abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

evolution equations frobenius integrability conservation laws and travelling waves we give new results concerning the frobenius integrability and solution of evolution equations admitting travelling wave solutions in particular we give a powerful result which explains the extraordinary integrability of some of these equations we also discuss local conservations laws for evolution equations in general and demonstrate all the results for the korteweg de vries equation

comparative lagrangian formulations for localized buckling an energy functional for a strut on a nonlinear softening foundation is worked into two different lagrangian forms in fast and slow space respectively the developments originate independently of the underlying differential equation and carry some quite general features in each case the kinetic energy is an indefinite quadratic form in fast space this leads to an escape phenomenon with fractal properties in slow space kinetic energy is added to a potential contribution that is familiar from modal formulations together and in conjunction with a recent set of numerical experiments they illustrate the extra complexities of localized as opposed to distributed periodic buckling

galerkin approximations to static and dynamic localization problems abstract a new galerkin type procedure is established which unlike the classical approach does not rely on the final shape being composed of linearly independent modes the procedure is applied to the evolution of a localized buckle of a thin elastic strip within a visco elastic medium unlike the related elastic problem no clear cut linear eigenvalues exist to model wavelength and exponential growth decay in the tails of the buckle pattern the new procedure introduces variables to measure these effects and allows them to change in time this results in a more natural evolutionary process than with fixed mode shapes analysis is run within an algebraic manipulator m aple and checked against that of a numerical boundary value solver colpar

asymptotic and rayleigh ritz routes to localized buckling solutions in an elastic instability problem two complementary approximate techniques are developed to describe the subcritical localized deflection patterns of elastic struts resting on elastic foundations one is a doublescale perturbatio

heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible hamiltonian hopft bifurcation this paper considers an unfolding of a degenerate reversible 1 1 resonance or hamiltonian hopf bifurcation for four dimensional systems of time reversible ordinary differential equations odes this bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs the degeneracy occurs via the vanishing of a normal form coefficient q2 0 that determines whether the bifurcation is super or subcritical of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits a partial unfolding of such solutions already occurs in the work of dias and iooss eur j mech b fluids 15 1996 367 393 given a sign of the coefficient of a higher order term q4 0 and 1 q2 0 the normal form then shows a small amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit a genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies this is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies at each limit point the homoclinic orbit generates another bump close to the periodic orbit as q2 is further decreased from zero the snake widens until at a critical value a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium a kink for q2 less than this value the kinks heteroclinic tangencies and snake like curves no longer occur

complicated dynamics of scalar reaction diffusion equations with a nonlocal term we consider the dynamics of scalar equations ut uxx f x u c x u 0 x l where denotes some weighted spatial average and dinchlet boundary conditions are assumed prescribing f c appropriately it is shown that complicated dynamics can occur specifically linearisations at equilibria can have any number of purely imaginary eigenvalues moreover the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily up to any finite order these results are in marked contrast with the case 0 where bounded solutions are known to converge to equilibrium

arnold tongues and mode jumping in the supercritical post buckling of an archetypal elastic structure the buckling and post buckling of a classical simply supported elastic strut on a stiffening elastic foundation is treated as a nonlinear dynamical system for supercritical loads this fourth order system has a pair of imaginary eigenvalues indicating the possibility of either a double or a quasi periodic response finite lengths impose constraints that restrict such responses to the rational positions on the real line of the ratio of competing wavelengths for irrational positions the response locks onto a single periodic wave the extent of the mode locking is described in parameter space by the geometry of arnold tongues critical positions on the tongues are identified which mark conditions for mode jumping from one wavelength to another

geometric aspects of reduction of order using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry in particular we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds in so doing some new results which generalise the class of lie group actions which can be used to solve ordinary differential equations are developed

effects of exponentially small terms in the perturbation approach to localized buckling localized buckling solutions are known to exist in the heuristic model of an elastic strut resting on an elastic winkler foundation the primary localized solution emerging from the critical buckling state consists of an amplitude envelope of approximately the form of a hyperbolic secant function which modulates a fast varying sinusoidal oscillation in previous works such solutions have been tracked for the entire post buckling regime both numerically and through a rayleigh ritz approach a very rich structure is known to exist in the subcritical load range but has been proved to exist for only a certain family of reversible systems studies have concentrated on symmetric homoclinic solutions and the asymmetric solutions which bifurcate from these solution paths the primary solution is known to exist for the entire subcritical parameter range and all other symmetric and associated asymmetric solutions exist strictly for values less than critical here we uncover a new family of antisymmetric solutions and some asymptotic analysis suggests that the primary antisymmetric solution exists over the same range as does the primary symmetric solution a perturbation approach can be used so as to describe the bifurcation hierarchy for the novel antisymmetric forms we illustrate a unified approach which is able to predict the circumstances under which non divergent localized solutions are possible and the results of the analysis are compared with some numerical solutions

