain that there exist at least 13 limit cycles with the distribution c 7 1 2 c 3 2 2 c 1 2 in the hamiltonian system under the perturbed term of degree 5 additionally various distributions of limit cycles are given by numerical exploration

the cyclicity of the elliptic segment loops of the reversible quadratic hamiltonian systems under quadratic perturbations abstract denote by q h and q r the hamiltonian class and reversible class of quadratic integrable systems there are several topological types for systems belong to q h q r one of them is the case that the corresponding system has two heteroclinic loops sharing one saddle connection which is a line segment and the other part of the loops is an ellipse in this paper we prove that the maximal number of limit cycles which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems called in reversible direction is two we also give the corresponding bifurcation diagram

a unified proof on the weak hilbert 16th problem for n 2 abstract the weak hilbert 16th problem for n 2 was solved by horozov and iliev proc london math soc 69 1994 198 244 zhang and li adv in math 26 1997 445 460 gavrilov invent math 143 2001 449 497 and li and zhang nonlinearity 15 2002 1775 1992 by using different methods for different cases the aim of this paper is to give a unified and easier proof for all cases the proof is restricted to the real domain combines geometric and analytical methods and uses deformation arguments

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

nine limit cycles around a singular point by perturbing a cubic hamiltonian system with a nilpotent center in this paper we study bifurcation of limit cycles in planar cubic near hamiltonian systems with a nilpotent center we use normal form theory to compute the generalized lyapunov constants and show that there exist at least 9 limit cycles around the nilpotent center this is a new lower bound on the number of limit cycles in planar cubic near hamiltonian systems with a nilpotent center

limit cycle bifurcations by perturbing a quadratic integrable system with a triangle abstract finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem which is related to the second part of hilbert s 16th problem this paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle it is proved that the cyclicity of the triangle in quadratic systems is at least three

derived invariance of hochschild mitchell co homology and one point extensions in this article we prove derived invariance of hochschild mitchell homology and cohomology and we extend to k linear categories a result by barot and lenzing concerning derived equivalences and one point extensions we also prove the existence of a long exact sequence a la happel and we give a generalization of this result which provides an alternative approach

modeling cell to cell spread of hiv 1 with logistic target cell growth abstract in this paper we consider a model containing two modes for hiv 1 infection and spread one is the diffusion limited cell free virus transmission and the other is the direct cell to cell transfer of viral particles we show that the basic reproduction number is underestimated in the existing models that consider only the cell free virus transmission or the cell to cell infection ignoring the other assuming logistic growth for target cells we find that if the basic reproduction number is greater than one the infection can persist and the hopf bifurcation can occur from the positive equilibrium within certain parameter ranges

the distribution of limit cycles in quadratic systems with four finite singularities abstract in this paper we prove that if a quadratic system with four finite singularities contains a weak singularity then there exists at most one limit cycle not surrounding this singularity we also show that if the limit cycles of quadratic systems with four finite singularities are distributed over two nests then each nest contains exactly one limit cycle

bifurcations of limit cycles in complex networks of hamiltonian systems and lloyd s conjecture abstract this paper is motivated by hilbert s sixteenth problem concerning the number and distribution of limit cycles in planar polynomial vector fields we propose an original method inspired by the complex systems framework in order to construct a family of hamiltonian systems admitting non degenerate centers located on the vertices of a planar network we introduce two novel polynomial perturbations adapted to those systems with the aim to produce a high number of limit cycles the first one is oriented along the gradient of the unperturbed system and the second one is again constructed with a complex systems approach using the melnikov integral we reach in a small number of particular cases the lower bound of cubical order for the hilbert number h n conjectured by lloyd in 1988

higher order bifurcations of limit cycles abstract this paper shows that asymmetrically perturbed symmetric hamiltonian systems of the formx y y x x3 1y 2x2 3xy 4x2y with analytic j e o e have at most two limit cycles that bifurcate for smalle 0 from any period annulus of the unperturbed system this fact agrees with previous results of petrov dangelmayr and guckenheimer and chicone and iliev but shows that the result of three limit cycles for the asymmetrically perturbed exterior duffing oscillator recently obtained by jebrane and oladek is incorrect the proofs follow by deriving an explicit formula for thekth order melnikov function mk h and using a picard fuchs analysis to show that in each case mk h has at most two zeros moreover the method developed in this paper for determining the higher order melnikov functions also applies to more general perturbations of these systems

quadratic systems with an integrable saddle a complete classification in the coefficient space r 12 a quadratic polynomial differential system can be identified with a single point of r12 through the coefficients using the algebraic invariant theory we classify all the quadratic polynomial differential systems of r12 having an integrable saddle we show that there are only 47 topologically different phase portraits in the poincar e disc associated to this family of quadratic systems up to a reversal of the sense of their orbits moreover each one of these 47 representatives is determined by a set of affine invariant conditions

