global existence and nonexistence of solution for cauchy problem of two dimensional generalized boussinesq equations abstract in this paper we consider the cauchy problem of two dimensional generalized boussinesq type equation u t t u u t t 2 u f u 0 under the assumption that f u is a function with exponential growth at infinity and under some assumptions on the initial data we prove the existence and nonexistence of global weak solution there are very few works on boussinesq equation with nonlinear exponential growth term by potential well theory

global existence and exponential growth of solution for the logarithmic boussinesq type equation abstract in this paper we consider the boussinesq type equation associated with logarithmic nonlinear term and initial boundary value conditions by using potential well method combined with the logarithmic sobolev inequality we obtain the existence of global solution at last we show that the l 2 norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions for initial data the result shows that the polynomial nonlinearity is a critical condition of blow up in finite time for the solutions of boussinesq equations

global existence and blow up of solutions for cauchy problem of generalized boussinesq equation abstract we study the cauchy problem of generalized boussinesq equation u t t u x x u x x f u x x 0 where f u u p or u p 1 u p 1 by introducing a family of potential wells we obtain invariant sets vacuum isolating and threshold result of global existence and nonexistence of solution

on the determination of the basin of attraction of a periodic orbit in two dimensional systems we consider the general nonlinear differential equation x f x with x epsilon r 2 and develop a method to determine the basin of attraction of a periodic orbit borg s criterion provides a method to prove existence uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction in order to use the criterion one has to find a function w epsilon c 1 r 2 r such that l w x w x l x is negative for all x epsilon k where k is a positively invariant set here l x is a given function and w x denotes the orbital derivative of w in this paper we prove the existence and smoothness of a function w such that l w x mu parallel to f x parallel to we approximate the function w which satisfies the linear partial differential equation w x mu parallel to f x parallel to l x using radial basis functions and obtain an approximation omega such that l omega x 0 using radial basis functions again we determine a positively invariant set k so that we can apply borg s criterion as an example we apply the method to the van der pol equation

some physical structures for the 2 1 dimensional boussinesq water equation with positive and negative exponents in this paper a mathematical method is constructed to study two variants of the two dimensional boussinesq water equation with positive and negative exponents in terms of travelling wave solutions the partial differential equations are transformed to nonlinear ordinary differential equations exact solutions are then derived for various cases to describe the different physical structures such as compactons solitons solitary patterns and periodic solutions the exponent of the wave function u and the ratio of the two coefficients a and b in the boussinesq equation are shown to qualitatively determine the physical structures of the solutions

heteroclinic bifurcation and singularly perturbed boundary value problems abstract we study a singularly perturbed boundary value problem in r m n x f x y e e y g x y e b 1 x 0 y 0 e 0 b 2 x 0 y 0 e 0 given a candidate for the 0th order approximation which exhibits both boundary layers and interior layers we present a complete procedure to compute higher order expansions and a procedure to compute the real solution near a truncated asymptotic expansion assuming the hyperbolicity of the regular layers and some generic assumptions similar results concerning the existence of periodic solutions relaxation oscillations are also presented several ideas from dynamical systems theory are employed e g exponential dichotomies fredholm alternatives and heteroclinic bifurcations

on the determination of the basin of attraction of periodic orbits in three and higher dimensional systems the determination of the basin of attraction of a periodic orbit can be achieved using a lyapunov function a lyapunov function can be constructed by approximation of a first order linear pde for the orbital derivative via meshless collocation however if the periodic orbit is only accessible numerically a different method has to be used near the periodic orbit borg s criterion provides a method to obtain information about the basin of attraction by measuring whether adjacent solutions approach each other with respect to a riemannian metric using a numerical approximation of the periodic orbit and its first variation equation a suitable riemannian metric is constructed

the cauchy problem for a class of the multidimensional boussinesq type equation abstract in this paper we study the cauchy problem for a class of the multidimensional boussinesq type equation u t t u 2 u 2 u t t f u where f u a u p or a u p 1 u a 0 is a constant first we establish a local existence theorem for the solution then for m 1 n 1 m 2 3 4 n 3 we prove the existence of a global h m solution finally we prove the global nonexistence and finite time blow up of the solution

uniqueness and stability of periodic bursting solutions a detailed analysis of so called square bursting oscillators is given an interesting feature of these models is that the bursting solution need not be unique or stable for arbitrarily small values of a singular perturbation parameter this is a global phenomenon due to interactions between a homoclinic orbit and other invariant manifolds using geometric singular perturbation methods we characterize the set of parameters for which the bursting solution is uniquely determined and asymptotically stable

converse theorems on contraction metrics for an equilibrium the stability and basin of attraction of an equilibrium can be determined by a contraction metric a contraction metric is a riemannian metric with respect to which the distance between adjacent trajectories decreases the advantage of a contraction metric over e g a lyapunov function is that the contraction condition is robust under perturbations of the system while the sufficiency of a contraction metric for the existence stability and basin of attraction of an equilibrium has been extensively studied in this paper we will prove converse theorems showing the existence of several different contraction metrics this will be useful to develop algorithms for the construction of contraction metrics