on the periodic orbits of the fourth order differential equation u qu u ef u u u u we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu u ef u u u u where q and e are real parameters e is small and f is a nonlinear function

global asymptotics in some quasimonotone reaction diffusion systems with delays abstract both dichotomy and trichotomy on the global asymptotics of some quasimonotone reaction diffusion systems with delays are established in terms of the principal eigenvalue of linear weakly coupled elliptic systems applications to a class of delayed reaction diffusion models of single species growth and to a reaction diffusion system with delay modelling the spread of bacterial infections are provided

hidden symmetry concepts in the elastic buckling of axially loaded cylinders abstract the recognized harmonic buckling modes for the elastic axially loaded cylinder are by nature symmetric equal and opposite amplitudes giving equal energy levels on the other hand in combination they account for considerable asymmetry inwards deflection being fundamentally different from outwards this asymmetry is also apparent in the underlying differential equations and in the final large deflection yoshimura pattern taking the view that underlying symmetries are closely linked to the form of bifurcation experienced on buckling and hence to the gross instability of the phenomenon the paper thus explores a classic problem in a new light the wellknown donnell equations are first employed the analysis being neatly written in terms of the two variables radial displacement w and stress function an extension is then presented derived from the full strain displacement equations appropriate to genuinely large deflections this makes use of a suite of computer programs written for a small micro which handle the required manipulations of multiplication and integration of harmonic functions in algebraic rather than numerical fashion

a multiplicity of localised buckling modes for twisted rod equations the kirchhoff love equations governing the spatial equilibria of long thin elastic rods subject to end tension and moment are reviewed and used to examine the existence of localized buckling solutions the effects of shear and axial extension are not considered but the model does additionally allow for nonlinear constitutive laws under the assumption of infinite length the dynamical phase space analogy allows one to use techniques from dynamical systems theory to characterize many possible equilibrium paths localizing solutions correspond to homoclinic orbits of the dynamical system under non dimensionalization the twisted rod equations are shown to depend on a single load parameter and the bifurcation behaviour of localizing solutions of this problem is investigated using analytical and numerical techniques first in the case of a rod with equal principal bending stiffnesses where the equilibrium equations are completely integrable a known one parameter family of localizing solutions is computed for a variety of subcritical loads load deflection diagrams are computed for this family and certain materially nonlinear constitutive laws are shown to make little difference to the qualitative picture the breaking of the geometrical circular symmetry destroys complete integrability and in particular breaks the non transverse intersection of the stable and unstable manifolds of the trivial steady state the resulting transverse intersection which is already known to lead to spatial chaos is explicitly demonstrated to imply a multitude of localized buckling modes a sample of primary and multi modal solutions are computed numerically aided by the reversibility of the differential equations finally parallels are drawn with the conceptually simpler problem of a strut resting on a nonlinear elastic foundation for which much more information is known about the global behaviour of localized buckling modes

on the limit cycles for a class of fourth order differential equations we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation where p is a rational number different from 0 e is small and f is a nonlinear function

new symmetry solution techniques for first order non linear pdes using recent extensions of work of s lie vorlesungen uber differentialgleichungen mit bekannten infinitesimalen transformationen leipzig teubner 1891 and e cartan lecons sur les invariants integreaux hermann paris 1922 les systems differentials exterieures et leurs applications geometrique hermann paris 1945 for integrating frobenius integrable vector field distributions via symmetry we examine some symmetry techniques for finding local solutions of first order non linear partial differential equations pdes in the language of exterior differential systems we develop a technique for solving first order quasilinear pdes that is then applied to general first order non linear partial differential equations our results are significant inasmuch as we give algorithms for solving first order pdes in the presence of symmetry that are essentially mechanical in nature and done in the original coordinate system

structural localization phenomena and the dynamical phase space analogy the localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints a from a modal perspective b from a formulation which allows the amplitude to modulate in an asymptotically defined slow space and c from a dynamical analogy in phase space suggested by the form of the underlying differential equation a simple strut on an asymmetric nonlinear foundation provides a typical illustrative example the three approaches emphasize different features of the localization phenomenon the modal view illustrates the distinctive effects of boundary conditions the modulated approach generates a convenient second order differential equation in the amplitude function and the dynamical phase space analogy suggests a useful interpretation of localization as a homoclinic connection comparisons are also made with nonlinear numerical solutions as the strut length approaches infinity it is shown that the fully localized solution represents the unstable post buckled state with the lowest energy allowing evaluation of the minimum energy barrier relevant to dynamical impact studies attention is drawn to the possibility of spatial chaos becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