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

a two level time stepping method for layered ocean circulation models this paper describes a numerical method with two time levels for solving partial differential equations that represent layered models of ocean circulation the method is designed to be used with a barotropic baroclinic splitting that separates the fast and slow motions into subsystems that are solved by different techniques with this method some of the dependent variables are predicted and then corrected after the initial prediction steps all steps involve centered differencing and averaging about the midpoint of the time interval in question the coriolis and pressure terms are evaluated at the same time levels to avoid a first order truncation error in the geostrophic balance between those terms compared to the three level leapfrog method that is widely used in geophysical fluid dynamics the present method does not admit a computational mode and the maximum permissible time step is at least twice as large in addition with the two level method it is possible to use a nearly nonoscillatory advection algorithm to solve the equations for mass and momentum in a simple test problem the present method gives less phase error than the leapfrog method and two other methods and it does not give any amplitude error the differences are especially large when comparing the present method to the leapfrog method the two level method also gives good results in a nonlinear test problem involving vanishing layers at the top and bottom of the fluid domain and an interface moving along sloping bottom topography

bifurcations and distribution of limit cycles for near hamiltonian polynomial systems this paper concerns with limit cycles through hopf and homoclinic bifurcations for near hamiltonian systems by using the coefficients appeared in melnikov functions at the centers and homoclinic loops some sufficient conditions are obtained to find limit cycles

recollements of abelian categories and special types of comma categories abstract in this paper we first study conditions under which a recollement relative to abelian categories induces a new recollement relative to abelian categories and comma categories then we apply our results to deduce results about recollements relative to categories of modules furthermore if a ring r is morita equivalent to s we obtain that the upper triangular matrix ring r m 0 t is morita equivalent to s n 0 t for an arbitrary ring t r t bimodule m and s t bimodule n depending on m

homological epimorphisms recollements and hochschild cohomology with a conjecture by snashall solberg in view abstract we show that recollements of module categories give rise to homomorphisms between the associated hochschild cohomology algebras which preserve the strict gerstenhaber structure i e the cup product the graded lie bracket and the squaring map we review various long exact sequences in hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict gerstenhaber structure as well as a byproduct we generalise a known long exact cohomology sequence of koenig nagase to arbitrary surjective homological epimorphisms we use our observations to motivate and formulate a variation of the finite generation conjecture by snashall solberg

on the mahler measure of the coxeter polynomial of an algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is a basic connected and triangular algebra with n pairwise non isomorphic simple modules we consider the coxeter transformation a t as the automorphism of the grothendieck group k 0 a induced by the auslander reiten translation in the derived category der b mod a of the module category mod a of finite dimensional left a modules we say that a is of cyclotomic type if the characteristic polynomial a of a is a product of cyclotomic polynomials equivalently if the mahler measure m a 1 in 6 we have considered many examples of algebras of cyclotomic type in the representation theory literature in this paper we study the mahler measure of the coxeter polynomial of accessible algebras in 1933 d h lehmer found that the polynomial t 10 t 9 t 7 t 6 t 5 t 4 t 3 t 1 has mahler measure 0 1 176280 and he asked whether there exist any smaller values exceeding 1 in this paper we prove that for any accessible algebra a either m a 1 or m b 0 for some convex subcategory b of a we introduce interlaced tower of algebras a m a n with m n 2 satisfying a s 1 t 1 a s t a s 1 for m 1 s n 1 we prove that if spec a n s 1 r and a n is not of cyclotomic type then m a m m a n we construct several examples

bifurcation analysis of an sirs epidemic model with a generalized nonmonotone and saturated incidence rate abstract in this paper we study a susceptible infectious recovered sirs epidemic model with a generalized nonmonotone and saturated incidence rate k i 2 s 1 i i 2 in which the infection function first increases to a maximum when a new infectious disease emerges then decreases due to psychological effect and eventually tends to a saturation level due to crowding effect it is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values and the model undergoes saddle node bifurcation bogdanov takens bifurcation of codimension two hopf bifurcation and degenerate hopf bifurcation of codimension two as the parameters vary it is shown that there exists a critical value 0 for the psychological effect and two critical values k k 0 k 1 k 0 k 1 for the infection rate such that i when 0 or 0 and k k 0 the disease will die out for all positive initial populations ii when 0 and k 0 k k 1 the disease will die out for almost all positive initial populations iii when 0 and k k 1 the disease will persist in the form of a positive coexistent steady state for some positive initial populations and iv when 0 and k k 0 the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations numerical simulations including the existence of one or two limit cycles and data fitting of the influenza data in mainland china are presented to illustrate the theoretical results

a recursive approach to compute normal forms normal forms are instrumental in the analysis of dynamical systems described by ordinary differential equations particularly when singularities close to a bifurcation are to be characterized however the computation of a normal form up to an arbitrary order is numerically hard this paper focuses on the computer programming of some recursive formulas developed earlier to compute higher order normal forms a computer program to reduce the system to its normal form on a center manifold is developed using the maple symbolic language however it should be stressed that the program relies essentially on recursive numerical computations while symbolic calculations are used only for minor tasks some strategies are proposed to save computation time examples are presented to illustrate the application of the program to obtain high order normalization or to handle systems with large dimension

