the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

on spacelike hypersurfaces with constant mean curvature in the de sitter space soit n n n 1 c g une variete de lorentz a n 1 dimensions de courbure constante positive c soit m une hypersurface du genre espace complete a courbure moyenne constante h dans n on suppose que 1 h c 1 2 quand n 2 2 h 2 n 1 c 1 2 n quand n 3 alors m est totalement ombilicale

global hopf bifurcation of two parameter flows the behavior of center indices as introduced by j mallet paret j yorke is analyzed for two parameter flows the integer sum of center indices along a one dimensional curve in parameter space is called the h index a nonzero h index implies global hopf bifurcation the index h is not a homotopy invariant this fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero which we call b points to each b point we assign an integer b index such that the h index relates to the b indices by a formula such as occurs in the calculus of residues

period doubling and the lefschetz formula on donne une application du theoreme de point fixe de lefschetz pour demontrer sous certaines hypotheses l existence d une famille d orbites periodiques pour une application lisse

numerical semi global analysis of a 1 2 resonant hopf hopf bifurcation abstract in this paper a numerical semi global analysis of the dynamics near a 1 2 resonant hopf hopf bifurcation on a four dimensional mathematical model of a simple nonlinear oscillator is performed the 1 2 resonant hopf hopf bifurcation is a codimension three singularity denoted by two pairs of purely imaginary eigenvalues with frequency ratio 1 2 a structure involving 1 1 and 1 2 resonant neimark sacker bifurcations is clearly identified both resonances are coupled by lower codimension singularities such as generalized hopf and period doubling cusp points and cyclic fold curves a three parameter semi global analysis is performed and some of the codimension two singularities unfolded by the 1 2 resonant hopf hopf bifurcation are identified several codimension three points are also detected the obtained results can be useful for further theoretical analysis of the corresponding normal form

homoclinic doubling cascades cascades of period doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos in this paper we consider routes to onset of chaos involving homoclinic doubling bifurcations we show the existence of cascades of homoclinic doubling bifurcations which occur persistently in two parameter families of vector fields on 3 the cascades are found in an unfolding of a codimension three homoclinic bifurcation which occur an orbit flip at resonant eigenvalues we develop a continuation theory for homoclinic orbits in order to follow homoclinic orbits through infinitely many homoclinic doubling bifurcations

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

chaotic systems counting the number of periods abstract characterization of chaotic motion may proceed both at an averaged macroscopic level using such notions as lyapunov exponents dimensions and entropies and at a microscopic level in the latter case the number of periodic orbits being a topological invariant plays an important role for various one dimensional mappings the counting problem itself has many interesting facets and may be solved more or less completely in different ways recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher dimensional dissipative chaotic systems

computing hopf bifurcations ii three examples from neurophysiology in guckenheimer myers and sturmfels siam j numer anal 34 1997 we present algorithms for detecting hopf bifurcations in two parameter families of vector fields based on classical algebraic constructions in addition to their utility as augmented systems for use with standard newton type continuation methods they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension the present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for hopf pathfollowing that appear in the literature

classification and unfoldings of degenerate hopf bifurcations abstract this paper initiates the classification up to symmetry covariant contact equivalence of perturbations of local hopf bifurcation problems which do not satisfy the classical non degeneracy conditions the only remaining hypothesis is that i should be simple eigenvalues of the linearized right hand side at criticality then the lyapunov schmidt method allows a reduction to a scalar equation g x 0 where g x g x a definition is given of the codimension of g and a complete classification is obtained for all problems with codimension 3 together with the corresponding universal unfoldings the perturbed bifurcation diagrams are given for the cases with codimension 2 and for one case with codimension 3 for this last case one of the unfolding parameters is a modal parameter such that the topological codimension equals in fact 2 formulas are given for the calculation of the taylor coefficients needed for the application of the results and finally the results are applied to two simple problems a model of glycolytic oscillations and the fitzhugh nerve equations

global hopf bifurcation in a neural netlet sufficient conditions are obtained for the global hopf bifurcation of periodic solutions in the dynamics of a neural netlet of activation and inhibition with continuously distributed delays the orbital asymptotic stability of the bifurcating periodic solutions is investigated the results are illustrated with numerical simulations a few problems are formulated in the form of conjectures