group foliation and non invariant solutions of the heavenly equation the main physical results of this paper are new exact analytical solutions of the heavenly equation of importance in the general theory of relativity these solutions are not invariant under any subgroup of the symmetry group of the equation the main mathematical result is a new method of obtaining non invariant solutions of partial differential equations with infinite dimensional symmetry groups the method involves the compatibility of the given equations with a differential constraint which is automorphic under a specific symmetry subgroup the latter acting transitively on the submanifold of the common solutions by studying the integrability of the resulting conditions one can provide an explicit foliation of the entire solution manifold of the considered equations

finite element modelling of spatially chaotic structures a simple non linear mechanical system comprising a pin jointed string of finite length links supported by elastic springs at the pins and compressed by an axial load is viewed from two perspectives when seen as an initial value problem equilibrium equations provide an iterative non linear mapping when seen as a boundary value problem it becomes a simple finite element model at loads less than the critical buckling load a preferred buckling configuration is found that is localized along the length in the limit of infinite length this is described as a homoclinic connection in phase space joining the flat equilibrium state to itself the infinite sequence of homoclinic points thus defined embeds within the complex topological structure of a homoclinic tangle within which also appear periodic quasi periodic and chaotic spatial solutions implications in the finite element setting are discussed 1997 john wiley sons ltd

rayleigh ritz analysis for localized buckling of a strut on a softening foundation by hermite functions this paper proposes a rayleigh ritz procedure for localized buckling of a strut on a non linear elastic foundation firstly the deflected shape of a strut is expanded into a series of hermite orthogonal functions which are proved energy integrable in an infinite region secondly the errors of the numerical integrations of hermite functions on the infinite region are investigated and the suitable integral limit is proposed through the numerical investigation it is demonstrated that the first thirty hermite functions are usually enough to approximate the localized buckling pattern the proposed method overcomes the disadvantages of the traditional methods in which the trial functions in either rayleigh ritz or galerkin analysis are based on the perturbation analyses of the corresponding non linear differential equation

symmetry and uniqueness of the solitary wave solution for the ostrovsky equation we consider herein the ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water or the capillary waves in a plasma with the effects of rotation using the modified sliding method we prove that the solitary wave moving to the left to the ostrovsky equation is symmetric about the origin and unique up to translations we also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed weak rotation and dispersive parameter

the evolution of invariant manifolds in hamiltonian hopf bifurcations we consider the evolution of the stable and unstable manifolds of an equilibrium point of a hamiltonian system of two degrees of freedom which depends on a parameter n the eigenvalues of the linearized system are complex for no0 and pure imaginary for n40 thus for no0 the equilibrium has a two dimensional stable manifold and a two dimensional unstable manifold but for n40 these stable and unstable manifolds are gone if the sign of a certain term in the normal form is positive then for small negative n the stable and unstable manifolds of the system are either identical or must have transverse intersection thus either the system is totally degenerate or the system admits a suspended smale horseshoe as an invariant set r 2002 elsevier science usa all rights reserved

a nonlocal sturm liouville eigenvalue problem a nonlocal eigenvalue problem of the form u a x u bu u with homogeneous dirichlet boundary conditions is considered where b is a rank one bounded linear operator and x belongs to some bounded interval on the real line the behaviour of the eigenvalues is studied using methods of linear perturbation theory in particular some results are given which ensure that the spectrum remains real a sturm type comparison result is obtained finally these results are applied to the study of some nonlocal reaction diffusion equations

flow invariance for nonlinear partial differential delay equations several fundamental results on existence flow invariance regularity and linearized stability of solutions to the nonlinear partial differential delay equation u t bu t f u t t 0 uo with b c x x x w accretive are developed for a general banach space x in contrast to existing results with the history response f globally defined and at least lipschitz on bounded sets the results are tailored for situations with f defined on possibly thin subsets of the initial history space e only and are applied to place several classes of population models in their natural l 1 setting the main result solves the open problem of a subtangential condition for flow invariance of solutions in the fully nonlinear case paralleling those known for the cases of a no delay b ordinary delay equations with b 0 and c the semilinear case

