antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

traveling wave solutions in a diffusive system with two preys and one predator abstract this paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator by constructing upper and lower solutions the existence of nontrivial traveling wave solutions is established the asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles applying the theory of asymptotic spreading the nonexistence of traveling wave solutions is proved

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

existence and qualitative properties of travelling waves for an epidemiological model with mutations in this article we are interested in a non monotone system of logistic reaction diffusion equations this system of equations models an epidemics where two types of pathogens are competing and a mutation can change one type into the other with a certain rate we show the existence of minimal speed travelling waves that are usually non monotonic we then provide a description of the shape of those constructed travelling waves and relate them to some fisher kpp fronts with non minimal speed

on the curvature of piecewise flat spaces we consider analogs of the lipschitz killing curvatures of smooth riemannian manifolds for piecewise flat spaces in the special case of scalar curvature the definition is due to t regge considerations in this spirit date back to j steiner we show that if a piecewise flat space approximates a smooth space in a suitable sense then the corresponding curvatures are close in the sense of measures

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

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

alternating adaptive projections in antenna synthesis the projection operator is a basic building block in the application of the alternating projections method to antenna synthesis in general it is a non linear operator that is repeatedly applied in the course of a single synthesis process thus having a considerable impact on the convergence properties of the resulting algorithm a novel approach to the computation of these projections is presented which exploits a simple definition of the relevant spaces particularly that of radiated fields characterization of field mask specification as scale invariant under this definition adds a further degree of freedom namely reference level which impacts on the sensitive to scaling projector in order to compute the optimum reference level an iterative procedure is proposed which is simple to implement easily integrable in standard alternating projection routines and adds negligible computational burden numerical tests confirm an improved performance with respect to the fixed scaling projection operator in terms of convergence rate and robustness against the initial guess supporting our approach as a valid aid in overcoming the drawbacks of the alternating projections based antenna synthesis

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

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

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

optimal narrow beam low sidelobe synthesis for arbitrary arrays the classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios the proposed approach allows also to handle arbitrary arrays this extended basic synthesis problem is first rewritten as a convex optimization problem this problem is then transformed into either a linear program or a second order cone program that can be solved efficiently by readily available software the convex formulation is not strictly equivalent to the initial synthesis problem but arguments are given that establish that the obtained optimum is either identical or can be made as close as wanted to the desired one finally numerical applications and a comparison with a classical solution are presented to both confirm the optimality of the solution and illustrate the potentialities of the proposed approach

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

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

optimal far field focusing of uniformly spaced arrays subject to arbitrary upper bounds in nontarget directions a new approach providing new theoretical results to the optimal far field focusing of uniformly spaced arrays subject to a completely arbitrary mask for sidelobe bounds is presented and discussed in both cases of linear or planar arrays with rectangular boundaries it is first shown that the problem can be formulated without any loss to performances on the maximum as a linear programming one guaranteeing a globally optimal solution second a sufficient uniqueness criterion for the solution of the overall problem is also developed which shows how the solution may not be unique as is actually the case when planar arrays are considered in addition further globally effective optimization procedures are proposed for the latter case in order to optimize directivity smoothness of excitations or other performance parameters in the set of equivalent solutions last an extension to planar arrays with a nonrectangular boundary is also given a thorough numerical analysis confirms the effectiveness of the approach proposed and of the solution codes developed

a simple algorithm to achieve desired patterns for arbitrary arrays a simple iterative algorithm which can be used to find array weights that produce array patterns with a given look direction and an arbitrary sidelobe specification is presented the method can be applied to nonuniform array geometries in which the individual elements have arbitrary and differing radiation patterns the method is iterative and uses sequential updating to ensure that peak sidelobe levels in the array meet the specification computation of each successive pattern is based on the solution of a linearly constrained least squares problem the constraints ensure that the magnitude of the sidelobes at the locations of the previous peaks takes on the prespecified values phase values for the sidelobes do not change during this process and problems associated with choosing a specific phase value are therefore avoided experimental evidence suggests that the procedure terminates in remarkably few iterations even for arrays with significant numbers of elements

