computing two dimensional poincar maps for hyperchaotic dynamics poincar map pm is one of the felicitous discrete approximation of the continuous dynamics to compute pm the discrete relation s between the successive point of interactions of the trajectories on the suitable poincar section ps are found out these discrete relations act as an amanuensis of the nature of the continuous dynamics in this article we propose a computational scheme to find a hyperchaotic pm hpm from an equivalent three dimensional 3d subsystem of a 4d or higher hyperchaotic model for the experimental purpose a standard four dimensional 4d hyperchaotic lorenz stenflo system hlss and a five dimensional 5d hyperchaotic laser model hlm is considered equivalent 3d subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3d subsystems the quantitative measurement of this comparison is made promising by recurrence quantification analysis rqa various two dimensional 2d poincar mas are computed for several suitable poincar sections for both the systems but only some of them are hyperchaotic in nature the hyperchaotic behavior is verified by positive values of both one dimensional 1d lyapunov exponent le i and 2d lyapunov exponent le ii at the end similarity of the dynamics between the hyperchaotic systems and their 2d hyperchaotic poincar maps hpm has been established through mean recurrence time mrt statistics for both of 4d hlss and 5d hlm and the best approximated discrete dynamics for both the hyperchaotic systems are found out

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

an accelerated poincare map method for autonomous oscillators a novel time domain method for finding the periodic steady state of a free running electrical oscillator is introduced the method is based on the extrapolation technique mpe the new method is applied to a family of artificial benchmark problems and to a real circuit colpitt s oscillator it turns out to have super linearly convergence properties in both cases full implementational details are provided

computing the minimum distance between two bezier curves a sweeping sphere clipping method is presented for computing the minimum distance between two bezier curves the sweeping sphere is constructed by rolling a sphere with its center point along a curve the initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve the nearest point on a curve must be contained in the sweeping sphere along the other curve and all of the parts outside the sweeping sphere can be eliminated a simple sufficient condition when the nearest point is one of the two end points of a curve is provided which turns the curve curve case into a point curve case and leads to higher efficiency examples are shown to illustrate efficiency and robustness of the new method

some considerations on poincare maps for chaotic flows the standard way to reduce a flow to a time discrete dynamical system is by the technique of the poincaresurface of section we discuss the relationship between poincaremaps obtained from different surfaces of section and give some considerations for practical application for this method

a preparation theorem for codimension one foliations after gluing foliated complex manifolds we derive a preparation like theorem for singularities of codimension one foliations and planar vector fields in the real or complex setting without computation we retrieve and improve results of levinson moser for functions dufour zhitomirskii for nondegenerate codimension one foliations proving in turn the analyticity str oyzyna y zo ladek for non degenerate planar vector fields and bruno ecalle for saddle node foliations in the plane

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

a subdivision based algorithm for the sparse resultant multivariate resultants generalize the sylvester resultant of two polynomials and characterize the solvability of a polynomial system they also reduce the computation of all common roots to a problem in linear algebra we propose a determinantal formula for the sparse resultant of an arbitrary system of n 1 polynomials in n variables this resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their bezout number our algorithm uses a mixed polyhedral subdivision of the minkowski sum of the newton polytopes in order to construct a newton matrix its determinant is a nonzero multiple of the sparse resultant and the latter equals the gcd of at most n 1 such determinants this construction implies a restricted version of an effective sparse nullstellensatz for an arbitrary specialization of the coefficients there are two methods that use one extra variable and yield the sparse resultant this is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n we conjecture its extension to producing an exact rational expression for the sparse resultant

macaulay style formulas for sparse resultants we present formulas for computing the resultant of sparse polynomials as a quotient of two determinants the denominator being a minor of the numerator these formulas extend the original formulation given by macaulay for homogeneous polynomials

slow fast bogdanov takens bifurcations abstract in this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of bogdanov takens bt bifurcations we introduce among other precise definitions the notion of slow fast bt bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits based on geometric singular perturbation theory including blow up we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

cyclicity of a fake saddle inside the quadratic vector fields abstract this paper concerns the study of small amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity this degenerate singularity is also known as an impassable grain the canonical form of the unperturbed vector field is like a degenerate flow box near the singularity the phase portrait consists of parallel fibers all but one of which have no singular points and at the singular fiber there is one node we demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

is one dimensional poincare map sufficient to describe the chaotic dynamics of a three dimensional system study of continuous dynamical system through poincare map is one of the most popular topics in nonlinear analysis this is done by taking intersections of the orbit of flow by a hyper plane parallel to one of the coordinate hyper planes of co dimension one naturally for a 3d attractor the poincare map gives rise to 2d points which can describe the dynamics of the attractor properly in a very special case sometimes these 2d points are considered as their 1d projections to obtain a 1d map however this is an artificial way of reducing the 2d map by dropping one of the variables sometimes it is found that the two coordinates of the points on the poincare section are functionally related this also reduces the 2d poincare map to a 1d map this reduction is natural and not artificial as mentioned above in the present study this issue is being highlighted in fact we find out some examples which show that even this natural reduction of the 2d poincare map is not always justified because the resultant 1d map may fail to generate the original dynamics properly this proves that to describe the dynamics of the 3d chaotic attractor the minimum dimension of the poincare map must be two in general