dynamics of a reaction diffusion equation with nonlocal inhibition abstract in this paper we investigate the dynamics of a nonlocal scalar reaction diffusion equation which corresponds to the limiting equation for a singularly perturbed activator inhibitor system the equation is a nonlocal perturbation of the chafee infante problem it is shown that the nonlocal equation generates a gradient flow whose dynamics can differ from the local problem in several interesting ways stable heterogeneous equilibria having any number of zeros on the spatial domain can be obtained by an appropriate choice of parameters the steady state equation exhibits an interesting family of secondary bifurcations near double critical points by analyzing the center manifolds dynamics near the double critical points it is found that a sequence of secondary bifurcations can result in a transfer of stability between two equilibria by reversing a heteroclinic connection between them as a consequence the number of zeros of a solution the lap number can increase along trajectories

travelling wave fronts in reaction diffusion systems with spatio temporal delays this paper deals with the existence of travelling wave fronts in reaction diffusion systems with spatio temporal delays our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system new iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions following this the existence of travelling wave fronts for reaction diffusion systems with spatio temporal delays is established finally we apply the main results to a single species diffusive model with spatio temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels

coupled nonlinear parabolic systems with time delays a coupled system of parabolic equations with time delays is investigated by the method of upper lower solutions it is shown by using upper and lower solutions as coupled initial iterations that the corresponding sequences converge monotonically to a unique solution of the system this convergence leads to an existence comparison theorem which is useful for studying the asymptotic behavior of the solution this method is extended to a coupled system of integro differential equations and systems of parabolic ordinary equations applications are given to some model problems in ecology and nuclear engineering including the stability property of a steady state solution

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

localization and mode interaction in sandwich structures compressed sandwich structures comprising two stiff face plates separated by a softer core material while designed principally as efficient integral structures can lose this quality when faces buckle locally interaction between overall euler buckling and local buckling of one face suggests that failure will localize into the centre a variational formulation leading to a pair of nonlinear differential equations subject to integral constraints describes the post buckling response these are solved by a combination of numerical shooting and continuation techniques such that the response far into the unstable post buckling regime can be portrayed solutions with both linear and nonlinear constitutive core relations are compared with the results of an engineering body force approach and with those of earlier periodic rayleigh ritz analyses the latter demonstrate the extra destabilization that comes with localization

convergence to equilibrium in degenerate parabolic equations with delay abstract in 11 busenberg huang 1996 showed that small positive equilibria can undergo supercritical hopf bifurcation in a delay logistic reaction diffusion equation with dirichlet boundary conditions consequently stable spatially inhomogeneous time periodic solutions exist previously in 12 badii diaz tesei 1987 considered a similar logistic type delay diffusion equation but differing in two important respects firstly by the inclusion of nonlinear degenerate diffusion of so called porous medium type and secondly by the inclusion of an additional dominating instantaneous negative feedback where terms local in time majorize the delay terms in some sense sufficient conditions were given ensuring convergence of non negative solutions to a unique positive equilibrium a natural question to ask and one which motivated the present work is can one still ensure convergence to equilibrium in delay logistic diffusion equations in the presence of nonlinear degenerate diffusion but in the absence of dominating instantaneous negative feedback the present paper considers this question and provides sufficient conditions to answer in the affirmative in fact the results are much stronger establishing global convergence for a much wider class of problems which generalize the porous medium diffusion and delay logistic terms to larger classes of nonlinearities furthermore the results obtained are independent of the size of the delay

homoclinic bifurcations in reversible hamiltonian systems abstract we study the existence of homoclic solutions for reversible hamiltonian systems taking the family of differential equations u iv au u f u b 0 as a model where f is an analytic function and a b real parameters these equations are important in several physical situations such as solitons and in the existence of finite energy stationary states of partial differential equations but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others hamiltonian systems and successfully employed in situations where standard methods fail we reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations we also plot the homoclinic values configuration in parameters space giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations

global asymptotic behaviour in some functional parabolic equations the global asymptotic behavior in some functional parabolic equations was presented the sufficient condition for the global convergence to a spatially uniform equilibrium in the system of functional partial differential equations was provided results were observed to be valid for all non negative spatio temporal kernels which also satisfied a uniform normailzation condition

superposition formulas for exterior differential systems abstract in this article we solve an inverse problem in the theory of quotients for differential equations we characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original the fact that a system can be written as a quotient can be used to find the general solution to these equations some examples are given to demonstrate the theory

flow invariance for solutions to nonlinear nonautonomous partial differential delay equations we investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations u t b t u t f t ut t s us where b t is a family of nonlinear multivalued accretive operators with d b t possibly depending on t and the operators f t being defined and lipschitz continuous possibly only on thin subsets of the initial history space e the results are applied to population dynamics models we also study the asymptotic behavior of solutions to this equation our analysis will be based on the evolution operator associated to the equation in the initial history space e