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

s1 degree and global hopf bifurcation theory of functional differential equations abstract the recently developed s 1 degree and bifrucation theory are applied to provide a purely topological argument of a global hopf bifurcation theory for functional differential equations of mixed type in the special case where the equation is of retarded type the established result represents an analog of alexander and yorke s global hopf bifurcation theorem which has been obtained by chow fiedler mallet paret and nussbaum using different approaches

patterns at primary hopf bifurcations at a plexus of identical oscillators mathematically an oscillator is a system of ordinary differential equations with a periodic limit cycle usually arising from a hopf bifurcation for a stationary solution a plexus is a collection of such oscillators coupled to each other via one way or two way communication channels plexuses model a wide number of phenomena in the mathematically oriented sciences the primary hopf bifurcation theory of such a plexus is developed there are features of the bifurcating solutions which depend only on the combinatorics of the plexus and not on the details of the individual oscillators such features are called the pattern of the bifurcation the relation of patterns to the modes of small oscillations of mechanical systems is discussed more detailed information is developed for weak coupling the theory is used to analyze two particular types of plexuses viz rings and lines of oscillators it is then shown that the structure of a plexus permits numerical bifurcation computations to be made more efficien

dense heteroclinic tangencies near a bykov cycle abstract this article presents a mechanism for the coexistence of hyperbolic and non hyperbolic dynamics arising in a neighbourhood of a bykov cycle where trajectories turn in opposite directions near the two nodes we say that the nodes have different chirality we show that in the set of vector fields defined on a three dimensional manifold there is a class where tangencies of the invariant manifolds of two hyperbolic saddle foci occur densely the class is defined by the presence of the bykov cycle and by a condition on the parameters that determine the linear part of the vector field at the equilibria this has important consequences the global dynamics is persistently dominated by heteroclinic tangencies and by newhouse phenomena coexisting with hyperbolic dynamics arising from transversality the coexistence gives rise to linked suspensions of cantor sets with hyperbolic and non hyperbolic dynamics in contrast with the case where the nodes have the same chirality we illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three dimensional sphere

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3

homotopy invariants of time reversible periodic orbits ii towards applications abstract for a reversible periodic orbit we apply the sequence of homotopy invariants deg n n 1 2 3 defined in fiedler heinze 1996 we use this sequence to prove a global bifurcation result for reversible periodic orbits with prescribed minimal period this result will be applied to second order systems with neumann boundary conditions a discussion and remarks on the sequence of degrees concludes the paper

counting the number of periods in one dimensional maps with multiple critical points abstract the problem of counting the number of different periodic orbits in continuous maps of an interval is solved a map with m monotone pieces laps and m 1 critical points may have at most m 1 independent kneading sequences which provide the most convenient parameters for the map when one or more kneading sequences are kept constant or bound to vary simultaneously various degenerated cases of the map arise the number of period n orbits of the general m lap map as various degenerated cases is given by combinations of nink n with k m where nm n is the number of period n orbits of a particular kind of degenerated m lap map with only one kneading sequence the quantity nm n may be calculated or enumerated by many different methods which are all discussed in this paper we also give tabulated values for nm n for m 2 to 7

multiple eigenvalue bifurcation for fredholm operators on fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

solvability of general differential algebraic equations in the last few years there has been considerable research on differential algebraic equations daes f t x x 0 where f x is identically singular most of this effort has focused on computing a solution that is assumed to exist that is the dae is assumed solvable more recently there have been existence results developed using differential geometry for complex higher index systems these characterizations can be hard to verify in practice in this paper the computational verification of solvability is investigated this first requires developing an alternative set of sufficient conditions for solvability which are more amenable to computation verification of these conditions using readily available numerical and symbolic software is then discussed an example from robotics where classical graph theoretical approaches give an incorrect answer is worked to illustrate the usefulness of the sufficient condition and the computational approach

space like submanifolds in de sitter spaces let m n be a space like submanifold in a de sitter space mpn p c with flat normal bundle this paper gives some intrinsic conditions for m n to be totally umbilical