the application of a modified differential evolution strategy to some array pattern synthesis problems a high convergence rate and a robust exploration ability constitute a contradiction in modern heuristic optimization techniques this paper describes a modified differential evolution strategy mdes that builds up a balanced relationship between the two contradictive elements by introducing several modifications into the conventional differential evolution strategy des the novel mdes opens up an effective and robust approach for global optimization problems several representative mathematical functions are minimized using various optimization methods and the convergence rates are compared to evaluate the performance of mdes moreover array synthesis examples which can be formulated as non convex problems are presented including the optimal synthesis of sum and difference patterns and the synthesis of unequally spaced arrays numerical results demonstrate that the proposed approach improves the performance of the algorithm significantly in terms of both the convergence rate and the exploration ability

a joint optimization approach for the synthesis of large 4 d heterogeneous antenna arrays due to the inherent nonlinearity in the synthesis of large four dimensional 4 d heterogeneous arrays conventional pattern synthesis techniques are no longer suitable in this paper an effective joint optimization approach based on the combination of convex programming cp and differential evolution de algorithm is proposed by designing suitable time sequences of a 4 d heterogeneous array with uniform static amplitude excitations the original synthesis problem is simplified into two independent small problems which can be easily solved in two steps respectively in the first step the approach takes definite advantages from the convexity of the problem with respect to the static phase excitations and the switch on duration times these optimization variables become known and are fixed in the second step then the de algorithm is used to optimize the switch on time instants owing to the efficiency of the cp only 2 3 of the optimization variables need be solved in the first step while even fewer number of optimization variables are solved in the second step consequently the joint optimization of cp and de algorithm significantly improves the overall synthesis efficiency two numerical examples are presented to demonstrate the effectiveness of the proposed approach

spreading speeds of a lotka volterra predator prey system the role of the predator abstract this paper is concerned with the spreading speeds of a modified lotka volterra predator prey system the intention is to describe the propagation mode whereby the two species coinvade a new habitat under some assumptions it is proved that the spreading speed of the prey is slower than in the case where the predator vanishes and the density of the prey on the coexistence domain is also smaller compared with the case where interspecies actions disappear therefore the predator has a negative effect on the evolution of the prey by slowing the spreading speed and decreasing the population density of the prey

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

a variational principle for weighted delaunay triangulations and hyperideal polyhedra we use a variational principle to prove an existence and uniqueness theorem for planar weighted delaunay triangulations with non intersecting site circles with prescribed combinatorial type and circle intersection angles such weighted delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles more generally we consider weighted delaunay triangulations in piecewise flat surfaces allowing cone singularities with prescribed cone angles in the vertices the material presented here extends work by rivin on delaunay triangulations and ideal polyhedra

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

duality structures and discrete conformal variations of piecewise constant curvature surfaces abstract a piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry euclidean hyperbolic or spherical with the specified edge lengths additional geometric structure leads to a notion of discrete conformal structure generalizing circle packings and their generalizations as studied by thurston and others we analyze discrete conformal variations of piecewise constant curvature 2 manifolds giving particular attention to the variation of angles formulas are derived for the derivatives of angles in each background geometry which yield formulas for the derivatives of curvatures and to curvature functionals finally we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries

on the semi monotone operator theory and applications abstract in this paper eis a real banach space a e e e is a semi monotone operator we first study the variational inequality problem findu k such that a u u v u 0 wherek e is a closed convex subset then the operator equationa u u p and we also construct a degree theory for demi continuous semi monotone operators in reflexive banach spaces

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

invasion traveling wave solutions of a predator prey system abstract this paper is concerned with the traveling wave solutions of a predator prey system in which intra specific conflictions are involved by using the comparison principle and the asymptotic speed of spreading the existence nonexistence and minimal wave speed of non negative traveling wave solutions are established such a traveling wave solution can model the spatial temporal process where the predator invades the territory of the prey and they eventually coexist

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

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

gauss images of hyperbolic cusps with convex polyhedral boundary we prove that a 3 dimensional hyperbolic cusp with convex poly hedral boundary is uniquely determined by its gauss image furthermore any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2 is the metric of the gauss image of some convex polyhedral cusp this result is an analog of the rivin hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their gauss images the proof uses a variational method namely a cusp with a given gauss image is identified with a critical point of a functional on the space of cusps with cone type singularities along a family of half lines the functional is shown to be concave and to attain maximum at an interior point of its domain as a byproduct we prove rigidity statements with respect to the gauss image for cusps with or without cone type singularities in a special case our theorem is equivalent to existence of a circle pattern on the torus with prescribed combinatorics and intersection angles this is the genus one case of a theorem by thurston in fact our theorem extends thurston s theorem in the same way as rivin hodgson s theorem extends an dreev s theorem on compact convex polyhedra with non obtuse dihedral angles the functional used in the proof is the sum of a volume term and curvature term we show that in the situation of thurston s theorem it is the potential for the combinatorial ricci flow considered by chow and luo our theorem represents the last special case of a general statement about isometric immersions of compact surfaces

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

an effective deterministic procedure for the synthesis of shaped beams by means of uniform amplitude linear sparse arrays a rigorous deterministic approach is proposed for the mask constrained power pattern synthesis of shaped beams through one dimensional sparse arrays having uniform amplitude excitations and variable phases the approach includes the well known density taper procedure as a special case the corresponding technique which exploits an analytical formulation of the problem and convex programming routines results in fast computation and is able to achieve optimal solutions without recurring to any global optimization numerical results assessing the capabilities of the proposed design method are provided opening the way to the development of an effective approach for the synthesis of circularly symmetric shaped beams via uniformly excited circular ring arrays

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

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

optimal synthesis of reconfigurable planar arrays with simplified architectures for monopulse radar applications a new approach to the power synthesis of fixed geometry reconfigurable planar arrays radiating sum and difference patterns is presented the proposed design technique allows to maximize the radiation performance field slope amplitude or even directivity of both beam patterns over assigned directions subject to completely arbitrary masks for sidelobe bounds whatever the elements disposal and the array boundary the overall problem is solved through a convex quadratic programming procedure moreover if centrosymmetric antenna layouts are adopted the synthesis is reduced to an even more powerful linear programming routine which preserves the solution uniqueness and optimality guaranteed by quadratic programming codes while dramatically decreasing their computational weight and hence enabling the design of much larger planar arrays the proposed approach also takes into account the need of reducing the complexity of the beam forming network bfn which is fulfilled by sharing part of the excitation amplitudes between the two radiation modalities a set of numerical examples is reported and discussed to show the versatility and effectiveness of the proposed approach

a hybrid approach to the synthesis of subarrayed monopulse linear arrays in this paper a hybrid approach for the synthesis of the ldquooptimalrdquo compromise between sum and difference patterns for subarrayed monopulse antennas is presented first the subarray configuration is determined by exploiting the knowledge of the optimum difference mode coefficients to reduce the dimension of the searching space in the second step the subarray weights are computed by means of a convex programming procedure which takes advantages from the convexity for a fixed clustering of the problem at hand a set of representative results is reported to assess the effectiveness of the proposed approach comparisons with state of the art techniques are also presented

linear antenna array pattern synthesis with prescribed broad nulls a technique of synthesizing an antenna array pattern with prescribed broad nulls is presented the array pattern synthesis problem is formulated as a least square null constrained optimization problem numerical techniques based on the matrix factorization method are developed for reducing the computational complexity of determining the optimal weight vector subsequently a set of eigenvector constraints are used to approximate the effect of the quadratic constraint numerical results are presented to illustrate the performance achievable

a numerical pattern synthesis algorithm for arrays a numerical technique for pattern synthesis in arrays is presented for a given set of elements the technique allows one to find a set of array coefficients that steer the main beam in a given direction and yield sidelobes meeting a specified criterion if such a set of array coefficients exists if the pattern specifications cannot be met with the given elements the algorithm finds the best attainable pattern the advantage of this technique is that it can be used with an arbitrary set of array elements different elements in the array can have different element patterns and the array can have arbitrary nonuniform spacing between elements the synthesis technique is based on adaptive array theory the given array elements are assumed to be used as the elements of an adaptive array the main beam is pointed in the proper direction by choosing the steering vector for that direction and the sidelobes are controlled by introducing a large number of interfering signals at many angles throughout the sidelobe region the algorithm iterates on the interference powers until a suitable pattern is obtained