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

on the cauchy problem for a generalized boussinesq equation in this paper we consider the cauchy problem for a generalized boussinesq equation we show that under suitable conditions a global solution for the initial value problem exists in addition we derive the sufficient conditions for the blow up of the solution to the problem

linear b spline finite element method for the improved boussinesq equation in this paper we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved boussinesq equation the finite element method with linear b spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives it is shown that the coefficient matrix for the second order term in this system is invertible consequently for the first time the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting wave interaction and blow up behavior

low and high frequency oscillations in three dimensional food chain systems the dynamic behavior of a three trophic level food chain prey predator superpredator is analyzed in this paper prey is logistic while predator and superpredator have functional response of the holling type moreover the trophic levels are assumed to be characterized by increasing and quite diversified time responses using a singular perturbation approach explicit conditions for the persistence of the three populations are derived and the structure of the corresponding attractors is noted as well as the nature of transients the analysis shows that the system can have low frequency cycles due to the interactions between predator and superpredator nevertheless the interactions between prey and predator can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low frequency cycle this periodic burst of high frequency oscillations develops in particular when the two predators are fairly efficient these results note in a very concise w

necessary conditions for a limit cycle and its basin of attraction in this paper we consider a general differential equation of the form x f x with f epsilon c l r n r n and n greater than or equal to 2 borg hartman leonov and others have studied sufficient conditions for the existence uniqueness and exponential stability of a periodic orbit and for a set to belong to its basin of attraction they used a certain contraction property of the flow with respect to the euclidian or a riemannian metric in this paper we also prove sufficient conditions including upper bounds for the floquet exponents of the periodic orbit moreover we show the necessity of these conditions using floquet theory and a lyapunov function

construction of a cpa contraction metric for periodic orbits using semidefinite optimization abstract a riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction while the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit the explicit construction of the metric is a difficult problem in this paper the construction of such a contraction metric is achieved by formulating it as an equivalent problem namely a feasibility problem in semidefinite optimization the contraction metric a matrix valued function is constructed as a continuous piecewise affine cpa function which is affine on each simplex of a triangulation of the phase space the contraction conditions are formulated as conditions on the values at the vertices the paper states a semidefinite optimization problem we prove on the one hand that a feasible solution of the optimization problem determines a cpa contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough an objective function can be used to obtain a bound on the largest floquet exponent of the periodic orbit

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

global existence of cauchy problem for boussinesq paradigm equation in this paper we study the cauchy problem for the boussinesq paradigm equation u t email protected b email protected t email protected email protected email protected u where f u email protected u p new functionals are introduced and their sign preserving properties under the flow of the equation are studied the existence of a global weak solution with supercritical initial energy is proved the explicit value of the critical energy constant is found

meshless collocation error estimates with application to dynamical systems in this paper we derive error estimates for generalized interpolation in particular collocation in sobolev spaces we employ our estimates in collocation problems using radial basis functions and extend and improve previously known results for elliptic problems finally we use meshless collocation to approximate lyapunov functions for dynamical systems

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

propagation phenomena in a bistable reaction diffusion system consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical biochemical systems these equations are of the fitzhugh nagumo type and contain three parameters for certain ranges of the parameters the system exhibits two stable singular points a singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves a complete rigorous description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity numerical results of solutions to these equations are also presented these calculations illustrate how waves are generated from initial data how they interact during collisions and how they are influenced by local disturbances and boundary conditions in one example a righltward moving

an application of the separation principle for detecting slow fast limit cycles in a three dimensional system a geometric method called the separation principle is used in this paper for detecting limit cycles in a slow fast dynamical system describing a food chain with trophic levels with diversified time responses the analysis is very simple and based on singular perturbation arguments the limit cycle resulting from the method is composed of a concatenation of catastrophic transitions occuring at different speeds

