bi relative algebraic k theory and topological cyclic homology it is well known that algebraic k theory preserves products of rings however in general algebraic k theory does not preserve fiber products of rings and one defines bi relative algebraic k theory so as to measure the deviation it was proved recently by cortinas 6 that rationally bi relative algebraic k theory and bi relative negative cyclic homology agree in this paper we show that with finite coefficients bi relative algebraic k theory and bi relative topological cyclic homology agree as an application we show that for a possibly singular curve over a perfect field k of positive characteristic p the cyclotomic trace map induces an isomorphism of the p adic algebraic k groups and the p adic topological cyclic homology groups in non negative degrees as a further application we show that the difference between the p adic k groups of the integral group ring of a finite group and the p adic k groups of a maximal z order in the rational group algebra can be expressed entirely in terms of topological cyclic homology let f be a functor that to a unital associative ring a associates a symmetric spectrum f a if i a is a two sided ideal the relative term f a i is defined to be the homotopy fiber of the map f a f a i induced from the canonical projection if further f a b is a ring homomorphism such that f i f i is an isomorphism onto a two sided ideal of b there is an induced map f a i f b f i and the bi relative term f a b i is defined to be the homotopy fiber of this map then there is a distinguished triangle in the stable homotopy category

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

the obstruction to excision in k theory and in cyclic homology let f a b be a ring homomorphism of not necessarily unital rings and i triangleleft a an ideal which is mapped by f isomorphically to an ideal of b the obstruction to excision in k theory is the failure of the map between relative k groups k a i k b f i to be an isomorphism it is measured by the birelative groups k a b i similarly the groups hn a b i measure the obstruction to excision in negative cyclic homology we show that the rational jones goodwillie chern character induces an isomorphism ch k a b i otimes mathbb q overset sim to hn a otimes mathbb q b otimes mathbb q i otimes mathbb q

on the initial value problem for the modified benjamin ono equation the initial value problem for the modified benjamin ono equation is solved by exploiting the fokas ablowitz inverse scattering transform for a certain complex extension of the benjamin ono equation the asymptotics of the jost functions for small positive values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case

periodic ilw equation with discrete laplacian we study an integro differential equation which generalizes the periodic intermediate long wave ilw equation the kernel of the singular integral involved is an elliptic function written as a second order difference of the weierstrass function using sato s formulation we show the integrability and construct some special solutions an elliptic solution is also obtained we present a conjecture based on a poisson structure that gives an alternative description of this integrable hierarchy we note that this poisson algebra in turn is related to a quantum algebra related to the family of macdonald difference operators

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

linear stability analysis for the multikink solution of the sine hilbert equation the linear stability analysis is performed for the multikink solution of the sine hilbert sh equation we first solve the eigenvalue problem of the sh equation linearized about the multikink solution and then establish the completeness relation among the eigenfunctions this relation enables us to solve the initial value problem of the linearized sh equation proving the stability of the solution against infinitesimal perturbations as an application of the completeness relation we develop a direct multikink perturbation theory while employing the method of multiple time scales we derive the evolution equations that describe the slow change of the kink parameters as well as a nonsecular expression for the first order correction to the multikink solution a novelty of the present analysis is that the proof of the completeness relation is done by purely algebraic means with the help of the classical theory of algebraic equations

water propagation in two dimensional petroleum reservoirs abstract in the present paper we investigate the problem of water propagation in 2 dimensional 2d petroleum reservoir in which each site has the probability p of being occupied we first analyze this propagation pattern described by darcy equations by focusing on its geometrical features we find that the domain walls of this model at p p c 0 59 are schramm loewner evolution sle curves with 3 05 0 1 consistent with the ising universality class we also numerically show that the fractal dimension of these domain walls at p p c is d f 1 38 consistent with sle 3 along with this analysis we introduce a self organized critical soc model in which the water movement is modeled by a chain of topplings taking place when the water saturation exceeds the critical value we present strong indications that it coincides with the reservoir simulation described by darcy equation we further analyze the soc model and show numerically that for this model the spanning cluster probability has a maximum around p 0 65

final state problem for class of nonlinear nonlocal dispersive equation abstract we study the large time asymptotics of the solutions to nonlinear dispersive equations of the form 0 1 t u p i x x u x u 3 t 0 x r where p i x is a pseudo differential operator defined by the symbol p in this paper we consider the case that p is nonlocal and nonhomogeneous and show the existence of the solution to 0 1 around a given modified free solution

the interacton equation given the lie algebra lfp of vector fields of a free particle then in a joint field of interacting particles an interacton is defined as a direct summand isomorphic to lfp for several soliton equations it is shown how interactons can be obtained from various methods of symmetry analysis and how their dynamics can be expressed in terms of self interaction alone this procedure leads to new integrable systems furthermore in case of nonlinear schrodinger equations it is shown how the dynamics of interactons can be formulated in terms of self consistent potentials with antlinear parts

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

singularity confinement analysis of integrodifferential equations of benjamin ono type we analyse the integrability of various types of intregrodifferential equations belonging to the benjamin ono family the method used is a combination of two well known integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems we confirm the results of hietarinta based on the study of multisoliton solutions in particular we show that the third order extension to benjamin ono that he proposed does pass the test and is thus an excellent candidate for integrability

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

dynamics of solitons in a damped sine hilbert equation a damped sine hilbert sh equation is proposed it can be linearized by a dependent variable transformation which enables one to solve an initial value problem of the equation the n soliton solution is obtained explicitly and its properties are investigated in comparison with those of the n soliton solution of the sh equation in particular the interaction of the two solitons is explored in detail with the aid of the pole representation it is found that the interaction process is classified into the two types according to the initial amplitudes and positions of both solitons in the general n soliton case the long time behavior of the solution is shown to be characterized by the positive n zeros of the hermite polynomial of degree 2n finally a linearized version of the damped sh equation is briefly discussed

an application of the inverse scattering transform to the modified intermediate long wave equation the modified intermediate long wave milw equation is a 1 1 dimensional nonlinear singular integro differential equation that possesses soliton solutions in an appropriate limit the milw equation reduces to the well known modified korteweg de vries equation in this paper we solve the initial value problem for the milw equation through a suitable implementation of the inverse scattering transform and use of the miura type transformation that maps solutions of the milw equation into solutions of a complexified version of the standard intermediate long wave ilw equation the initial value used for the milw equation is assumed to be real valued sufficiently smooth and decaying to zero as the absolute value of the spatial variable approaches large values an interesting feature of the procedure we develop is that soliton solutions for the ilw and milw equations can be derived by appropriate specializations of a master set of equations

on the k theory and topological cyclic homology of smooth schemes over a discrete valuation ring we show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic 0 p the p adic etale k theory and p adic topological cyclic homology agree

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

on the initial value problem for a class of nonlinear integral evolution equations including the sine hilbert equation a method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented the underlying scattering problem is a matrix riemann hilbert problem scattering analysis shows that the spectrum is purely discrete an application is to the so called sine hilbert equation h t c sin where c is a constant and h denotes the hilbert transform

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

p adic deformation of algebraic cycle classes we study the p adic deformation properties of algebraic cycle classes modulo rational equivalence we show that the crystalline chern character of a vector bundle on the closed fibre lies in a certain part of the hodge filtration if and only if rationally the class of the vector bundle lifts to a formal pro class in k theory on the p adic scheme

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

bak tang wiesenfeld model on the square site percolation lattice the bak tang wiesenfeld btw model is considered on the site diluted square lattice tuned by the occupancy probability p various statistical observables of the avalanches are analyzed in terms of p e g the fractal dimension of their exterior frontiers gyration radius loop lengths and green s function the model exhibits critical behavior for all amounts of p and the exponents of the statistical observables are analyzed we find a distinct universality class at which is unstable towards a p 1 btw fixed point this universality class displays some common features such as a two dimensional 2d ising universality class e g the fractal dimension of loops in the thermodynamic limit is which is compatible with the fractal dimension of geometrical spin clusters of the 2d critical ising model with

uniqueness properties of solutions to the benjamin ono equation and related models abstract we prove that if u 1 u 2 are real solutions of the benjamin ono equation defined in x t r 0 t which agree in an open set r 0 t then u 1 u 2 we extend this uniqueness result to a general class of equations of benjamin ono type in both the initial value problem and the initial periodic boundary value problem this class of 1 dimensional non local models includes the intermediate long wave equation we relate our uniqueness results with those for a water wave problem finally we present a slightly stronger version of our uniqueness results for the benjamin ono equation

backlund transformation conservation laws and inverse scattering transform of a model integrodifferential equation for water waves the backlund transformation bt an infinite number of conservation laws and the inverse scattering transform ist of a model integrodifferential equation for water waves in fluids of finite depth y matsuno j math phys 29 49 1989 are constructed by employing the bilinear transformation method the model equation is also shown to pass the painleve test these facts prove the complete integrability of the equation both the deep and shallow water limits of various results thus obtained are then investigated in detail in addition a new method to evaluate conserved quantities for pure n soliton is developed by utilizing actively the time part of the bt it is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the benjamin ono and the korteweg de vries equations the most important problem left open in this paper is to solve various ist equations