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

symbolic computation of normal forms for semi simple cases this paper presents a method and computer programs for computing the normal forms of ordinary differential equations whose jacobian matrix evaluated at an equilibrium involves semi simple eigenvalues the method can be used to deal with systems which are not necessarily described on a center manifold an iterative procedure is developed for finding the closed form expressions of the normal forms and associated nonlinear transformations computer programs using a symbolic computer language maple are developed to facilitate the application of the method the programs can be conveniently executed on a main frame a workstation or a pc machine without any interaction a number of examples are presented to demonstrate the applicability of the method and the computation efficiency of the maple programs

the loop quantities and bifurcations of homoclinic loops abstract the stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles for a homoclinic loop of a given planar vector field a sequence of quantities the homoclinic loop quantities were defined to study the stability and bifurcations of the loop among the sequence of the loop quantities the first nonzero one determines the stability of the homoclinic loop there are formulas for the first three and the fifth loop quantities in this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops as applications we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability switching

on the perturbations of a hamiltonian system the perturbations of a hamiltonian system having compounded cycle are studied in this paper the existence theory and stability theory of singular closed orbits are applied to study the given perturbed systems by using the small parametric perturbation techniques of differential equations we study hopf bifurcation singular closed orbits bifurcation and give the number and distributions of limit cycles in the above perturbed near hamiltonian system

symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation technique abstract this paper presents a perturbation method for computing the normal forms of resonant double hopf bifurcations with the aid of a computer algebraic system this technique based on the method of multiple time scales can be used to deal with general n dimensional systems without the application of center manifold theory explicit iterative formulas have been derived for uniquely determining the coefficients of normal forms and associated non linear transformations up to an arbitrary order user friendly symbolic computer programs written in maple have been developed which can be easily applied for computing the normal forms of a given system examples are presented to show the applicability of the methodology and the convenience of using the computer software

a unified approach to energy conservation and potential vorticity dynamics for arbitrarily structured c grids a numerical scheme applicable to arbitrarily structured c grids is presented for the nonlinear shallow water equations by discretizing the vector invariant form of the momentum equation the relationship between the nonlinear coriolis force and the potential vorticity flux can be used to guarantee that mass velocity and potential vorticity evolve in a consistent and compatible manner underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field the numerical scheme also exhibits conservation of total energy to within time truncation error simulations of the standard shallow water test cases confirm the analysis and show convergence rates between 1st and 2nd order accuracy when discretizing the system with quasi uniform spherical voronoi diagrams the numerical method is applicable to a wide class of meshes including latitude longitude grids voronoi diagrams delaunay triangulations and conformally mapped cubed sphere meshes

a system with three limit cycles appearing in a hopf bifurcation and dying in a homoclinic bifurcation the cusp of order 4 abstract in this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a hopf bifurcation h of order 3 and vanishing simultaneously in a homoclinic loop bifurcation hl of order 3 the region with three limit cycles is a topological 3 simplex the system is a generalization of bogdanov s system at the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4 this bifurcation diagram is a cone it contains a cone on the bifurcation diagram with the three limit cycles inside a 3 simplex region plus saddle node and cusp bifurcations of lower order

the number and distributions of limit cycles for a class of quintic near hamiltonian systems this paper is about the number of limit cycles for a quintic near hamiltonian system it is proved that the system can have 20 22 24 limit cycles with different distributions of limit cycles for each case the limit cycles are obtained by using the methods of bifurcation theory and qualitative analysis

on the stability of double homoclinic and heteroclinic cycles in this paper we give a criterion for the stability of planar double homoclinic and heteroclinic cycles with one or two saddles in some degenerate case

the cyclicity of the period annulus of the quadratic hamiltonian systems with non morsean point abstract in this paper we investigate the quadratic hamiltonian systems with non morsean point it is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to two

derived equivalences for φ auslander yoneda algebras in this paper we introduce auslander yoneda algebras in a triangulated category with a parameter set in n and present a method to construct new derived equivalences between these auslander yoneda algebras not necessarily artin algebras or their quotient algebras from a given almost stable derived equivalence as consequences of our method we have 1 suppose that a and b are representation finite self injective artin algebras with ax and by additive generators for a and b respectively if a and b are derived equivalent then the auslander yoneda algebras of x and y are derived equivalent for every admissible set in particular the auslander algebras of a and b are both derived equivalent and stably equivalent 2 for a self injective

characterization of the finite weak singularities of quadratic systems via invariant theory abstract this article is about weak singularities of quadratic differential systems that is non degenerate singular points with traces of the corresponding linearized systems at such points equal to zero these could be foci centers or saddles necessary and sufficient conditions for a real quadratic system to possess a fixed number of weak singularities of a specific order are given the conditions are stated in terms of affine invariant polynomials in the 12 dimensional space of the coefficients