multiple topological structures of chaotic attractors ruling the emission of a driven laser we analyze the chaotic emission of an erbium doped fiber laser with sine wave pump power modulation for different modulation frequencies and modulation indexes for each working condition considered the template which summarizes the corresponding chaotic attractor has been determined by means of topological analysis techniques the variety of templates obtained is much wider than in previous works apart from horseshoes reverse horseshoes or jellyroll structures with different global torsions two more kinds of structures have been found particularly one is especially notable as its folding mechanism does not fit the rolling scheme valid for all templates found in former experimental studies apart from strengthening the usefulness of templates as convenient objects for characterization of chaotic attractors the interest of these results lies in the identification of folding mechanisms for chaos generation not found in nature up to now

from template analysis to generating partitions ii characterization of the symbolic encodings we give numerical evidence of the validity of a previously described algorithm for constructing symbolic encodings of chaotic attractors from a template analysis we verify that the different solutions that can be found are dynamically equivalent and that our approach yields results that are consistent with those obtained from methods based on homoclinic tangencies this is further confirmed by verifying directly that the computed partitions are generating to a high degree of accuracy and that they can be used to estimate precisely the metric entropy it is also shown that the correct number of symbols needed to describe the dynamics is naturally provided and that a compact parameterization of a partition can easily be determined which makes our algorithm suitable for applications such as real time encoding

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

braid analysis of a bouncing ball i identify the template organizing the chaotic dynamics of a bounding ball system i further show how to estimate the topological parameter values of the system directly from a time series a process i call topological time series analysis two distinct methods to determine the topological parameters are illustrated and compared the pruning front procedure and a braid analysis both procedures lead to compatible results

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

remarks on a nonlinear parabolic equation the equation ut u u r is studied in rn and in the periodic case it is shown that the equation is well posed in l1 and possesses regularizing properties for nonnegative initial data and 0 the solution decays in l1 rn as t in the periodic case it tends uniformly to a limit a consistent difference scheme is presented and proved to be stable and convergent

evaluation of three and two field finite element methods for the dynamic response of saturated soil the three field formulation u p u is presented and is applied to solve dynamic equations for saturated porous media in order to evaluate the accuracy and the performance of this three field approximation the u u and the u w formulations are also considered and a few one and two dimensional examples are solved using the three methods problems for which both the pore fluid and the solid grains are incompressible are also considered and are solved using the u p u approximation accuracy was determined by a comparison of the finite element results with the analytical solution given by gajo and mongiovi 19 for a wide range of permeability

homoclinic bifurcations for the henon map abstract chaotic dynamics can be effectively studied by continuation from an anti integrable limit we use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations we find a bound on the parameter range for which the henon map exhibits a complete binary horseshoe as well as a subshift of finite type we classify homoclinic bifurcations and study those for the area preserving case in detail simple forcing relations between homoclinic orbits are established we show that a symmetry of the map gives rise to constraints on certain sequences of homoclinic bifurcations our numerical studies also identify the bifurcations that bound intervals on which the topological entropy is apparently constant

estimating a generating partition from observed time series symbolic shadowing we propose a deterministic algorithm for approximating a generating partition from a time series using tessellations using data generated by henon and ikeda maps we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order and provide good estimates of the metric and topological entropies the algorithm gives useful results even with a short noisy time series

knotted periodic orbits in suspensions of smale s horseshoe extended families and bifurcation sequences abstract we consider knotted periodic orbits in the natural suspension of smale s horseshoe map as constructed by holmes and williams 1985 and holmes 1986 we identify quartets and octets of periodic orbits which are isotopic knots and indicate how they are related via the sequences of bifurcations in which they are created in two parameter families such as the henon map

estimating generating partitions of chaotic systems by unstable periodic orbits an outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1 it has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition here we present a general dimension independent and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits our algorithm allows us to obtain the approximate location of the generating partition for the ikeda hammel jones moloney map

topology from time series we describe methods for the study of topological properties of the invariant manifolds of experimental dynamical systems we explain how to compute such invariants as the euler characteristic and betti numbers using time series data and suggest a number of potential applications