on the limit from the intermediate long wave equation to the benjamin ono equation the intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter referred to here as for it reduces to the benjamin ono equation it has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences here it is shown that for the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the benjamin ono equation

geometrical clusters of darcy s reservoir model and ising universality class abstract in this paper the geometrical features of the fluid propagation in two dimensional petroleum reservoir described by darcy equations are studied the porous media are considered to be tuned by the occupancy parameter p being the probability that a pore is occupied we analyze the statistical geometrical observables of the darcy model to this end we let the water to be injected into random sites of the porous media and solve numerically the darcy equations to describe the flow motion pattern using the control volume finite difference cvfd method the fractal dimension of the frontier of the avalanches defined as the set of the sites through which the fluid passed and the distribution functions of gyration radius loop length and cluster mass are numerically obtained revealing that at p p c the critical occupancy parameter above which there is definitely a spanning cluster in the system this model lies within a universality class compatible with the ising model we observe that for p p c although the model shows critical behaviors this duality is broken the mentioned exponents are reported in this paper

on the vanishing of negative k groups we show that for a d dimensional scheme x essentially of finite type over an infinite perfect field k of characteristic p 0 the negative k groups k q x vanish for q d provided that strong resolution of singularities holds over the field k

predicting oil recovery using percolation one particular practical problem in oil recovery is to predict the time to breakthrough of a uid injected in one well and the subsequent decay in the production rate of oil at another well because we only have a stochastic view of the distribution of rock properties we need to predict the uncertainty in the breakthrough time and post breakthrough behaviour in order to calculate the economic risk in this paper we use percolation theory to predict i the distribution of the chemical path shortest path between two points representing well pairs at a given euclidean separation and present a scaling hypothesis for this distribution which is conrmed by numerical simulation ii the distribution of breakthrough times which can be calculated algebraically rather than by very time consuming direct numerical simulation of large numbers of realisations c 1999 elsevier science b v all rights reserved

kinks of the sine hilbert equation and their dynamical motions an exact method for constructing exact solutions of the sine hilbert equation is developed it is shown that the sh equation can be transformed into a system of non linear ordinary differential equations through a dependent variable transformation furthermore this system of equations is transformed into a system of decoupled linear ordinary differential equations exact solutions of the sh equation are then constructed by means of a simple integration the dynamical properties of the solutions are also studied in detail the solution is interpreted as a superposition of n pulses which may be termed kinks where n is a positive integer asymptotically for large time it is composed of a pulse with a constant velocity and n 1 static pulses this is a novel characteristic of the solution unlike the well known n soliton solution which behaves like n moving pulses for large time

cyclic polytopes and the k theory of truncated polynomial algebras this paper calculates the relative algebraic k theory k k x x n x of a truncated polynomial algebra over a perfect field k of positive characteristic p since the ideal generated by x is nilpotent we can apply mccarthy s theorem the relative algebraic k theory is isomorphic to the relative topological cyclic homology mc and it is the latter groups we actually evaluate the result is best expressed in terms of big witt vectors let wm k denote the big witt vectors in k of length m i e the multiplicative group wm k 1 xk x 1 x k x

pro unitality and pro excision in algebraic k theory and cyclic homology the purpose of this paper is to study pro excision in algebraic k theory and cyclic homology after suslin wodzicki cuntz quillen cortinas and geisser hesselholt as well as continuity properties of andre quillen and hochschild homology a key tool is first to establish the equivalence of various pro tor vanishing conditions which appear in the literature this allows us to prove that all ideals of commutative noetherian rings are pro unital in a suitable sense we show moreover that such pro unital ideals satisfy pro excision in derived hochschild and cyclic homology it follows hence and from the suslin wodzicki criterion that ideals of commutative noetherian rings satisfy pro excision in derived hochschild and cyclic homology and in algebraic k theory in addition our techniques yield a strong form of the pro hochschild kostant rosenberg theorem an extension to general base rings of the cuntz quillen excision theorem in periodic cyclic homology a generalisation of the f igin tsygan theorem a short proof of pro excision in topological hochschild and cyclic homology and new artin rees and continuity statements in andre quillen and hochschild homology msc 19d55 primary 16e40 13d03 secondary

toric varieties monoid schemes and cdh descent we give conditions for the mayer vietoris property to hold for the algebraic k theory of blow up squares of toric varieties and schemes using the theory of monoid schemes these conditions are used to relate algebraic k theory to topological cyclic homology in characteristic p to achieve our goals we develop many notions for monoid schemes based on classical algebraic geometry such as separated and proper maps and resolution of singularities

stability of nonlinear periodic internal waves in a deep stratified fluid stability of the nonlinear periodic solutions of the benjamin ono equation with respect to one dimensional perturbations and taking into account transverse modulations is analytically studied the full set of eigenfrequencies and of small perturbations is found in the one dimensional case the dispersion equation for long wave oblique perturbations is obtained these perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation the instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves

generalized intermediate long wave hierarchy in zero curvature representation with noncommutative spectral parameter the simplest generalization of the intermediate long wave hierarchy ilw is considered to show how to extend the zakharov shabat dressing method to nonlocal i e integro partial differential equations the purpose is to give a procedure of constructing the zero curvature representation of this class of equations this result obtains by combining the drinfeld sokolov formalism together with the introduction of an operator valued spectral parameter namely a spectral parameter that does not commute with the space variable x this extension provides a connection between the ilwk hierarchy and the saveliev vershik continuum graded lie algebras in the case of ilw2 the fairlie zachos sinh algebra was found

pulse formation in a dissipative nonlinear system a nonlocal nonlinear evolution equation is proposed that describes pulse formation in a dissipative system a novel feature of the equation is that it can be solved exactly through a linearization procedure the solutions are constructed under appropriate initial and boundary conditions and their properties are investigated in detail of particular interest is pulse formation which is caused by a balance between nonlinearity and dissipation the asymptotic behavior of the solution for large time is then represented by a train of moving pulses with equal amplitudes the corresponding position of each pulse is shown to be characterized by the zero of the hermite polynomial irrespective of initial conditions

algebraic k theory and descent for blow ups we prove that algebraic k theory satisfies pro descent for abstract blow up squares of noetherian schemes as an application we derive weibel s conjecture on the vanishing of negative k groups

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

linearization of novel nonlinear diffusion equations with the hilbert kernel and their exact solutions two novel nonlinear diffusion equations with the hilbert kernel are proposed the equations can be linearized by introducing appropriate dependent variable transformations the initial value problems for the proposed equations are then solved exactly through the linearization and explicit nonperiodic and periodic solutions are constructed the properties of solutions are investigated in detail it is found that the blow up of solutions occurs at a finite time for both the nonperiodic and periodic cases due to the breakdown of certain analytic conditions imposed on the dependent variables

uncertainty in oil production predicted by percolation theory in this paper we apply scaling laws from percolation theory to the problem of estimating the time for a fluid injected into an oilfield to breakthrough into a production well the main contribution is to show that when these previously published results are used on realistic data they are in good agreement with results calculated in a more conventional way but they can be obtained significantly and more quickly as a result they may be used in practical engineering circumstances and aid decision making for real field problems

on the k theory of local fields in this paper we establish a connection between the quillen k theory of certain local fields and the de rham witt complex of their rings of integers with logarithmic poles at the maximal ideal the fieldsk we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p 2 when k contains the pth roots of unity the relationship between the k theory with z p coefficients and the de rham witt complex can be described by a sequence

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

new integrable nonlinear integrodifferential equations and related solvable finite dimensional dynamical systems a new integrable nonlinear integrodifferential equation nide is proposed this equation may be interpreted as a model equation for deep water waves the n periodic and n soliton solutions for the equation are constructed by means of the bilinear transformation method these solutions have the same structure as that for the benjamin ono equation which describes internal waves in stratified fluids of great depth furthermore it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite dimensional dynamical systems described by first order nonlinear ordinary differential equations the discussion is also made on a more general nide that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth

on genus two solutions for the ilw equation the existence of theta function solutions of genus two for the intermediate long wave equation is established a numerical example is also presented the method basically goes along with krichever s construction of theta function solutions for soliton equations such as the kronecker product equation this idea leads us to a question whether a riemann surface exists which allows a peculiar abelian integral of the third kind the answer is affirmative at least for genus two curves

darboux transformations infinitesimal symmetries and conservation laws for the nonlocal two dimensional toda lattice the technique of darboux transformation is applied to the nonlocal partner of the two dimensional periodic an 1 toda lattice this system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with a quantized spectral parameter we give the generalization of the darboux transformation technique on linear equations of this type we present the connections between the solutions of overdetermined linear systems and their expansions in series in the neighbourhood of singular points the solutions of the nonlocal toda lattice and infinite hierarchies of the infinitesimal symmetries and conservation laws are obtained

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

two dimensional dynamical system associated with abel s nonlinear differential equation a two dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations odes with a complex parameter it reduces to abel s nonlinear ode of the first kind by an appropriate transformation using this fact the properties of solutions are investigated in detail with the aid of numerical computations it is found that various types of bifurcation phenomena occur depending on the values of the parameter in particular the solution is shown to blow up in finite time under certain conditions in order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane finally a discussion is made on some generalizations of the proposed system