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

an effective hybrid approach for the optimal synthesis of monopulse antennas the problem of synthesizing optimal sum and difference patterns subject to arbitrary sidelobe bounds by means of a simple feeding network is dealt with it is shown how a hybrid optimization procedure properly acting on both clustering into subarrays and excitations of these latter greatly improves performances of known feeding architectures with respect to existing procedures results of the proposed technique which significantly outperform existing methods renew the interest of these simple feeding architectures and emphasize the effectiveness of this kind of hybrid procedures exploiting the convexity of the problem with respect to a part of the unknowns

numerical synthesis of arbitrary discrete arrays a numerical algorithm for the synthesis of arbitrary discrete arrays is given the technique is based on the algorithm of c a olen and r t compton 1990 but employs a different iterative recursive algorithm yielding an optimum convergence rate it is shown that the new algorithm converges to the optimum solution in a significantly reduced number of iterations compared with the technique of olen and compton use of the technique is illustrated by applying the synthesis procedure to a ten element linear array and to a conformal array consisting of ten dipoles located on a parabolic curve

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

existence of multidimensional travelling wave solutions of an initial boundary value problem on considere le probleme suivant u t u xx u yy f u u x 0 t u x l t 0 u x y 0 u 0 x y ou x r 1 0 y l et f u u u 1 u avec 0 1 2

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

hybrid design of non regular linear arrays with accurate control of the pattern sidelobes the hybridization of an analytical thinning technique based on almost difference sets adss and a convex programming cp strategy is proposed for designing non regular linear arrays over a half wavelength lattice that fit arbitrary upper bounds on the far field pattern the proposed approach combines the efficiency of the ads procedure in defining a low sidelobe thinned arrangement with the effectiveness of the cp approach in deducing the optimal tapering to control the beam shape a set of numerical results is presented to assess features and potentialities of the ads cp method

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

the existence of periodic travelling waves for singularly perturbed predator prey equations via the conley index this paper is concerned with the existence of periodic travelling wave solutions of reactiondiffusion equations which arise as models of predator prey interactions in mathematical ecology the relevant equations are somewhat similar to the fitz hugh nagumo equations and our results apply to this latter system as well in studying the fitz hugh nagumo equations conley 3 i initiated a new topological approach to this problem his method depends on the construction of an isolating neighborhood cf section 2a about a singular periodic solution associated with such a neighborhood is a topological invariant the conley index which when nontrivial implies the existence of a smooth solution near the singular one the new aspects of the present discussion are in the method used to construct the isolating neighborhood in ifi4 and in the computation of the index with regard to the isolating neighborhood our construction is global in that a parameter taken to be small in conley s equations is now allowed to assume large values in order to compute the index we continue our equations to those studied by conley actually we go one step further and continue conley s equations to the van der pol equations crossed with a pair of linear equations admitting a repelling critical point the index for this latter system is easily computed and turns out to be non trivial it follows from the invariance of the index under continuation 21 that the desired

a universality of period doubling bifurcations abstract it is shown that the topology of periodic orbits belonging to different series of period doubling bifurcations can be distinguished by the structure of a template which represents a topological nature of orbits in order to do this a new quantity will be introduced the local crossing number which counts the number of half twists of a period doubled orbit along the tubular neighborhood of the orbit just before the bifurcation the local crossing number can be extracted by inspecting the power spectrum of period doubling orbits the topological nature of the orbits is further investigated by symbolic dynamics with the help of the template to find that any series of period doubling orbits can be characterized by a unique series of periodic blocks w y xy xy 3 xy 3 xy 2 a new concept irreducible and reducible templates is introduced with the irreducible template various models treated by others are reexamined