period doubling cascades of attractors a prerequisite for horseshoes this paper shows that if a horseshoe is created in a natural manner as a parameter is varied then the process of creation involves the appearance of attracting periodic orbits of all periods furthermore each of these orbits will period double repeatedly with those periods going to infinity

hopf bifurcation at 1 1 resonance on etudie un probleme concernant la bifurcation de hopf pour une famille a 1 parametre d equations differentielles dans un cas de resonance

nonlinear analysis of a maglev system with time delayed feedback control abstract this paper undertakes a nonlinear analysis of a model for a maglev system with time delayed feedback using linear analysis we determine constraints on the feedback control gains and the time delay which ensure stability of the maglev system we then show that a hopf bifurcation occurs at the linear stability boundary to gain insight into the periodic motion which arises from the hopf bifurcation we use the method of multiple scales on the nonlinear model this analysis shows that for practical operating ranges the maglev system undergoes both subcritical and supercritical bifurcations which give rise to unstable and stable limit cycles respectively numerical simulations confirm the theoretical results and indicate that unstable limit cycles may coexist with the stable equilibrium state this means that large enough perturbations may cause instability in the system even if the feedback gains are such that the linear theory predicts that the equilibrium state is stable

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

global families of limit cycles of planar analytic systems the global behavior of any one parameter family of limit cycles of a planar analytic system x f x a depending on a parameter a e r is determined it is shown that any one parameter family of limit cycles belongs to a maximal one parameter family which is either open or cyclic if the family is open then it terminates as the parameter or the orbits become unbounded or it terminates at a critical point or on a compound separatrix cycle of the system this implies that the periods in a one parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or compound separatrix cycle of the system this is a more specific result for planar analytic systems than wintner s principle of natural termination for n dimensional systems where the periods can become unbounded in strange ways this work generalizes duff s results for one parameter families of limit cycles generated by a one parameter family of rotated vector fields in particular it is shown that the behavior at a nonsingular multiple limit cycle of any one parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one parameter family of liniit cycles generated by a one parameter family of rotated vector fields

stability and bifurcations of equilibria in a multiple delayed differential equation the influence of multiple negative delayed feedback loops on the stability of a single action mechanism are considered a characteristic equation for the linearized stability of the equilibrium is completely analyzed as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops the bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold in particular the nature of hopf and more degenerate higher codimension bifurcations are explicitly determined

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

knotted periodic orbits in suspensions of smale s horseshoe period multiplying and cabled knots abstract following earlier work on knotted periodic orbits in a suspension of smale s 1 horseshoe diffeomorphism holmes and williams 2 we define a notion of iterated horseshoe knots we show that approximately half the horseshoe knots are cabled and that among these cablings are infinitely many non isotopic iterated torus knots we obtain uniqueness results for certain families of resonant torus knots related to cascades of period doubling and period multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the henon maps we give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots

self sustained oscillations of nonlinearly viscoelastic layers we study the motion of an incompressible nonlinearly viscoelastic layer subject to a slip stick frictional force applied to one of its faces by a moving belt this system is governed by a third order quasi linear parabolic hyperbolic partial differential equation subject to complicated boundary conditions we use a refined version of the hopf bifurcation theorem to show that this problem admits periodic solutions for the belt speed near critical values with the number of such solutions increasing as the viscosity of the layer decreases to verify the hypotheses of this theorem we must devote considerable effort to the tricky analysis of how the spectrum of the linearized problem depends on the belt speed and material parameters our analysis is complemented with both computations and a proof of the topological fact that the computed disposition of the eigenvalues does not omit any others we complete our study with a perturbation analysis justified by the hopf bifurcation theorem

global analysis of a system of predator prey equations we study a class of ode s modelling the interaction of one predator and one prey the prey is assumed to be asocial in the sense that there is a threshold below which it will die even in the absence of predators knowledge of the separatrices of the flow and the local nature of the equilibrium points yields an analogue of the kolmogorov theorem a more detailed study of the direction of hopf bifurcation and uniqueness of limit cycles together with recent work on their continuation allows us to exhibit a very rich dynamics for both this system as well as for the more classical case of a prey without threshold we briefly discuss the implications of this work for ecology

global generic dynamics close to symmetry abstract our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in s 3 the cycle involves two saddle foci of different type and is structurally stable within the class of z 2 z 2 symmetric vector fields the cycle contains a two dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry breaking perturbations gradually breaking the symmetry in a two parameter family we get a wide range of dynamical behaviour an attracting periodic trajectory other heteroclinic trajectories homoclinic orbits n pulses suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of shilnikov type we also show that generically the coexistence of linked homoclinic orbits at the two saddle foci has codimension 2 and takes place arbitrarily close to the symmetric cycle

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

a direct method for the computation of hopf bifurcation points we present a method for the numerical computation of a hopf bifurcation point of an evolution equation an augmented time independent system of equations has to be solved for which the hopf bifurcation point is an isolated solution in the case of a system of ordinary differential equations we propose an efficient implementation of newton s method for the solution of the resulting system of algebraic equations if the method is applied directly to a partial differential system with one space variable a two point boundary value problem is obtained we also compare this technique with a related method and results are given for two systems of partial differential equations describing chemical reacting systems

geometry and combinatorics of julia sets of real quadratic maps for real a correspondence is made between the julia setb forz z 2 in the hyperbolic case and the set of chains with the aid of cremer s theorem it is shown how a number of features ofb can be understood in terms of chains the structure ofb is determined by certain equivalence classes of chains fixed by orders of visitation of certain real cycles and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of chains the functional equations obeyed by attractive cycles are investigated and their relation to chains is given the first cascade of period doubling bifurcations is described from the point of view of the associated julia sets and chains certain julia sets associated with the feigenbaum function and some theorems of lanford are discussed

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

an index for global hopf bifurcation in parabolic systems on considere le systeme non lineaire d t u a u f u 0 et on cherche des solutions periodiques en temps u t qui bifurquent a partir d une solution stationnaire

the hopf bifurcation theorem for quasilinear differential algebraic equations we prove a generalization of the hopf bifurcation theorem for quasilinear differential equations daes i e equations of the form a g where the matrix a has constant but not full rank and hence the system cannot be made into an explicit ode the paper includes an appendix by j ernsthausen addressing the numerical calculation of the hopf points in the dae setting

on impasse points of quasilinear differential algebraic equations abstract this paper presents a mathematical characterization of the impasse points of quasilinear dae s a x g x where then a x is a nxn matrix having constant but not full r n in the domain of interest we show that a reduction procedure permits to reduce the dae to a quasilinear ode al xi xi g1 xi rr and that impasse points correspond to special singularities of the reduced ode impasse points of higher index dae s can also be defined the only difference is that the above reduction procedure takes more than one step

local levi flat hypersurfaces invariants by a codimension one holomorphic foliation in this paper we study codimension one holomorphic foliations leaving invariant real analytic hypersurfaces in particular we prove that a germ of real analytic levi flat hypersurface with sufficiently small singular set is given by the zeroes of the imaginary part of a holomorphic function

equivariant degree method for analysis of hopf bifurcation of relative periodic solutions case study of a ring of oscillators abstract the goal of this paper is to develop the equivariant degree based method for studying relative periodic solutions in the setiings with lack of smoothness and or genericity in this paper we consider an equivariant hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting s 1 spatial symmetries the existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted s 1 degree with one free parameter as a case study we consider a delay differential model of coupled identical passively mode locked semiconductor lasers with the dihedral symmetry group d 8 and a system of hysterestic electro mechanical oscillators coupled in the same symmetric fashion

snakes oriented families of periodic orbits their sources sinks and continuation abstract poincare observed that for a differential equation x x depending on a parameter each periodic orbit generally lies in a connected family of orbits in x space in order to investigate certain large connected sets denoted q of orbits containing a given orbit we introduce two indices an orbit index and a center index defined at certain stationary points we show that genetically there are two types of hopf bifurcation those we call sources 1 and sinks 1 generically if the set q is bounded in x space and if there is an upper bound for periods of the orbits in q then q must have as many source hopf bifurcations as sink hopf bifurcations and each source is connected to a sink by an oriented one parameter snake of orbits a snake is a maximal path of orbits that contains no orbits whose orbit index is 0 see fig 1 1

singular set of a levi flat hypersurface is levi flat we study the singular set of a singular levi flat real analytic hypersurface we prove that the singular set of such a hypersurface is levi flat in the appropriate sense we also show that if the singular set is small enough then the levi foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface

wild hyperbolic sets yet no chance for the coexistence of infinitely many klus simple newhouse attracting sets the phenomenon of the coexistence of infinitely many sinks for two dimensional dissipative diffeomorphisms is a result due to newhouse ne1 ne2 in fact for each parameter value at which a homoclinic tangency is formed nondegenerately there exist intervals in the parameter space containing dense sets of parameter values for which there are infinitely many coexisting sinks robinson r the structure of the sinks constructed by newhouse is limited simple newhouse parameter values are values at which there are infinitely many sinks having some special well defined property concerning the structure a result due to tedeschini lalli and yorke ty says that the lebesgue measure of the set of simple newhouse parameter values is zero when the tangencies are due to the standard affine horseshoe map it is argued in ty and pr that a more general derivation of this measure zero result would be desirable the main result of this paper is that the lebesgue measure of the set of klus simple parameter values including the simple newhouse parameter values is zero for saddle hyperbolic basic sets forming tangencies

discontinuous solutions of semilinear differential algebraic equations i distribution solutions abstract there is strong physical evidence that a full treatment of differential algebraic equations should be incorporate solutions with jump discontinuities it is shown here that for semilinear problems the setting of distributions allows for the development of a theory where indeed such discontinuities may occur this approach also settles the problem of inconsistent initial conditions in a very simple way on the other hand new issues arise as not only uniqueness but even countability of the number of solutions of initial value problems may now be lost a physically motivated but purely mathematical selection procedure to overcome this difficulty is discussed in part 2 of this paper an

a canonical partition of the periodic orbits of chaotic maps on montre que les orbites periodiques d une application en fer a cheval qui contracte l aire peuvent se partager en sous ensembles d orbites de periode minimum k 2k 4k 8k pour un k entier positif

derper an algorithm for the continuation of periodic solutions in ordinary differential equations abstract an algorithm for the continuation of periodic solutions of ode was derived and tested the algorithm is based on the shooting method and a standard continuation algorithm variational variables are used to compute the jacobi matrix the starting point on the periodic orbit is adaptively changed in the algorithm by fixing one component of the solution stability of periodic solutions along the continuous branch of solutions is determined by computing characteristic multipliers without difficulty the algorithm crosses limit points and at bifurcation points double period bifurcation points and points of tori bifurcation it proceeds along the original branch of solutions the usefulness of the algorithm is demonstrated in two examples one of them describes two stirred tank reactors with mutual mass exchange the other one is the lorenz model of turbulence

the double hopf bifurcation with 2 1 resonance many problems of physical interest involve the nonlinear interaction of two oscillators with different frequencies when these frequencies are incommensurate the interaction is straightforward involving the amplitude rather than the phase of each oscillator when the frequencies are in a ratio closely corresponding to a rational fraction of small denominator phase locking occurs and the dynamics is much richer in this paper we investigate fully the important case where the frequencies are in the ratio 2 1 the analysis allows for small departures from this ratio and treats not only the generic problem for small amplitudes but also an important degenerate case which appears to capture most of the dynamics at larger amplitudes

families of periodic orbits virtual periods and global continuability abstract for a differential equation depending on a parameter there have been numerous investigations of the continuation of periodic orbits as the parameter is varied mallet paret and yorke investigated in generic situations how connected components of orbits must terminate here we extend the theory to the general case dropping genericity assumptions

interaction of periodic and stationary bifurcation from multiple eigenvalues on considere des equations d evolution non lineaires du dt g u o dependant d un parametre reel dans un espace de hilbert reel e