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

a one dimensional nonlinear heat equation with a singular term abstract in this paper we are concerned with the dirichlet problem for the one dimensional nonlinear heat equation with a singular term u t u x x u m u x 2 f x t u 0 x t q t u a t u b t 0 t 0 t u x 0 u 0 x x i where t 0 q t i 0 t i a b with a b 0 1 m 2 we find that the problem may have multiple weak solutions for some initial data to prove this we need to study existence of positive classical solutions in addition we also discuss existence of a positive stationary solution for the above problem and relations between solutions of the above problem and the following problem u t u x x f x t x t q t u b t u a t 0 t 0 t u x 0 u 0 x x i

structure in the bifurcation diagram of the duffing oscillator we identify four levels of structure in the bifurcation diagram of the two well periodically driven duffing oscillator plotted as a function of increasing control parameter t the period of the driving term the superstructure or bifurcation peninsula repeats periodically as t increases by ensuremath sim 2 ensuremath pi beginning and ending with symmetric period one orbits whose local torsions differ by 2 within each bifurcation peninsula there is a systematic window structure the primary window structure is due to newhouse and newhouse like orbits fine structure is due to a farey sequence of well ordered orbits between the primary windows hyperfine structure consists of very narrow windows associated with non well ordered orbits we construct a template for the duffing oscillator a two dimensional return map and a one dimensional return map which describes the systematics of orbit creation and annihilation all structures are identified by topological indices our predictions are based on and compatible with numerical computations

existence uniqueness and asymptotic behavior of solutions for a singular parabolic equation abstract in this paper we are concerned with a singular parabolic equation v t v f x t v 2 v in a smooth bounded domain r n subject to zero dirichlet boundary condition and initial condition 0 under the assumptions on and f x t some existence and uniqueness results are obtained by applying parabolic regularization method and the sub supersolutions method we also discuss the asymptotic behaviors of solutions in the sense of l 2 0 t w 0 1 2 and l 0 t l 2 norms as 0 or as a byproduct we obtain the existence of solutions for some problems which blow up on the boundary

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

complex temporal patterns in molecular dynamics a direct measure of the phase space exploration by the trajectory at macroscopic time scales computer simulated trajectories of bulk water molecules form complex spatiotemporal structures at the picosecond time scale this intrinsic complexity which underlies the formation of molecular structures at longer time scales has been quantified using a measure of statistical complexity the method estimates the information contained in the molecular trajectory by detecting and quantifying temporal patterns present in the simulated data velocity time series two types of temporal patterns are found the first defined by the short time correlations corresponding to the velocity autocorrelation decay times 0 1 ps remains asymptotically stable for time intervals longer than several tens of nanoseconds the second is caused by previously unknown longer time correlations found at longer than the nanoseconds time scales leading to a value of statistical complexity that slowly increases with time a direct measure based on the notion of statistical complexity that describes how the trajectory explores the phase space and independent from the particular molecular signal used as the observed time series is introduced

structure of generating partitions for two dimensional maps we discuss the non uniqueness of the generating partitions of two dimensional maps and we give new explanations for ambiguities that are to be expected in the construction of generating partitions if a parameter is varied both points are discussed with the help of the connecting lines of homoclinic tangencies in the entire phase space i e also outside the attractor that can be calculated very easily with a new algorithm introduced recently

horseshoe templates with global torsion in a driven laser dimensional system with many longitudinal modes oscillating simultaneously hence in sec ii we describe the experimental system and check that topological analysis is applicable by estimating the lyapunov dimension of the attractors then sec iii is devoted to the extraction of upo from experimental data and the computation of their topological invariants we show there how careful signal processing can greatly help in extracting topological information from noisy time series of finite length and precision finally in sec iv we determine the templates of the different attractors compare them and relate them to the values of the modulation frequency in particular we show that these templates have a nonzero global torsion and that this torsion increases by one full turn when the period of modulation increases by 2p v r this observation is in close agreement with the theoretical studies of modulated nonlinear oscillators by gilmore and mccallum 10

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

variationally derived 3 field finite element formulations for quasistatic poroelastic analysis of hydrated biological tissues hydrated biological tissues are often modeled mechanically as poroelastic media with intrinsically incompressible solid and fluid constituents unlike many engineering materials these tissues may typically experience finite deformations during normal physiological activities accurate and efficient finite element formulations are required to solve large problems of experimental and clinical relevance this manuscript describes two new 3 field u p w mixed finite element formulations based on lagrange multiplier and augmented lagrangian representations of the saturation incompressibility constraint as a precursor the continuum mixture theory for finite deformation quasistatic poroelasticity with constituent incompressibility is first reformulated within the variational framework of the principal of virtual power in doing so the equivalence of the continuum mixture theory and biot formulations for this problem is established the improved performance of the mixed formulations over an analogous 2 field u w penalty formulation is demonstrated using axisymmetric numerical examples

homoclinic tangencies and non normal jacobians effects of noise in nonhyperbolic chaotic systems abstract a general study of the effects of noise interacting with the deterministic dynamics of nonhyperbolic chaotic 2 d maps and 3 flows is presented noise in these systems can have dramatic effects on the invariant measure of the system in regions of homoclinic tangencies perturbations are transported away from the neighborhood of the attractor leading to deformations which can be one to two orders of magnitude larger than the noise level a qualitative understanding is attained by a study of the universal properties of the unstable and stable foliation of the phase space in the vicinity of homoclinic tangencies through the investigation of nontrivial structures of the tangent space at homoclinic tangencies related to the non normal jacobians we obtain a quantitative description of these noise induced deformations of the attractor local expansion rates which are much larger than the maximal lyapunov exponent can be used as a measure of the system s structural instability under perturbations we exemplify our general results on the henon ikeda and duffing system a new and effective algorithm to calculate the homoclinic tangencies in the entire phase space based on the results of this paper is presented

bifurcational precedences in the braids of periodic orbits of spiral 3 shoes in driven oscillators we conjecture the existence of a 3 striped spiral horseshoe under one iterate of the poincare map that arises in the analysis of a class of driven oscillators these systems represent escape from a smooth potential well under periodic forcing we assume topological conjugacy between the flow of the differential equation and an idealized suspension of a spiral 3 shoe and deduce bifurcational precedences by consideration of intertwining in the braid of periodic orbits we argue that many significant subharmonic bifurcations observed in such systems can then be understood in terms of the creation of this 3 shoe

classification of strange attractors by integers we show how to characterize a strange attractor by a set of integers these are extracted from the chaotic time series data by first reconstructing the low period orbits and then determining the template or knot holder which supports all periodic orbits embedded in the strange attractor and the strange attractor itself the template is identified by a set of integers which therefore characterize the strange attractor this identification is explicitly demonstrated for the pirogon using a relatively small data set 5000 points

on the topology of the henon map topological invariants of the henon map are investigated by means of the pruning front first a long sequence of primary homoclinic tangencies is computed confirming the monotonicity of the front an algorithm to extract forbidden sequences is then introduced and discussed forbidden sequences of increasing lengths are used to construct a hierarchy of regular grammars represented by directed graphs which approximate the exact grammar arbitrarily well the topological entropy is estimated as the largest eigenvalue of their adjacency matrix it exhibits an exponential convergence towards the asymptotic value with an exponent in agreement with a previous conjecture based on the growth rate of the number of forbidden words

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

knotted periodic orbits in suspensions of annulus maps we consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3 manifold t 2 x i using results of birman williams topology 22 47 82 1983 contemp math 20 1 60 1983 we construct a knotholder or template that carries the set of periodic orbits of the flow we define rotation numbers and show that any orbit of period q and rotation number p q can be arranged as a positive braid on p strands this yields existence and uniqueness results for families of resonant torus knots p braids that are p q torus knots of period q p which correspond to order preserving birkhoff periodic orbits of the diffeomorphism we show that all other q periodic p braids have higher genus and we establish bounds on the genera of such knots we obtain existence and uniqueness results for a number of other non resonant torus knots including non order preserving q s q torus knots of rotation number 1

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

efficient detection of periodic orbits in chaotic systems by stabilizing transformations an algorithm for detecting periodic orbits in chaotic systems r l davidchack and y c lai phys rev e 3 60 1999 pp 6172 6175 which combines the set of stabilizing transformations proposed by schmelcher and diakonos phys rev lett 78 1997 pp 4733 4736 with a modified semi implicit euler iterative scheme and seeding with periodic orbits of neighboring periods has been shown to be highly efficient when applied to low dimensional systems the difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of the stabilizing transformations grows extremely fast with increasing system dimension here we analyze the properties of stabilizing transformations and propose an alternative approach for constructing a smaller set of transformations the performance of the new approach is illustrated on the four dimensional kicked double rotor map and the six dimensional system of three coupled henon maps

dynamics of porous media at finite strain we present a finite element model for the analysis of a mechanical phenomenon involving dynamic expulsion of fluids from a fully saturated porous solid matrix in the regime of large deformation momentum and mass conservation laws are written in lagrangian form by a pull back from the current configuration to the reference configuration following the solid matrix motion a complete formulation based on the motion of the solid and fluid phases is first presented then approximations are made with respect to the material time derivative of the relative flow velocity vector to arrive at a so called v p formulation which is subsequently implemented in a finite element model we show how the resulting finite element matrix equations can be consistently linearized using a compressible neo hookean hyperelastic material with a kelvin solid viscous enhancement for the solid matrix as a test function for the nonlinear constitutive model numerical examples are presented demonstrating the significance of large deformation effects on the transient dynamic responses of porous structures as well as the strong convergence profile exhibited by the iterative algorithm

from template analysis to generating partitions i periodic orbits knots and symbolic encodings abstract we present a detailed algorithm to construct symbolic encodings for chaotic attractors of three dimensional flows it is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by lefranc et al phys rev lett 73 1994 1364 for each orbit the symbolic names that are consistent with its knot theoretic invariants and with the topological structure of the attractor are first obtained using template analysis this information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations we provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations and obtain partitions whose border is localized with a precision of 0 01 a distinctive advantage of this approach is that the solution is progressively refined using higher period orbits which makes it robust to noise and suitable for analyzing experimental time series furthermore the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one dimensional and hyperbolic maps

on the mechanical energy and effective stress in saturated and unsaturated porous continua abstract this paper has two objectives a to formulate a mechanical theory of porous continua within the framework of strong discontinuity concept commonly employed for the analysis of strain localization in inelastic solids and b to introduce an effective stress tensor for three phase solid water air partially saturated porous continua emerging from principles of thermodynamics to achieve the first objective strong forms of the boundary value problems are compared between two formulations the first in which the velocity jump at the solid fluid interface is treated as a strong discontinuity problem and the second in which the strong discontinuity is smeared in the representative volume element as for the second objective an effective cauchy stress tensor of the form 1 k k s p 1 emerges from the formulation where is the total stress tensor k and ks are the bulk moduli of the solid matrix and solid phase respectively and p is the mean pore water and pore air pressures weighted according to the degree of saturation we show from the first and second laws of thermodynamics that this effective stress tensor is power conjugate to the solid rate of deformation tensor and that it includes the mechanical power required to compress the solid phase

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

existence of the maximal weak solution for a class of singular parabolic equations abstract in this paper we study the initial boundary value problem for a class of singular parabolic equations under some additional conditions we prove that there exists a weak solution for the problem by parabolic regularization moreover we show that the solution obtained by parabolic regularization is the maximal solution among all weak solutions

numerical solution of problems in incompressible finite elasticity by augmented lagrangian methods i two dimensional and axisymmetric problems we discuss in this paper a new approach to equilibrium problems in incompressible finite elasticity this approach which is based on the use of a convenient augmented lagrangian functional leads to a family of iterative methods which seem very effective for solving this type of elasticity problems once they have been approximated by appropriate finite element methods the possibilities of the above methods are illustrated by the numerical solution of several problems in two dimensional and axisymmetric geometries comparisons with available analytical results are also presented

event and time scale characteristics of heart rate dynamics the cardiac system shows various scale dynamic activities from secondly to yearly therefore multiple time scale characteristics of heart dynamics have received much attention for understanding and distinguishing healthy and pathological cardiac systems in this paper we expand the multiple time scale analysis into event and time scales to investigate scale characteristics in healthy and pathologic cardiac systems to do this we define a measure based on symbolic dynamics which calculates complexity at each time and event scale called the unit time block entropy utbe this measure allows a reliable comparison of experimental data through matching the number of words and the total measurement time at the same time for all rr interval sequences which are composed of the time durations between consecutive r waves of electrocardiograms we apply the utbe to the healthy heart rate hr group and pathological hr groups and find that the rr interval acceleration is more effective than the rr interval in distinguishing each group and we also find that the normal and pathological hr groups are clearly distinguished in some specific event and time scale regions

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

embeddings of low dimensional strange attractors topological invariants and degrees of freedom when a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding dependent we show that there are just three topological properties that depend on the embedding parity global torsion and knot type we discuss how they can change with the embedding finally we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings these results apply only to chaotic attractors of genus one which covers the majority of cases in which experimental data have been subjected to topological analysis this means that the conclusions drawn from previous analyses for example that the mechanism generating chaotic behavior is a smale horseshoe mechanism a reverse horseshoe a gateau roul e an s template branched manifold etc are not artifacts of the embedding chosen for the analysis

on the attracting set for duffing s equation ii a geometrical model for moderate force and damping abstract after a brief review of some earlier work on duffing s equation in the small force and damping regions we use the results of numerical integrations to construct a geometrically defined poincare map which captures the qualitative features of the attracting set of larger force and damping levels this map has a small constant jacobian determinant and can be regarded as a perturbation of a non invertible one dimensional map we give a partial analysis of the map and pose some important open questions regarding perturbations of one dimensional maps and the creation of strange attractors during bifurcation to horseshoes

on the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term we study questions of existence uniqueness and asymptotic behaviour for the solutions of u x t of the problem begin gathered text u t delta u lambda e u text lambda text 0 t 0 text x text varepsilon b hfill p text u x 0 u 0 x text x text varepsilon b hfill text u x t 0 text on text partial b times 0 infty hfill end gathered where b is the unit ball x varepsilon r n x text leqq text 1 text and n geqq 3 our interest is focused on the parameter 0 2 n 2 for which p admits a singular stationary solution of the form s x 2log x

single point gradient blow up on the boundary for diffusive hamilton jacobi equations in planar domains consider the diffusive hamilton jacobi equation u t u u p p 2 on a bounded domain with zero dirichlet boundary conditions which arises in the kpz model of growing interfaces it is known that u remains bounded and that u may blow up only on the boundary in this paper under suitable assumptions on omega subset mathbb r 2 and on the initial data we show that the gradient blow up singularity occurs only at a single point x 0 in partial omega this is the first result of this kind in the study of problems involving gradient blow up phenomena in general domains of mathbb r n we also obtain results on nondegeneracy and localization of blow up points

existence of solutions for degenerate parabolic equations with singular terms abstract in this paper we deal with parabolic problems whose simplest model is u p u b u p u 0 in 0 t u 0 x u 0 x in u t x 0 on 0 t where t 0 n 2 p 1 b 0 and u 0 is a positive function in l bounded away from zero

a mixture approach to the mechanics of skin skin can be considered to be a mixture of a solid and a fluid a general theory for the description of the behaviour of mixtures is presented and applied to a mixture of a solid and a fluid a numerical procedure is presented to solve the non linear field equations describing such a mixture the abilities of the procedure are demonstrated by means of a confined compression test

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

a nonlinear finite element formulation for axisymmetric torsion of biphasic materials in this paper we present a finite element formulation for describing the large deformation torsional response of biphasic materials with specific application to prediction of nonlinear coupling between torsional deformation and fluid pressurization in articular cartilage due to the use of a cylindrical coordinate system a particular challenge arises in the linearization of the weak form the torsional axisymmetric case considered gives rise to additional geometric terms which are important for the robustness of the numerical implementation and that would not be present in a cartesian formulation a detailed derivation of this linearization process is given couched in the context of a variational formulation suitable for finite element implementation a series of numerical parametric studies are presented and compared to experimental measurements of the time dependent response of cartilage

homoclinic tangencies generating partitions and curvature of invariant manifolds a method to compute the curvature of the unstable manifold is introduced and applied to henon map and duffing attractor showing that it allows the authors to locate the homoclinic tangencies and in turn to construct a generating partition the probability distribution of curvature values is investigated showing a power law decay finally the shape of the multifractal spectrum of effective liapunov exponents in nonhyperbolic systems is discussed

nonuniqueness of solutions for a singular diffusion problem in this paper we consider a singular diffusion problem and show by constructing a counterexample that the weak solution to the problem is not unique the proof consists of several steps first we prove that there exists a maximal weak solution to the problem we show that the support of the continuous maximal weak solution cannot decrease in time then we cite an example of a nonnegative continuous function with shrinking support that also solves the problem and therefore the problem possesses at least two weak solutions for some continuous nonnegative initial data