approximate discrete dynamics of emg signal approximation of a continuous dynamics by discrete dynamics in the form of poincare map is one of the fascinating mathematical tool which can describe the approximate behaviour of the dynamics of the dynamical system in lesser dimension than the embedding dimension the present article considers a very rare biomedical signal like electromyography emg signal it determines suitable time delay and reconstruct the attractor with embedding dimension three by measuring its lyapunov exponent the attractor so reconstructed is found to be chaotic naturally the poincare map obtained by corresponding poincare section has to be chaotic too this may be verified by calculation of lyapunov exponent of the map the main objective of this article is to show that poincare map exists in this case as a 2d map for a suitable poincare section only in fact the article considers two poincare sections of the attractor for construction of the poincare map it is seen that one such map is chaotic but the other one is not so both are verified by calculation of lyapunov exponent of the map

adjusted least squares fitting of algebraic hypersurfaces we consider the problem of fitting a set of points in euclidean space by an algebraic hypersurface we assume that points on a true hypersurface described by a polynomial equation are corrupted by zero mean independent gaussian noise and we estimate the coefficients of the true polynomial equation the adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator the adjusted least squares estimator is based on constructing a quasi hankel matrix which is a bias corrected matrix of moments for the case of unknown noise variance the estimator is defined as a solution of a polynomial eigenvalue problem in this paper we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation

normal forms near a symmetric planar saddle connection abstract the present paper studies vector fields of the form x q 2 o 1 x 2 1 x 2 o y y p x o 1 x 2 y o y 2 which contain a separatrix connection between hyperbolic saddles with opposite eigenvalues where the connection is fixed smooth semi local normal forms are provided in vicinity of the connection both in the resonant and non resonant case first a formal conjugacy is constructed near the separatrix then a smooth change of coordinates is realized by generalizing known local results near the hyperbolic points

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

on the numerical computation of poincare maps abstract this note describes a method for finding simply and accurately the intersections of a numerically integrated trajectory with a surface of section

lagrange interpolation on subgrids of tensor product grids this note shows that a wide class of algebraically motivated constructions for lagrange interpolation polynomials always yields a tensor product interpolation space as long as the nodes form a tensor product grid or a lower subset thereof

normal linearization and transition map near a saddle connection with symmetric resonances abstract we consider a heteroclinic connection in a planar system between two symmetric hyperbolic saddles of which the eigenvalues are resonant starting with a c normal form defined globally near the connection we normally linearize the vector field by using finitely smooth tags of logarithmic form we furthermore define an asymptotic entry exit relation and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

effective polynomial computation 1 euclid s algorithm 2 continued fractions 3 diophantine equations 4 lattice techniques 5 arithmetic functions 6 residue rings 7 polynomial arithmetic 8 polynomial gcd s classical algorithms 9 polynomial elimination 10 formal power series 11 bounds on polynomials 12 zero equivalence testing 13 univariate interpolation 14 multivariate interpolation 15 polynomial gcd s interpolation algorithms 16 hensel algorithms 17 sparse hensel algorithms 18 factoring over finite fields 19 irreducibility of polynomials 20 univariate factorization 21 multivariate factorization list of symbols bibliography index

implicitization of curves and hyper surfaces using predicted support we reduce implicitization of rational planar parametric curves and hyper surfaces to linear algebra by interpolating the coefficients of the implicit equation given a superset of its terms for predicting these terms we focus on methods that exploit input and output structure in the sense of sparse or toric elimination theory namely by computing the newton polytope of the implicit polynomial via sparse resultant theory our algorithm works even in the presence of base points but in this case the implicit equation shall be obtained as a factor of the produced polynomial we implement our methods in maple and some in matlab as well and study their numerical stability and efficiency on several classes of curves and surfaces we apply our approach to approximate implicitization and quantify the accuracy of the approximate output which turns out to be satisfactory on all tested examples in building a square or rectangular interpolation matrix an important issue is over sampling the given curve or surface we conclude that unitary complex numbers offer the best tradeoff between speed and accuracy when numerical methods are employed namely svd whereas for exact kernel computation random integers is the method of choice we compare our prototype to existing software and find that it is rather competitive

versal unfoldings of predator prey systems with ratio dependent functional response abstract in this paper we study the versal unfolding of a predator prey system with ratio dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations we first construct the unfolding and prove its versality and degeneracy of codimension 2 then we discuss all its possible bifurcations including transcritical bifurcation hopf bifurcation and heteroclinic bifurcation give conditions of parameters for the appearance of closed orbits and heteroclinic loops and describe the bifurcation curves phase portraits for all possible cases are presented

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

rational hausdorff divisors a new approach to the approximate parametrization of curves in this paper we introduce the notion of rational hausdorff divisor we analyze the dimension and irreducibility of its associated linear system of curves and we prove that all irreducible real curves belonging to the linear system are rational and are at finite hausdorff distance among them as a consequence we provide a projective linear subspace where all irreducible elements are solutions of the approximate parametrization problem for a given algebraic plane curve furthermore we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization therefore we present a generic answer to the approximate parametrization problem in addition we introduce the notion of hausdorff curve and we prove that every irreducible hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve