subharmonic solutions in singular systems abstract we consider the problem of bifurcation from homoclinic towards periodic orbits for a periodic singular system of differential equations we use a functional analytic approach and the lyapunov schmidt method to obtain a bifurcation function for this problem which tends as e 0 to the well known melnikov function for the bifurcation towards transverse homoclinic orbits we also study the behavior of the bifurcation functions as the period of the periodic orbit tends to infinity

analytical approximation of cuspidal loops using a nonlinear time transformation method abstract in this work we consider cuspidal loops i e homoclinic orbits to cuspidal singular points we develop an iterative procedure founded on the nonlinear time transformation method to estimate such codimension three global bifurcations up to any wanted order not only in the space of parameters but also in the phase plane as far as we know this is the first time in the literature that this theoretical result is achieved for these global connections the existence and uniqueness of the perturbed solution obtained are proved to illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate takens bogdanov bifurcations excellent agreement is found between our analytical predictions and the corresponding numerical continuations

homoclinic bifurcations in chua s circuit abstract in this paper we study the possible relationship between the birth of the double scroll l o chua et al ieee cas 33 11 1986 1073 and the homoclinic bifurcations in the traditional chua s equations using a one dimensional poincare map we determine the existence of secondary symmetric homoclinic orbits of shil nikov type born with the chua s attractor connecting unstable and stable manifolds of the trivial equilibrium point in addition taking into account the presence of other homoclinic orbits for the asymmetric attractor and heteroclinic orbits for the symmetric attractor connecting unstable and stable manifold of the non trivial equilibrium points we suggest a hypothesis about the birth of double scroll structure on the plane

multiple internal layer solutions generated by spatially oscillatory perturbations abstract for a singularly perturbed system of reaction diffusion equations we study the bifurcation of internal layer solutions due to the addition of a spatially oscillatory term in the singular limit the existence and stability of internal layer solutions are determined by the intersection of a fast jump surface 1 and a slow switching curve c the case when the intersection is transverse was studied by x b lin construction and asymptotic stability of structurally stable internal layer solutions preprint in this paper we show that when 1 intersects with c tangentially saddle node or cusp type bifurcation may occur higher order expansions of internal layer solutions and eigenvalue eigenfunctions are also presented to find a true internal layer solution and true eigenvalue eigenfunctions we use a newton s method in functions spaces that is suitable for numerical computations

sbil nikov hopf bifurcation abstract we derive a 2 parameter poincare map modeling a 3 dimensional vector flow near the codimension 2 point defined by a sil nikov homoclinic orbit to a fixed point undergoing a hopf bifurcation in one parameter regime the map reproduces glendinning and sparrow s results for the sil nikov bifurcation while in another regime it provides new results for a small hopf cycle with a global reinjection mechanism the map is analyzed finding primary homoclinic tangencies to the small hopf cycle parameter dependence of periodic orbits associated with these tangencies is described global behavior of the tube like unstable manifold of the hopf cycle is studied in depth analytically and numerically locating its simplest subsidiary tangencies

a numerical bifurcation function for homoclinic orbits we present a numerical method to locate periodic orbits near homoclinic orbits using a method of x b lin proc roy soc edinburgh 116a 1990 pp 295 325 and solutions of the adjoint variational equation we get a bifurcation function for periodic orbits whose periods are asymptotic to infinity on approaching a homoclinic orbit as a bonus a linear predictor for continuation of the homoclinic orbit is easily available numerical approximation of the homoclinic orbit and the solution of the adjoint variational equation are discussed we consider a class of methods for approximating the latter equation such that a scalar quantity is preserved we also consider a context where the effects of continuous symmetries of equations can be incorporated applying the method to an ordinary differential equation on r3 studied by e freire a rodriguez luis and e ponce phys d 62 1993 pp 230 253 we show the bifurcation function gives good agreement with path followed solutions even down to low period as an example application to a parabolic partial differential equation pde we examine the bifurcation function for a homoclinic orbit in the kuramoto sivashinsky equation

when gap solitons become embedded solitons a generic unfolding a two parameter unfolding is considered of single pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth order reversible dynamical systems one parameter controls the linearisation with a transition occurring between a saddle centre and a hyperbolic equilibrium in the saddle centre region the homoclinic orbit is of codimension one which is controlled by the second generic parameter whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable a geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold depending on the sign of the perturbation of the second parameter special cases of different actions of z2 symmetry are considered as is the case of the system being hamiltonian application of these results is considered to the transition between embedded solitons corresponding to the codimension one homoclinic orbits and gap solitons the structurally stable ones in nonlinear wave systems the theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of fifth order korteweg de vries equation

differential harnack inequalities for the backward heat equation with potential under the harmonic ricci flow abstract in this paper a perelman type harnack inequality for the conjugate heat equation under the harmonic ricci flow is proved differential harnack inequalities for the positive solutions of the backward heat equation with potential under the harmonic ricci flow are also considered

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

heteroclinic and periodic cycles in a perturbed convection model abstract vivancos and minzoni new choatic behaviour in a singularly perturbed model preprint proposed a singularly perturbed rotating convection system to model the earth s dynamo process numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions in this paper we show that if the perturbation is sufficiently small the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions one looks like a figure delta and the other looks like a figure eight due to the fast slow characteristic of the system the reduced slow system has a relay nonlinearity asymptotic method in singularly perturbed systems consultants bureau new york and london 1994 solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces we develop new types of melnikov integral and lyapunov schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity

forced symmetry breaking of homoclinic cycles in a pde with o 2 symmetry we perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a pde with o 2 symmetry to one with so 2 symmetry taking particular care of the consequences of the continuous group action we concentrate on the kuramoto sivashinsky equation with spatially periodic boundary conditions the breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry in particular we note that although chossat 1993 has proved that generic perturbations cause the appearance of quasiperiodic orbits for the simplest possible flux terms this is not the case we compare these results with numerical simulations of a galerkin approximation of the equations

bifurcation behavior of a superlattice model we present a complete description of the stationary and dynamical behavior of semiconductor superlattices in the framework of a discrete drift model by means of numerical continuation singular perturbation analysis and bifurcation techniques the control parameters are the applied dc voltage phi and the doping nu in nondimensional units we show that the organizing centers for the long time dynamics are takens bogdanov bifurcation points in a broad range of parameters and we cast our results in a phi nu phase diagram for small values of the doping the system has only one uniform solution where all the variables are almost equal for high doping we find multistability corresponding to domain solutions and the stationary solutions may exhibit chaotic spatial behavior in the intermediate regime of nu the solution can be time periodic depending on the bias the oscillatory regions are related to the appearance and disappearance of hopf bifurcation tongues which can be sub or supercri

a numerical liapunov schmidt method with applications to hopf bifurcation on a square we discuss an iterative method for calculating the reduced bifurcation equation of the liapunov schmidt method and its numerical approximation using appropriate genericity assumptions with symmetry we derive a taylor series for the reduced equation where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation this method is used to calculate reduced equations at hopf bifurcation of the two dimensional brusselator equations on a square with neumann and dirichlet boundary conditions we examine several hopf bifurcations within the three parameter space there are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

further results on the minimum rank of regular classes of 0 1 matrices abstract let b n k be the set of all 0 1 matrices of order n with constant line sum k and let n k be the minimum rank over b n k it is known that n k n k n k n k k where n k is the rank of a recursively defined matrix a b n k brualdi manber and ross showed that n k n k if and only if k n in this paper we show that n k n k k if and only if n k satisfies one of the following three relations i n 1 mod k k 2 or 3 ii n k 1 k 2 iii n 4 q 3 k 4 and q 1 moreover we obtain the exact values of n 4 for all n 4 and determine all the possible ranks of regular 0 1 matrices in b n 4 we also present some positive integer pairs n k such that n k n k n k k which gives a positive answer to a question posed by pullman and stanford

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

applications of catastrophe theory to the physical sciences abstract catastrophe theory was introduced by rene thom in the late 1960 s as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations such as differential and difference equations and in particular discussed the special case of elementary catastrophe theory singularities of smooth real valued functions popular expositions have tended to overemphasize this special case but it remains the major source of ideas and methods here we survey the applications of catastrophe theory to the physical sciences physics chemistry engineering fluid mechanics etc for brevity we confine attention to an area lying between elementary and general catastrophe theory usually known as singularity theory this is the theory of singularities of smooth vector valued functions which mathematically is a straightforward though non trivial generalization of the real valued case in the last few years it has developed into a powerful and useful technique in several areas of theoretical physics notably optics and bifurcation theory equivariant catastrophe theory taking account of symmetry is likely to prove especially interesting

the origins of chaos in a modified van der pol oscillator abstract an experimental and theoretical investigation of a modified van der pol oscillator has led to the discovery of degenerate bifurcations which control the global dynamics of the system a two dimensional parameter space for the oscillator is studied and the measured loci of critical points where a qualitative change of behaviour is observed are given in detail it is shown explicitly by analysis of the experimental data that chaos arises through the silnikov mechanism

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

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

back in the saddle again a computer assisted study of the kuramoto sivashinsky equation a numerical and analytical study of the kuramoto sivashinsky partial differential equation pde in one spatial dimension with periodic boundary conditions is presented the structure stability and bifurcation characteristics of steady state and time dependent solutions of the pde for values of the parameter alpha less than 40 are examined the numerically observed primary and secondary bifurcations of steady states as well as bifurcations to constant speed traveling waves limit cycles are analytically verified persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of o 2 their effect on the system dynamics is discussed and several tertiary bifurcations observed numerically are presented

boolean rank of upset tournament matrices abstract the boolean rank of an m n 0 1 matrix m is the minimum k for which matrices a and b exist with m ab a is m k b is k n and boolean arithmetic is used the intersection number of a directed graph d is the minimum cardinality of a finite set s for which each vertex v of d can be represented by an ordered pair s v t v of subsets of s such that there is an arc from vertex u to vertex v in d if and only if s u t v o the intersection number of a digraph is equal to the boolean rank of its adjacency matrix using this fact we show that the intersection number of an upset tournament equivalently the boolean rank of its adjacency matrix is equal to the number of maximal subpaths of certain types in its upset path

construction and asymptotic stability of structurally stable internal layer solutions we introduce a geometric asymptotic method to treat structurally stable internal layer solutions we consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability proofs of the existence of exact solutions and eigenvalue eigenfunctions are outlined multi layered solutions are constructed by a new shooting method through a sequence of pseudo poincare mappings that do not require the transversality of the flow to cross sections the critical eigenvalues are determined by a coupling matrix that generates the slep matrix the transversality of the shooting method is related to the nonzeroness of the critical eigenvalues an equivalent approach is given to mono layer solutions they can be determined by the intersection of a fast jump surface and a slow switching curve which reduces fenichel s transversality condition to the slow manifold the critical eigenvalue is determined by the angle of the intersection we present three examples the first treats the critical eigenvalues of the system studied by angenent mallet paret peletier the second shows that a key lemma in the slep method may not hold the third is a perturbed activator inhibitor system that can have any number of mono layer solutions some of the solutions can only be found with the new shooting method

minimum matrix rank of k regular 0 1 matrices abstract in this paper we present results on minimum boolean and nonnegative integer rank of k regular 0 1 matrices results on these semiring ranks give equivalent results on minimum biclique covering numbers and partition numbers for k regular bipartite graphs and digraphs

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

an analytical and numerical study of a modified van der pol oscillator abstract a three dimensional system of differential equations that models an electronic oscillator is considered the equations allow a variety of periodic orbits that originate from a degenerate hopf bifurcation which is analytically studied numerical results are presented that show the existence of saddle node cusps of periodic orbits as well as period doubling bifurcations that result in the coexistence of multiple canard orbits if one of the parameters is small the presence of chaotic attractors is also detected

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

singular 0 1 matrices with constant row and column sums abstract we address the problem of explicitly constructing an n n 0 1 matrix with constant line sum k and rank r for each r k 2 mod k for all even k we also solve the problem for all but at most the first 2k 4 possible ranks r 3k

viscous singular shock profiles for a system of conservation laws modeling two phase flow abstract this paper is concerned with singular shocks for a system of conservation laws via the dafermos regularization u t f u x t u x x for a system modeling incompressible two phase fluid flow the existence of viscous profiles is proved using geometric singular perturbation theory the weak convergence and the growth rate of the viscous solution are also derived the weak limit is the sum of a piecewise constant function and a measure supported on a shock line and the maximum value of the viscous solution is of order exp 1

primes in the semigroup of boolean matrices abstract let a b c be n n matrices of zeros and ones using boolean addition and multiplication we say that a is prime if it is not a permutation matrix and if a bc implies that b or c must be a permutation matrix conditions sufficient for a matrix to be prime are provided and a characterization of primes in terms of a nation of rank is given finally an order property of primes is used to obtain a result on prime factors

the non transverse shil nikov hopf bifurcation uncoupling of homoclinic orbits and homoclinic tangencies abstract it is known that in a neighbourhood of a codimension two shil nikov hopf bifurcation primary periodic orbits lie on a single wiggly curve in period parameter space with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit in contrast it has recently been shown by hirschberg and laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit this paper analyses the codimension three bifurcation caused by a non transverse i e degenerately parametrically unfolded shil nikov hopf bifurcation which contains nearby dynamics akin to both degeneracies two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle focus homoclinic orbits to equilibria or the uncoupling of this locus from the locus of hopf bifurcations we undertake a heuristic analysis of the unfolding showing that in both cases it contains codimension two non transversal homoclinic orbits to equilibria and non transversal homoclinic tangencies to periodic orbits unfolding the former non transverse orbit is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas asymptotic expressions are given for the accumulation of two types of isola forming bifurcations the implications of z 2 equivariance on the unfolding is discussed finally numerical evidence is presented for both upward and downward pointing non transverse shil nikov hopf bifurcations occuring in a model of an autonomous non linear electronic circuit both with respect to a z 2 equivariant and a z 2 non equivariant equilibrium numerical computation of curves of periodic orbits homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications

one mechanism of appearance of a spiral quasiattractor involving heteroclinic contours abstract bifurcations and the structure of limit sets are studied for a three dimensional van der pol duffing system with a cubic nonlinearity on a base of both computer simulations and theoretical results a model map is proposed which allows one to follow the evolution in the phase space from a simple morse smale structure to chaos it is established that appearance of complex multistructural set of double scroll type is stipulated by the presence of a heteroclinic orbit of intersection of the unstable manifold of a saddle periodic orbit and stable manifold of an equilibrium state of saddle focus type

bifurcation from homoclinic to periodic solutions in singular ordinary differential equations abstract problems of bifurcations from homoclinic to periodic orbits are considered for periodic singular differential equations a functional analytic approach together with the method of lyapunov schmidt is used to obtain bifurcation functions for these problems which yield periodic solutions of all integer multiples of the period of the vector field

bifurcation and chaos near sliding homoclinics we study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface we assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations following a functional analytic approach we construct a melnikov like function m in such a way that if m has a simple zero at some point then the system has solutions that behave chaotically applications of this result to quasi periodic systems are also given

bifurcation from codimension one relative homoclinic cycles we study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group the relative homoclinic cycles we consider are not robust but have codimension one we assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions we show how suspensions of subshifts of finite type generically appear in the unfolding descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

discretizing dynamical systems with generalized hopf bifurcations we consider the discretizations of parameter dependent continuous time dynamical systems we show that the general one step methods shift a generalized hopf bifurcation and turn it into a generalized neimark sacker point we analyze the effect of discretization methods on the emanating hopf curve in particular we obtain estimates for the eigenvalues of the discretized system along this curve a detailed analysis of the discretized first lyapunov coefficient is also given the results are illustrated by a numerical example dynamical consequences are discussed

long time existence and bounded scalar curvature in the ricci harmonic flow abstract in this paper we study the long time existence of the ricci harmonic flow in terms of scalar curvature and weyl tensor which extends cao s result 6 in the ricci flow in dimension four we also study the integral bound of the riemann curvature for the ricci harmonic flow generalizing a recently result of simon 38

discrete dichotomies and bifurcations from critical homoclinic orbits abstract perturbed discrete systems likexn 1 f xn g xn xn r n n z when the associated unperturbed map 0 is not invertible and has acriticalorbit n homoclinic to a hyperbolic fixed pointpare studied bycriticalwe mean that thef n are invertible for any integern 0 butf 0 is not invertible the main goal is to give sufficient conditions for a bifurcationfrom zero to many homoclinicswhen the parameter crosses zero we also give a melnikov like result assuring thepersistence of homoclinicsin a complete neighborhood of 0 this result is similar to the ones obtained for diffeomorphisms and flows

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

differential harnack estimates for backward heat equations with potentials under the ricci flow abstract in this paper we derive a general evolution formula for possible harnack quantities as a consequence we prove several differential harnack inequalities for positive solutions of backward heat type equations with potentials including the conjugate heat equation under the ricci flow we shall also derive perelman s harnack inequality for the fundamental solution of the conjugate heat equation under the ricci flow

multi hump pulses in systems with reflection and phase invariance abstract we investigate the existence and stability of standing and travelling multi hump waves in partial differential equations with reflection and phase symmetries we focus on 2 and 3 pulse solutions that arise near bi foci and apply our results to the complex cubic quintic ginzburg landau equation

the logarithmic sobolev inequalities along geometric flow and applications abstract for some class of geometric flows we obtain the logarithmic sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non collapsing and non inflated properties which generalize the results in the case of ricci flow or list ricci flow or harmonic ricci flow as applications for mean curvature flow in lorentzian space with nonnegative sectional curvature and twisted kahler ricci flow on fano manifolds we get the results above

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

homoclinic snaking near a heteroclinic cycle in reversible systems snaking curves of homoclinic orbits have been found numerically in a number of ode models from water wave theory and structural mechanics along such a curve infinitely many fold bifurcation of homoclinic orbits occur thereby the corresponding solutions spread out and develop more and more bumps oscillations about their own centre a common feature of the examples is that the systems under consideration are reversible in this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria one of which is a bi focus using lin s method a snaking of 1 homoclinic orbits is proved to occur in an unfolding of such a cycle further dynamical consequences are discussed as an application a system of boussinesq equations is considered where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi focus equilibrium

an algorithm for melnikov functions and application to a chaotic rotor in this work we study a dynamical system with a complicated nonlinearity which describes oscillation of a turbine rotor and give an algorithm to compute melnikov functions for analysis of its chaotic behavior we first derive the rotor model whose nonlinear term brings difficulties to investigating the distribution and qualitative properties of its equilibria this nonlinear model provides a typical example of a system for which the homoclinic and heteroclinic orbits cannot be analytically determined in order to apply melnikov s method to make clear the underlying conditions for chaotic motion we present a generic algorithm that provides a systematic procedure to compute melnikov functions numerically substantial analysis is done so that the numerical approximation precision at each phase of the computation can be guaranteed using the algorithm developed in this paper it is straightforward to obtain a sufficient condition for chaotic motion under damping and periodic external excitation whenever the rotor parameters are given

multiple homoclinic orbits in conservative and reversible systems we study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows we consider the existence of two homoclinic orbits in the bellows configuration where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times in conservative systems one finds one parameter families of suspended horseshoes parameterized by the level of the first integral a somewhat similar picture occurs in reversible systems with two homoclinic orbits that are both symmetric the lack of a first integral implies that complete horseshoes do not exist we provide a description of orbits that necessarily do exist a second possible configuration in reversible systems occurs if a non symmetric homoclinic orbit exists and forms a bellows together with its symmetric image we describe the nonwandering set in an unfolding the nonwandering set is shown to simultaneously contain one parameter families of periodic orbits hyperbolic periodic orbits of different index and heteroclinic cycles between these periodic orbits

analysis of the t point hopf bifurcation abstract in a parameterized three dimensional system of autonomous differential equations a t point is a point of the parameter space where a special kind of codimension 2 heteroclinic cycle occurs if the parameter space is three dimensional such a bifurcation is located generically on a curve a more degenerate scenario appears when this curve reaches a surface of hopf bifurcations of one of the equilibria involved in the heteroclinic cycle we are interested in the analysis of this codimension 3 bifurcation which we call t point hopf in this work we propose a model based on the construction of a poincare map that describes the global behavior close to a t point hopf bifurcation the existence of certain kinds of homoclinic and heteroclinic connections between equilibria and or periodic orbits is proved the predictions deduced from this model strongly agree with the numerical results obtained in a modified van der pol duffing electronic oscillator

multipulses of nonlinearly coupled schrodinger equations abstract the capacity of coupled nonlinear schrodinger nls equations to support multipulse solutions multibump solitary waves is investigated a detailed analysis is undertaken for a system of quadratically coupled equations that describe the phenomena of second harmonic generation and parametric wave interaction in non centrosymmetric optical materials utilising the framework of homoclinic bifurcation theory and employing a lyapunov schmidt reduction method developed by hale lin and sandstede a novel mechanism for the generation of multipulses is identified which arises from a resonant semi simple eigenvalue configuration of the linearised steady state equations conditions for the existence of multipulses as well as a description of their geometry are derived from the analysis

exchange lemmas 1 deng s lemma abstract deng s lemma gives estimates on the behavior of solutions of ordinary differential equations in the neighborhood of a partially hyperbolic equilibrium we prove a generalization in which partially hyperbolic equilibrium is replaced by normally hyperbolic invariant manifold

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

steady solutions of the kuramoto sivashinsky equation abstract steady solutions of the kuramoto sivashinsky equation are studied these solutions are defined on the whole x line and propagate with a constant speed c 2 in time for large c 2 it is shown that the solution is unique and has a conical form for small c 2 there is a periodic solution and an infinite set of quasi periodic solutions as asserted by moser s twist map theorem numerical computations for intermediate values of c 2 suggest that below c 1 6 of every speed there is a continuum of odd quasi periodic solutions or a cantor set of chaotic solutions wrapped by infinite sequences of conic solutions

a gradient estimate for all positive solutions of the conjugate heat equation under ricci flow abstract we establish a point wise gradient estimate for all positive solutions of the conjugate heat equation this contrasts to perelman s point wise gradient estimate which works mainly for the fundamental solution rather than all solutions like perelman s estimate the most general form of our gradient estimate does not require any curvature assumption moreover assuming only lower bound on the ricci curvature we also prove a localized gradient estimate similar to the li yau estimate for the linear schrodinger heat equation the main difference with the linear case is that no assumptions on the derivatives of the potential scalar curvature are needed a classical harnack inequality follows

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

an adaptive newton picard algorithm with subspace iteration for computing periodic solutions this paper is concerned with the efficient computation of periodic orbits in large scale dynamical systems that arise after spatial discretization of partial differential equations pdes a hybrid newton picard scheme based on the shooting method is derived which in its simplest form is the recursive projection method rpm of shroff and keller siam j numer anal 30 1993 pp 1099 1120 and is used to compute and determine the stability of both stable and unstable periodic orbits the number of time integrations needed to obtain a solution is shown to be determined only by the system s dynamics this contrasts with traditional approaches based on newton s method for which the number of time integrations grows with the order of the spatial discretization two test examples are given to show the performance of the methods and to illustrate various theoretical points

using lin s method to solve bykov s problems abstract we consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria the constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions and called t points in this context we study codimension two t points and their unfoldings in r n in our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria in doing so we establish lin s method as a unified approach to re gain and extend results of bykov s seminal studies and related works to a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane

analytic and algebraic conditions for bifurcations of homoclinic orbits i saddle equilibria abstract we study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four dimensional systems which may be hamiltonian or not only one parameter is enough to treat these types of bifurcations in hamiltonian systems but two parameters are needed in general systems we apply a version of melnikov s method due to gruendler to obtain saddle node and pitchfork types of bifurcation results for homoclinic orbits furthermore we prove that if these bifurcations occur then the variational equations around the homoclinic orbits are integrable in the meaning of differential galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds we illustrate our theories with an example which arises as stationary states of coupled real ginzburg landau partial differential equations and demonstrate the theoretical results by numerical ones

lyapunov schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies abstract this article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits in addition tangencies in the intersection of un stable manifolds are allowed the main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time and their parameter values a leading order expansion is given in terms of the time spent near vertices and if applicable the location on the non trivial tangent directions the only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit the essential assumptions are hyperbolicity of the vertices and transversality of parameters using the result conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven finally equilibrium to periodic orbit heteroclinic cycles of various types are considered

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number