oscillation of a class of partial functional population model abstract this paper is concerned with the oscillation of a class of partial functional population model and some sufficient conditions for all positive solutions of the model to oscillate about the positive equilibrium are obtained the approach is based on the upper and lower solution method of the partial functional differential equations and the oscillation theory of the functional differential equation moreover some numerical simulations are also given to illustrate our results

explicit solutions of pde via vessiot theory and solvable structures we consider the problem of computing the integrable sub distributions of the non integrable vessiot distribution of multi dimensional second order partial differential equations pdes we use vessiot theory and solvable structures to find the largest integrable distributions contained in the vessiot distribution associated to second order pdes in particular we show how the solvable symmetry structure of the original pde can be used to construct integrable sub distributions leading to group invariant solutions of the pde in two and more than two independent variables

integration of pdes by differential geometric means we use vessiot theory and exterior calculus to solve partial differential equations pdes of the type uyy f x y u ux uy uxx uxy and associated evolution equations these equations are represented by the vessiot distribution of vector fields we develop and apply an algorithm to find the largest integrable sub distributions and hence solutions of the pdes we then apply the integrating factor technique sherring and prince 1992 trans am math soc 433 453 to integrate this integrable vessiot sub distribution the method is successfully applied to a large class of linear and nonlinear pdes

solitary wave interaction phenomena in a strut buckling model incorporating restabilisation the archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure the primary solution can be detected by a double scale perturbation procedure and is reminiscent of a solitary wave essentially it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions particular interest is paid to the process of adapting the procedure to account for the post buckling behaviour of two packet or double humped solitary waves in this model we employ the methods of beyond all orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post buckling behaviour of the system including the interaction phenomenon of interest the analysis is reinforced by direct numerical computations which reveal the so called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions however additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram

a galerkin procedure for localized buckling of a strut on a nonlinear elastic foundation working from the governing differential equation a galerkin procedure has been developed to model the localized buckle pattern of an elastic strut on a nonlinear elastic softening winkler foundation the choice of trial functions comes directly from a perturbation study about the critical point the actual shape being determined by the eigenvalues of the linearized equation a good comparison has been achieved with independently obtained numerical solutions using only a few modes other measures of the accuracy of the approximate solutions have also proven to be favourable

a picture of the global bifurcation diagram in ecological interacting and diffusing systems abstract this paper concerns global phenomena of pattern formation in stationary reaction diffusion equations possessing turing s diffusion induced instability which appear typically in mathematical biology global bifurcation diagrams with respect to two diffusion parameters are presented by integrating two complementary approaches analytical and numerical simple and double bifurcation analysis using group theoretic methods for the compact lie group d x results from singular perturbations when one diffusion constant is sufficiently small and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large are the main analytical results a new numerical method is used to trace all bifurcating branches we examine interrelations between those local and semi global results under the light of global pictures obtained by numerical studies a variety of interesting and new diagrams near double eigenvalues are observed coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

the spatial complexity of localized buckling in rods with noncircular cross section we study the postbuckling behavior of long thin elastic rods subject to end moment and tension this problem in statics has the well known kirchhoff dynamic analogy in rigid body mechanics consisting of a reversible three degrees of freedom hamiltonian system for rods with noncircular cross section this dynamical system is in general nonintegrable and in dimensionless form depends on two parameters a unified load parameter and a geometric parameter measuring the anisotropy of the cross section previous work has given strong evidence of the existence of a countable infinity of localized buckling modes which in the dynamic analogy correspond to n pulse homoclinic orbits to the trivial solution representing the straight rod this paper presents a systematic numerical study of a large sample of these buckling modes the solutions are found by applying a recently developed shooting method which exploits the reversibility of the system subsequent continuation of the homoclinic orbits as parameters are vari

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

systems of parabolic equations with continuous and discrete delays abstract in this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear reaction function may depend on both continuous infinite or finite and discrete delays it is shown that if the reaction function is locally lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function for systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system this approach yields a global attractor of the parabolic system and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system application to three model problems in population dynamics and chemical reactions is given

stability and bifurcation of a predator prey model with disease in the prey and temporal spatial nonlocal effect in this paper we consider the dynamics of a predator prey model with disease in the prey and ratio dependent michaelis menten functional response the model is a reaction diffusion system with a nonlocal term representing the temporal spatial weighted average for the prey density the limiting case of the system reduces to the lotka volterra diffusive system with logistic growth of the prey we study the linear stability of the two non trivial steady states either with or without nonlocal term the bifurcations to three types of periodic solutions occurring from the coexistence steady state are investigated for two particular kernels which reveal the important significance of temporal spatial nonlocal effects