double hopf bifurcation of coupled dissipative stuart landau oscillators with delay in this paper the double hopf bifurcation of delay coupled dissipative stuart landau oscillators models are considered where the time delay is regarded as one of the parameters and the other is the natural frequency it is found that as the values of the two parameters vary a number of double hopf bifurcation points arise consequently it has been found that the system exhibits very rich complex dynamics some resonant and non resonant double hopf bifurcation phenomena can be observed some numerical simulations support our analysis result

bifurcation to chaotic scattering abstract we investigate a novel type of bifurcation to chaos which occurs in the context of chaotic scattering in chaotic scattering the deflection angle versus impact parameter is singular on a set of impact parameters which is fractal this behavior is caused by the presence of a chaotic invariant set of unstable bounded orbits in the bifurcation considered here the chaotic set arises abruptly in a sense to be discussed as the particle energy e decreases from above a critical value em to below we call this transition an abrupt bifurcation to fully developed chaotic scattering numerical computation of the dimension d of the chaotic set shows that in agreement with theoretical prediction d 1 in em e 1 near the abrupt bifurcatio

on the continuability of periodic orbits of parametrized three dimensional differential equations on considere un systeme autonome a n dimensions d equations differentielles ordinaires a 1 parametre x f x on suppose que f est definie et continue sur un sous ensemble ouvert o de r r n et que f x 0 sur o on etudie les orbites periodiques du systeme contenues dans o

classification and unfoldings of 1 2 resonant hopf bifurcation in this paper we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1 2 ratio this corresponds to a hopf hopf mode interaction with 1 2 resonance as occurs in the context of dissipative mechanical systems using an approach based on liapunov schmidt reduction and singularity theory we give a framework in which to study these problems and their perturbations in two cases no distinguished parameter and one distinguished bifurcation parameter we give a complete classification of the generic cases and their unfoldings

singular levi flat real analytic hypersurfaces we initiate a systematic local study of singular levi flat real analytic hypersurfaces concentrating on the simplest nontrivial case of quadratic singularities we classify the possible tangent cones to such hypersurfaces and prove the existence and convergence of a rigid normal form in the case of generic morse singularities we also characterize when such a hypersurface is defined by the vanishing of the real part of a holomorphic function the main technique is to control the behavior of the homorphic segre varieties contained in such a hypersurface finally we show that not every such singular hypersurface can be defined by the vanishing of the real part of a holomorphic or meromorphic function and give a necessary condition for such a hypersurface to be equivalent to an algebraic one

implicit differential equations near a singular point abstract this paper presents general and precise results on existence and number of solutions to implicitly defined ordinary differential equations in the vicinity of a singular point where the equation is not equivalent to an explicit one unlike in most if not in all other studies devoted to this problem in the literature it is not assumed that either rank drop occurs on an entire neighborhood of the singular point or that the equation is a scalar one attention is confined to the simplest but most frequently encountered kind of singular points it is shown that such standard singular points split into two complementary classes those from which exactly two distinct solutions emanate and those at which exactly two distinct solutions terminate whether a point of the latter class is eventually encountered is unaffected by slight modifications of the nonsingular initial condition and further evolution of the system governed by the singular ode then requires using a suitable jump condition according to appropriate physical criteria points of the former class are also shown to play a more subtle but equally important role in the dynamics the phenomena described here are relevant in various problems from the sciences such as phase transitions or plasticity they should also be relevant in some aspects of the classical domain of application of differential algebraic equations singular perturbations of ode s

quasiperiodic bifurcation in nonlinearly coupled oscillators near a point of strong resonance a pair of uncoupled linear oscillators have natural frequencies delta and gamma and damping coefficient proportional to lambda when they are coupled through a class of cubic nonlinearities the null solution bifurcates at lambda lambda c the bifurcating solutions are analyzed near the strong resonance delta gamma by relating these delta gamma frac 1 2 k varepsilon 2 to the amplitude varepsilon and using multiple scale theory the resulting approximation is uniformly valid even for the far from resonance case k o varepsilon 2 an entrainment parameter rho is defined rho 2 delta 2 lambda lambda c for which a critical value rho c exists for rho leqq rho c there are mixed mode frequency locked periodic oscillations rho rho c the oscillations depend on two frequencies and so are periodic or quasiperiodic depending on the ratio delta gamma