topological dimension and local coordinates from time series data a method for the estimation of the topological dimension of a manifold from time series data is presented it is based on the approximation of the manifold near a point chi by its tangent space at chi the dimension of the tangent space is estimated by constructing a maximal set of linearly independent vectors from the data near chi using the method of singular value decomposition the method is used to analyse experimental data obtained from a nonlinear electronic oscillator in a chaotic state

rigidity of infinite disk patterns let p be a locally flnite disk pattern on the complex plane c whose combinatorics are described by the one skeleton g of a triangulation of the open topological disk and whose dihedral angles are equal to a function e 0 2 on the set of edges let p be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function we show that p and p difier only by a euclidean similarity in particular when the dihedral angle function is identically zero this yields the rigidity theorems of b rodin and d sullivan and of o schramm whose arguments rely essentially on the pairwise disjointness of the interiors of the disks the approach here is analytical and uses the maximum principle the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern a similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane where the proof follows by a simple use of the maximum principle also we have a uniformization result for disk patterns in a future paper the techniques of this paper will be extended to the case when 0 in particular we will show a rigidity property for a class of inflnite convex polyhedra in the 3 dimensional hyperbolic space

a new formulation for the design of chebyshev arrays a new formulation for the design of chebyshev arrays is presented which makes no direct use of chebyshev polynomials the array factor is expressed in terms of cosine cosine hyperbolic functions based on which all analysis and design steps are carried out zeros of the array factor are used to obtain a system of equations for excitation currents solving this system of equations current magnitudes are determined in terms of the current of one element chosen as the independent variable a general formulation for an even or odd but otherwise arbitrary number of elements is presented the minimum number of elements required to achieve the desired beamwidth and side lobe level is obtained in a single step without resorting to an iterative process the optimum spacing between elements for broadside and end fire arrays is also addressed numerical results for example cases are provided

on the degrees of freedom of scattered fields starting from the observation that fields differing less than a prescribed error cannot be resolved as distinct entities the degrees of freedom of the scattered field are introduced and then computed the degrees of freedom are shown to be practically equal to the nyquist number appropriate to the effective spatial bandwidth of the scattered field and to the extension of the observation domain accordingly a finite number of elements of information can be used to determine the scattered field to a prescribed approximation error it is also shown that the field representation can be made in terms of field values and simple sampling functions provided that a marginal increase in the approximation error is tolerated the results not only completely justify the use of sampling interpolation for representing scattering fields but also demonstrate that such representation is practically an optimal one an algorithm for the reconstruction of scattered fields given the maximum allowed error is then produced

a new low side lobe pattern synthesis technique for conformal arrays the application of the new nonlinear optimization algorithms to array design is demonstrated by the low side lobe pattern synthesis of conformal arrays by adopting the information of array element realized gain patterns the authors formulate synthesis problems as nonlinearly constrained optimization problems and propose a direct method to solve them the technique allows them to find a set of array coefficients that yield a pattern meeting a specified side lobe level and achieving the maximum directivity if such a set of array coefficients exists if the side lobe specifications cannot be met with the given array the technique will result in a set of coefficients that yield a pattern meeting the best attainable side lobe level and having directivity as high as possible simplified synthesis problems for an axial dipole array in an infinite perfectly conducting cylinder are discussed and numerical results are given the synthesis technique works for general array patterns

an effective approach for the optimal focusing of array fields subject to arbitrary upper bounds a new approach to the optimal focusing of array fields subject to arbitrary upper bounds is presented the approach formulates the problem as the minimization of a linear function in a convex set unlike other approaches this one guarantees the achievement of the global optimum by using local optimization techniques and can moreover deal with any convex constraint on the unknowns such as near field constraints optimization is performed by two ad hoc developed solution algorithms which exploit the geometrical characteristics of the problem at hand thus leading to extremely effective and computationally efficient numerical codes an extensive numerical analysis has been performed in all cases of linear planar and circular arc arrays the enhanced performance of the proposed technique with respect to the solution algorithms available in the literature fully confirms the effectiveness of the approach

stability of travelling wave solutions of diffusive predator prey systems the stability of travelling wave solutions of singularly perturbed diffusive predator prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at a 0 is simple the proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues

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

knotted periodic orbits in suspensions of smale s horseshoe torus knots and bifurcation sequences we construct a suspension of smale s horseshoe diffeomorphism of the two dimensional disc as a flow in an orientable three manifold such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the duffing equation from this suspension we construct a knot holder or template a branched two manifold with a semiflow in such a way that the periodic orbits are isotopic to those in the full three dimensional flow we discuss some of the families of knotted periodic orbits carried by this template in particular we obtain theorems of existence uniqueness and non existence for families of torus knots we relate these families to resonant hamiltonian bifurcations which occur as horseshoes are created in a one parameter family of area preserving maps and we also relate them to bifurcations of families of one dimensional quadratic like maps which can be studied by kneading theory thus using knot theory kneading theory and hamiltonian bifurcation theory we are able to connect a countable subsequence of one dimensional bifurcations with a subsequence of area preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary one implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area preserving family there are no universal routes to chaos

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

a universal departure from the classical period doubling spectrum abstract the topology of the period doubling attractor at the onset of chaos and its implications for the structure of the power spectrum are discussed the presence of a seemingly anomalous peak at a non 2n frequency is explained in terms of a topological invariant of the attractor whose systematics is shown to be universal an experiment on an electronic circuit is performed and its results are shown to be fully consistent with the theory

on the global structure of period doubling flows abstract for a wide class of period doubling flows on r3 we analyze the global structure of the invariant manifolds and the topology of the bifurcating periodic orbits we emphasize aspects of the dynamics which are not visible in an analysis of the associated poincare return map the global manifold structure implies constraints for the subsequent bifurcational behavior of the flow the period doubled orbits are classified using the theory of iterated torus knots this classification reveals an infinite number of topologically distinct period doubling flows and raises the important question as to which sequences of torus knots appear in the asymptotic limit of a period doubling cascade if the asymptotic sequence of torus knots produced is not unique then the self similarity of the poincare map does not imply a self similar structure for the flow in particular at the accumulation point for the cascade one would not expect to observe a universal power spectrum for the flow this possibility is of some experimental interest

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

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

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

the nonlinear dynamics of the damped and driven toda chain iii classification of the nonlinear resonances and local bifurcations abstract the dynamics of damped and driven toda chains is investigated it is shown that the resonance and bifurcation scenario of alternating saddle node and period doubling bifurcations presented in the first part of this series on the nonlinear dynamics of the damped and driven toda chain physica d 31 1988 103 116 can be explained by the specific structure of the eigenvalue spectrum of the associated linearized flow map dx t

existence of travelling wave solutions of predator prey systems via the connection index we consider a reaction diffusion system which arises as a model of predator prey interactions in mathematical ecology the system admits four constant equilibria two of which are stable the main result is an existence theorem for travelling wave solutions which connect the two stable equilibria most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit the new aspect of the present discussion lies in devising some novel methods for locating the connecting solution in phase space which do not require that the equations be singularly perturbed in certain cases the main result holds for an arbitrary choice of diffusion coefficients

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

the conley index for fast slow systems ii multidimensional slow variable we develop a qualitative theory for fast slow systems with a one dimensional slow variable using conley index theory for singularity perturbed systems conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0 these orbits persist when the derivative of the slow variable is small and nonzero

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

topological invariants in period doubling cascades topological characterization is a useful description of dynamical behaviours as exemplified by templates which synthesize the topological properties of very dissipative chaotic attractors embedded in tri dimensional phase spaces such a description relies on topological invariants such as linking numbers between two periodic orbits which may be viewed as knots these invariants may therefore be used to understand the structure of dynamical behaviours nevertheless as an example the celebrated period doubling cascade is usually investigated by using total twists which are not topological invariants instead we introduce linking numbers between an orbit viewed as the core of a small ribbon and the edges of the ribbon such a linking number which is in fact the calugareanu invariant is related to the total twist number and the number of writhes of the ribbon a second topological invariant called the effective twist number is also introduced and is useful for investigating period doubling cascades in the case of a trivial suspension of a horseshoe map this topological invariant may be predicted from a symbolic dynamics with the aid of framed braid representations