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

instability of standing waves for a class of nonlinear schrodinger equations this paper discusses a class of nonlinear schrodinger equations with different power nonlinearities we first establish the existence of standing wave associated with the ground states by variational calculus then by the potential well argument and the concavity method we get a sharp condition for blowup and global existence to the solutions of the cauchy problem and answer such a problem how small are the initial data the global solutions exist at last we prove the instability of standing wave by combing those results

continuation and bifurcations numerical techniques and applications bifurcation to rotating waves from non trivial steady states use of approximate inertial manifolds in bifurcation calculations understanding steady state bifurcation diagrams for a model reaction diffusion system bifurcations chaos and self organization in reaction diffusion systems eigenvalue problems with the symmetry of a group and bifurcations steady state steady state mode interaction in nonlinear equations with z2 symmetry symbolic computation and bifurcation methods bifurcation analysis a combined numerical and analytical approach to the numerical solution of symmetry breaking bifurcation problems a computational method and path following for periodic solutions with symmetry global bifurcations and their numerical computation computation of invariant manifold bifurcations a method for homoclinic and heteroclinic continuation in two and three dimensions the global attractor under discretisation the numerical detection of hopf bifurcation points a newton like method for simple bifurcation problems with application to large sparse systems aspects of continuation software interactive system for studies in nonlinear dynamics linlbf a program for continuation and bifurcation analysis of equilibria up to codimension three on the topology of three dimensional separations a guide for classification the construction of cubature formulae using continuation and bifurcation software determining an organizing center for passive optical systems optimization by continuation stability of marangoni convection in a microgravity environment computing with reaction diffusion systems applications in image processing bifurcation of codimension 2 for a discrete map bifurcation of periodic solutions in pde s numerical techniques and applications abstract bifurcation and chaos in chua s circuit family abstract continuation and collocation for parameter dependent boundary value problems abstract porous medium combustion abstract block elimination and the computation of simple turning points abstract bifurcation into gaps in the essential spectrum abstract application of numerical continuation in aerospace problems abstract application of a reduced basis method in structural analysis abstract some applications of bifurcation theory in engineering abstract higher order predictors in numerical path following schemes abstract

global existence to generalized boussinesq equation with combined power type nonlinearities abstract the cauchy problem to the generalized boussinesq equation with combined power type nonlinearities is studied global solvability or finite time blow up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the nehari functional for generalized lienard or generalized bernoulli nonlinear terms the critical energy constant is explicitly evaluated a new method that can be considered as a modification of the potential well method is developed the performed numerical experiments support the theoretical results

a dynamical system with hopf bifurcations and catastrophes a well known second order positive dynamical system describing the dynamics of a renewable resource together with its consumers prey predator model is first considered such a dynamical system is recognized to be a subtle hopf bifurcation without catastrophes then the model is slightly modified by including a control parameter which represents the indirect exploitation of the resource this new system is shown to have a much richer and interesting dynamics in particular catastrophes of the fold type hysteresis may arise moreover when the control value corresponding to the hopf bifurcation is greater than the lower threshold of the hysteresis one of the two transitions characterizing the catastrophe is a transition between a stableequilibrium and a limit cycle the results are illustrated in the control phase portrait

on the asymptotic behavior of solution for the generalized ibq equation with hydrodynamical damped term abstract we study the cauchy problem for the generalized ibq equation with hydrodynamical damped term in n dimensional space we observe that the dissipative structure of the linearized equation is of the regularity loss type this means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data under smallness condition on the initial data we prove the global existence and decay of the small amplitude solution in the sobolev space

decay estimates for a dissipative dispersive linear semigroup and application to the viscous boussinesq equation abstract at the core of this article is the new estimate for a class of dissipative dispersive linear semigroup e t i p t arising in the study of the viscous boussinesq equation we combine the decay estimate with introducing a set of time weighted sobolev spaces where the time weights and the regularity of the sobolev spaces are determined by our decay estimate to show the global existence and asymptotic behavior of solutions to the viscous boussinesq equation in r n

single and multiple pulse waves for the fitzhugh nagumo the fitzhugh nagumo equations describe the propagation of impulses along a nerve axon we study traveling wave solutions which satisfy a set of ordinary differential equations with a cubic nonlinearity f v v 1 v v a our emphasis is on solutions which tend to zero at pm infty an improved proof of a previous result that there are at least two such solutions is presented further by allowing the middle root v a of the nonlinearity to be a small parameter better estimates of the parameter range where such solutions exist are given in view of recent work of evans fenichel feroe and sil nikov our results imply that there are a great many more bounded traveling waves than was known previously in particular there is an infinite family of periodic waves all traveling with the same speed a reasonable transversality hypothesis introduced by evans fenichel and feroe implies also that there are multiple pulse waves at speeds close to that of a single pulse

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz