functional dynamics i articulation process abstract the articulation process of dynamical networks is studied with a functional map a minimal model for the dynamic change of relationships through iteration the model is a dynamical system of a function f not of variables having a self reference term f f introduced by recalling that operation in a biological system is often applied to itself as is typically seen in rules in the natural language or genes starting from an inarticulate network two types of fixed points are formed as an invariant structure with iterations the function is folded with time until it has finite or infinite piecewise flat segments of fixed points regarded as articulation for an initial logistic map attracted functions are classified into step folded step fractal and random phases according to the degree of folding oscillatory dynamics are also found where function values are mapped to several fixed points periodically the significance of our results to prototype categorization in language is discussed

semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

nonlinear dynamics in coupled fuzzy control systems i coherence and chaos frustration in triangle configuration abstract nonlinear dynamics and chaos are studied in a system for which a complete set of equations of motion such as equations of newton navier stokes and van der pol is not available as a very general system as such we consider coupled classical spins pendulums each of which is under control by a fuzzy system that is designed to align the spin to an unstable fixed point the fuzzy system provides a deterministic procedure to control an object without use of a differential equation the positions and velocities of the spins are monitored periodically and each fuzzy control gives a momentum to its associated spin in the reverse directions if the monitoring is made with an interval short enough the spin spin interactions are overwhelmed by the fuzzy control and the system converges to a state as designed however a long interval monitoring induces dynamics of too late response and thereby results in chaos a great variety of dynamics are generated under very delicate balance between the fuzzy control and the spin spin interaction in which two independent mechanisms of creating negative and positive liapunov exponents interact with each other

border collision period doubling scenario using a one dimensional dynamical system representing a poincare return map for dynamical systems ofthe lorenz type we investigate the border collision period doubling bifurcation scenario in contrast to theclassical period doubling scenario this scenario is formed by a sequence of pairs of bifurcations whereby eachpair consists of a border collision bifurcation and a pitchfork bifurcation the characteristic properties of thisscenario like symmetry breaking and symmetry recovering as well as emergence of coexisting attractors andnoninvariant attractive sets are investigated doi 10 1103 physreve 70 026222 pacs number s 05 45 ac 82 40 bj 95 10 fhi introduction

maximum norm estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

discontinuity induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle abstract this paper applies methods of numerical continuation analysis to document characteristic bifurcation cascades of limit cycles in piecewise smooth hybrid dynamical system models of the eukaryotic cell cycle and associated period adding cascades in piecewise defined maps with gaps a general theory is formulated for the occurrence of such cascades for example given the existence of a period two orbit with one point on the system discontinuity and with appropriate constraints on the forward trajectory for nearby initial conditions in this case it is found that the bifurcation cascade for nearby parameter values exhibits a scaling relationship governed by the largest in magnitude floquet multiplier here required to be positive and real in complete agreement with the characteristic scaling observed in the numerical study a similar cascade is predicted and observed in the case of a saddle node bifurcation of a period two orbit away from the discontinuity provided that the associated center manifold is found to intersect the discontinuity transversally

pointwise error estimates of linear finite element method for neumann boundary value problems in a smooth domain pointwise error analysis of the linear finite element approximation for delta u u f in omega partial n u tau on partial omega where omega is a bounded smooth domain in mathbb r n is presented we establish o h 2 log h and o h error bounds in the l infty and w 1 infty norms respectively by adopting the technique of regularized green s functions combined with local h 1 and l 2 estimates in dyadic annuli since the computational domain omega h is only polyhedral one has to take into account non conformity of the approximation caused by the discrepancy omega h ne omega in particular the so called galerkin orthogonality relation utilized three times in the proof does not exactly hold and involves domain perturbation terms or boundary skin terms which need to be addressed carefully a numerical example is provided to confirm the theoretical result

chaos in a neural network circuit abstract we have constructed a neural network circuit of four clipped high grain integrating operational amplifiers coupled to each other through an array of digitally programmable resistor ladders mdacs in addition to fixed point and cyclic behavior the circuit exhibits chaotic behavior with complex strange attractors which are approached through period doubling intermittent attractor expansion and or quasiperiodic pathways couplings between the nonlinear circuit elements are controlled by a computer which can automatically search through the space of couplings for interesting phenomena we report some initial statistical results relating the behavior of the network to properties of its coupling matrix through these results and further research the circuit should help resolve fundamental issues concerning chaos in neural networks

fractality of deterministic diffusion in the nonhyperbolic climbing sine map abstract the nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter we show that on a suitable coarse scale this map generates an oscillating parameter dependent diffusion coefficient similarly to hyperbolic maps whose asymptotic functional form can be understood in terms of simple random walk approximations on finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter by using a green kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized takagi functions obeying generalized de rham type functional recursion relations we finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive and that for normal diffusion it increases by increasing the parameter value

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

control of rare intense events in spatiotemporally chaotic systems we address the problem of using feedback control for the purpose of suppressing rare intense events in spatially extended systems as an example we investigate the use of control to suppress turbulent spikes in the complex ginzburg landau equation in the limit of small dissipation we explore how information obtained by forecasting can be used to implement spatially and temporally localized control parameter changes and how control strength and cost are related to effectiveness in this framework the effects of model error and imperfect state measurement are also considered

multi order power spectrum and its application abstract a new technique was explored to investigate the time correlations in time series this is an extension of the ordinary power spectrum which is called the order q power spectrum i have extended the order q power spectrum further and made the multi order version of it this technique is able to be applied to various processes which have more than two states this approach is applied to a chaotic system which exhibits intermittent switching and shows us a surprising result appropriate choices for parameters show us various correlations that are not able to be observed in the ordinary power spectra

analysis of fully discrete fem for miscible displacement in porous media with bear scheidegger diffusion tensor fully discrete galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly used bear scheidegger diffusion dispersion tensor begin aligned d mathbf u gamma d m i mathbf u bigg alpha t i alpha l alpha t frac mathbf u otimes mathbf u mathbf u 2 bigg end aligned previous works on optimal order l infty 0 t l 2 norm error estimate required the regularity assumption nabla x partial td mathbf u x t in l infty 0 t l infty omega while the bear scheidegger diffusion dispersion tensor is only lipschitz continuous even for a smooth velocity field mathbf u in terms of the maximal l p regularity of fully discrete finite element solutions of parabolic equations optimal error estimate in l p 0 t l q norm and almost optimal error estimate in l infty 0 t l q norm are established under the assumption of d mathbf u being lipschitz continuous with respect to mathbf u

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

synchronization multistability and basin crisis in coupled pendula abstract the synchronization dynamics of two linearly coupled pendula is studied in this paper based on the lyapunov stability theory and linear matrix inequality lmi some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k th for the on set of full synchronization is obtained the numerical value of k th determined from the average energies of the systems is in good agreement with theoretical analysis prior to the on set of synchronization the boundary crisis of the chaotic attractor is identified in the bistable states where two asymmetric periodic attractors co exist it is shown that the coupled pendula can attain multistable states via a new dynamical transition the basin crisis that occur prior to the on set of stable synchronization the essential feature of basin crisis is that the two co existing attractors are destroyed while new three or more co existing attractors of the same or different periodicity are created in addition the linear perturbation technique and the routh hurwitz criteria are employed to investigate the stability of steady states and clearly identify the different types of bifurcations likely to be encountered finally two parameter phase plots show various regions of chaos hyperchaos and periodicity

fractal boundaries in open hydrodynamical flows signatures of chaotic saddles we introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently it is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors the fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set we point out that these boundaries coincide with streak lines passing through appropriately chosen points as an illustrative numerical experiment we consider a model of the von k arm an vortex street a time periodic two dimensional flow of a viscous fluid around a cylinder

globally coupled maps phase transitions and synchronization abstract bifurcations in a system of coupled maps are investigated using symbolic dynamics it is shown that for coupled shift maps the well known space time mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state for coupled non hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour the anomalous dependence of fluctuations on the system size is attributed to these bifurcations

discrete maximal regularity and the finite element method for parabolic equations maximal regularity is a fundamental concept in the theory of partial differential equations in this paper we establish a fully discrete version of maximal regularity for parabolic equations on a polygonal or polyhedral domain varomega we derive various stability results in the discrete l p 0 t l q varomega norms for the finite element approximation with the mass lumping to the linear heat equation our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of the result of dore and venni as an application optimal order error estimates in those norms are proved furthermore we study the finite element approximation for semilinear heat equations with locally lipschitz continuous nonlinear terms and offer a new method for deriving optimal order error estimates some interesting auxiliary results including discrete gagliardo nirenberg and sobolev inequalities are also presented

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

chaotic combustion of solids and high density fluids near points of strong resonance the loss of stability of steady planar combustion waves propagating through solids and high density fluids occurs via hopf bifurcation for sufficiently large values of a parameter related to the activation energy of the reaction near multiple eigenvalues corresponding to points of strong resonance the dynamical behaviour of an asymptotic model derived from the original set of conservation equations is governed by an ordinary dynamical system for the amplitudes of the eigenmodes of the linearized problem an analysis of these amplitude equations then completely determines the local bifurcation structure it is shown that in certain parameter regimes these equations admit finite amplitude chaotic solutions following the loss of stability of a secondary bimodal solution branch of entrained time periodic combustion waves thus in contrast to direct numerical integration of the original system of conservation equations far above the neutral stability boundary that marks the transition from steady to non steady burning it is shown that multi dimensional chaotic flame propagation can be predicted from nonlinear stability theory in the neighbourhood of a multiple bifurcation point

gronwall bellman type nonlinear delay integral inequalities on time scales abstract in this paper some gronwall bellman type nonlinear delay integral inequalities on time scales are established which provide a handy tool in deriving boundedness of solutions of certain delay dynamic equations on time scales our results generalize some of the main results in lipovan 2006 1 pachpatte 2000 2 ferreira and torres 2009 3 zhang and meng 2008 4 cheung and ren 2006 5 kim 2009 6 and some of our results unify continuous and discrete analysis in the literature

scaling laws for noise induced superpersistent chaotic transients a super persistent chaotic transient is characterized by the following scaling law for its average lifetime tau approximately exp c p p c alpha where c 0 and alpha 0 are constants p or p c is a bifurcation parameter and p c is its critical value as p approaches p c from above the exponent in the exponential dependence diverges leading to an extremely long transient lifetime historically the possibility of such transient raised the question of whether asymptotic attractors are relevant to turbulence here we investigate the phenomenon of noise induced super persistent chaotic transients in particular we construct a prototype model based on random maps to illustrate this phenomenon we then approximate the model by stochastic differential equations and derive the scaling laws for the transient lifetime versus the noise amplitude epsilon for both the subcritical p p c cases our results are the following in the subcritical case where a chaotic attractor exists in the absence of noise noise induced transients can be more persistent in the following sense of double exponential and algebraic scaling tau approximately exp k0 exp k1epsilon gamma for small noise amplitude epsilon where k0 0 k1 0 and gamma 0 are constants the longevity of the transient lifetime in this case is striking for the supercritical case where there is already a super persistent chaotic transient noise can significantly reduce the transient lifetime these results add to the understanding of the interplay between random and deterministic chaotic dynamics with surprising physical implications

the bandcount increment scenario ii interior structures bifurcation structures in the two dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely in a series of three papers we investigate th

discrete maximal parabolic regularity for galerkin finite element methods the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the time discontinuous galerkin solution of linear parabolic equations such estimates have many applications they are essential for example in establishing optimal a priori error estimates in non hilbertian norms without unnatural coupling of spatial mesh sizes with time steps

nonlinear electronic circuit part i multiple routes to chaos abstract a nonlinear electronic oscillator suitable for synchronized chaotic communication is studied this circuit is capable of transmitting discrete chaotic signals although the chaotic modes of operation are controlled in an analog way in part i of this review paper the three routes to chaotic operation that appear namely the period doubling intermittency and crisis induced intermittency are thoroughly studied and discussed in all three routes to chaos the appropriate experimental distributions were calculated moreover the chaotic nature of the circuit operation was evaluated by using the grassberger procaccia method calculation of the corresponding minimum embedding dimension the kolmogorov sinai entropy as well as the maximal lyapunov exponent give useful information in order to fully characterize the circuit operation

double precision errors in the logistic map statistical study and dynamical interpretation the nature of the round off errors that occur in the usual double precision computation of the logistic map is studied in detail different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined histograms of errors in trajectories given and for the case of fully developed chaos an explicit formula is found it is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram both jumps and locations seem to present geometric convergence characterized by the two first feigenbaum constants eventually a comparison with benford s law for the distribution of the leading digit of compilation of numbers is discussed

the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator abstract using recently developed techniques of wiggins 1988 we study the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator we show how homoclinic tori give rise to chaotic dynamics for single degree of freedom quasiperiodically forced oscillators in much the same way as smale horseshoes in the periodically forced case we give a complete description of the bifurcation set in the five dimensional parameter space for the two frequency forced duffing oscillator and then show how the two frequency results can be used to understand the effects of an arbitrary but finite number of frequencies

some new delay integral inequalities and their applications the main objective of this paper is to establish some new delay integral inequalities which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations

validity of numerical trajectories in the synchronization transition of complex systems we investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly nonhyperbolic behavior the system undergoes on off intermittency with respect to the synchronization state there are potentially severe consequences of these facts on the validity of the computer generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties

intermingled basins in coupled lorenz systems we consider a system of two identical linearly coupled lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength we verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa we investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system

double crises in two parameter dynamical systems a crisis is a sudden discontinuous change in a chaotic attractor as a system parameter is varied we investigate phenomena observed when two parameters of a dissipative system are varied simultaneously following a crisis along a curve in the parameter plane two such curves intersect at a point we call a double crisis vertex the phenomena we study include the double crisis vertex at which an interior and a boundary crisis coincide and related forms of double crisis we show how an experimenter can infer a crisis from observations of other related crises at a vertex

phase control of intermittency in dynamical systems we present a nonfeedback method to tame or enhance crisis induced intermittency in dynamical systems by adding a small harmonic perturbation to a parameter of the system the intermittent behavior can be suppressed or enhanced depending on the value of the phase difference between the main driving and the perturbation the validity of the method is shown both in the model and in an experiment with a co2 laser an analysis of this scheme applied to the quadratic map near crisis illustrates the role of phase control in nonlinear dynamical systems in this paper we show that the intermittency at an interior crisis in a dynamical system can be controlled by a phase control scheme we give experimental and numerical evi dence of the validity of the method for the co2 laser close to an interior crisis we show that if we apply a small harmonic perturbation to the laser an accurate selection of allows us to place the laser in the desired dynamical state which can be either the intermittent regime or the precrisis regime three different values of the frequency of the perturbation are used the frequency of the main forcing f 0 the frequency of the period doubling bifurcation f 0 2 and finally f 0 3 which is the frequency of the unstable orbit responsible for the interior crisis in order to have a deeper insight into the role of we also present an analysis of phase control of the quadratic map close to a crisis and we show that influences drastically the global dynamics of the system all this evi dence confirms that the phase plays an important role in the final state of the system so it can be used to control the intermittency the structure of the paper is as follows in sec ii we describe the laser and the phase control scheme in secs iii and iv we show the effectiveness of the method in control ling the intermittency when the unperturbed laser is operat ing before and after the crisis respectively the phase control of the quadratic map is presented in sec v finally sec vi is devoted to the conclusions

on detection of multi band chaotic attractors in this work we present two numerical methods for the detection of the number of bands of a multi band chaotic attractor the first method is more efficient but can be applied only for dynamical systems with a continuous system function whereas the second one is applicable for dynamical systems with a discontinuous system function as well using the developed methods we investigate a one dimensional piecewise linear map and report for both cases of a continuous and a discontinuous system functions some new bifurcation scenarios involving multi band chaotic attractors

one dimensional dynamics for a discontinuous map abstract a discontinuous two parameter family of one dimensional maps is shown to posses many of the features of the quadratic logistic map for instance a unique twice differentiable maximum however paths may be chosen in the two dimensional parameter space which give rise to a wealth of possible routes to chaos in particular paths may be found for the feigenbaum route to chaos period doubling cascade of bifurcations with a resulting single parameter in the map converging geometrically with a constant n directly to chaotic motion

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

unfolding a chaotic bifurcation the differential equation of the sinusoidally forced pendulum is studied by digital simulation in a regime where two simple symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased by introducing a small constant bias in the forcing to break the symmetry two discontinuous bifurcations unfold from the single merging event considering both forcing amplitude and bias together as controls a codimension two bifurcation of chaotic attractors is defined whose geometric structure in control phase space is closely related to the elementary cusp catastrophe the chaotic bifurcations are explained in terms of homoclinic structures smale cycles in the poincare map

a generalization of retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

systematic template extraction from chaotic attractors ii genus one attractors with multiple unimodal folding mechanisms asymmetric and symmetric chaotic attractors produced by the simplest jerk equivariant system are topologically characterized in the case of this system with an inversion symmetry it is shown that symmetric attractors bounded by genus one tori are conveniently analyzed using a two components poincare section resulting from a merging attractor crisis these attractors can be easily described as being made of two folding mechanisms here described as mixers one for each of the two attractors co existing before the crisis symmetric attractors are thus described by a template made of two mixers we thus developed a procedure for concatenating two mixers here associated with unimodal maps into a single one allowing the description of a reduced template that is a template simplified under an isotopy the so obtained reduced template is associated with a description of symmetric attractors based on one component poincare section as suggested by the corresponding genus one bounding torus it is shown that several reduced templates can be obtained depending on the choice of the retained one component poincare section

stability and analyticity in maximum norm for simplicial lagrange finite element semidiscretizations of parabolic equations with dirichlet boundary conditions stability and analyticity estimates in maximum norm are shown for spatially discrete finite element approximations based on simplicial lagrange elements for the model heat equation with dirichlet boundary conditions the bounds are logarithm free and valid in arbitrary dimension and for arbitrary polynomial degree the work continues an earlier study by schatz et al 5 in which neumann boundary conditions were considered

role of invariant and minimal absorbing areas in chaos synchronization in this paper the method of critical curves a tool for the study of the global dynamical properties of two dimensional noninvertible maps is applied to the study of chaos synchronization and related phenomena of riddling blowout and on off intermittency a general procedure is suggested in order to obtain the boundary of a particular two dimensional compact trapping region called absorbing area containing the onedimensional chaotic set on which synchronized dynamics occur the main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a milnor attractor with locally riddled basin or a chaotic saddle exists and may strongly influence the effects of riddling and blowout bifurcations some examples are given for a system of two coupled logistic maps and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area some general considerations are given concerning the transition from locally riddled to globally riddled basins and the role of the absorbing area in the occurrence of such transition is discussed s1063 651x 98 11611 2

discrete maximal l p regularity for finite element operators let a h h 0 be a family of elliptic finite element operators let i 0 t and consider the problem u h t a hu h t f h t t in i u h 0 0 in this paper we show that for 1 0 is said to have discrete maximal l p regularity

period doubling mode interactions with circular symmetry abstract a two parameter analysis of the interaction between two period doubling modes with azimuthal wavenumbers k and l l k 1 is carried out for systems with circular symmetry the problem is formulated in terms of an o 2 equivariant map on c 2 in the generic case all primary secondary and tertiary solution branches and their stability properties are classified the results depend on whether k l is odd or even when one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection symmetric mixed modes which in turn can undergo a tertiary hopf bifurcation to a quasiperiodic reflection symmetric pattern we conjecture that this invariant circle can break down with increasing amplitude to produce reflection symmetric chaos additional phase instabilities may occur in which the reflection symmetry is broken and the resulting pattern drifts either clockwise or counterclockwise the results explain a number of observations by ciliberto and gollub on parametrically excited surface waves in circular geometry and imply several new predictions for such experiments in an appendix the theory is compared with previous attempts to model the experiments

the bandcount increment scenario i basic structures bifurcation structures in two dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely in a series of three papers we investigate the chaotic domain without periodic inclusions for a map which is considered by many authors as some kind of one dimensional canonical form for discontinuous maps in the first part we report a novel bifurcation scenario formed by crises bifurcations which includes multi band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain

integral inequalities for retarded volterra equations integral inequalities are very useful in the qualitative analysis of differential and integral equations starting with o lipovan a retarded gronwall like inequality and its applications j math anal appl 252 2000 389 401 several recent investigations see o lipovan a retarded integral inequality and its applications j math anal appl 285 2003 436 443 b g pachpatte explicit bounds on certain integral inequalities j math anal appl 267 2002 48 61 b g pachpatte on some retarded integral inequalities and applications j inequal pure appl math 3 2002 article 18 b g pachpatte on a certain retarded integral inequality and its applications j inequal pure appl math 5 2004 article 19 b g pachpatte on some new nonlinear retarded integral inequalities j inequal pure appl math 5 2004 article 80 were devoted to retarded integral inequalities in this paper we consider the case of retarded volterra integral equations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the asymptotic behaviour of the solutions

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

crisis and stochastic resonance in shinriki s circuit abstract in nonlinear electronic oscillators of the chua type the merging of two precritical attractors leads to crisis induced chaos chaos intermittency this behavior can be induced either by addition of noise to the precritical system or by variation of a control parameter close to such a crisis small periodic signals can be enhanced by the addition of noise via the mechanism of stochastic resonance we present an experimental observation of this effect as well as a thorough description of the critical behavior of the system

a model for the transition to baroclinic chaos abstract the transition to chaos in a 2 layer baroclinic fluid with a slightly different viscosity in each layer is investigated a low order model is constructed by truncating the quasi geostrophic partial differential equations to include one azimuthal wave in each layer while retaining a general zonal flow correction when the viscosities are equal the model reduces to a modified form of the lorenz equations and displays a cusped return map for slightly different viscosities the transition to chaos follows the feigenbaum scenario the return map in the chaotic regime is nearly parabolic and highly contracted multiple attractors are found for the same control parameters the predicted route to chaos is similar to that observed in laboratory experiments although significant quantitative differences remain

two state on off intermittency caused by unstable dimension variability in periodically forced drift waves certain high dimensional dynamical systems present two or more attractors characterized by different energy branches for some parameter values the dynamics oscillates between these two branches in a seemingly random fashion a phenomenon called two state on off intermittency in this work we show that the dynamical mechanism underlying this intermittency involves the severe breakdown of hyperbolicity of the attractors through a mechanism known as unstable dimension variability we characterize the parametric evolution of this variability using statistical properties of the finite time lyapunov exponents as a model system that exhibits this behavior we consider periodically forced and damped drift waves in this spatiotemporal example there is a low dimensional chaotic attractor that is created by an interior crisis already presenting unstable dimension variability

integral inequalities in two independent variables for retarded volterra equations in this paper we obtain some retarded volterra integral inequalities in two independent variables which can be used as tools in the theory of partial differential and integral equations with time delays

explosions of chaotic sets abstract large scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied these bifurcations include sudden changes in the size and or type of the set an explosion is a bifurcation in which new recurrent points suddenly appear at a non zero distance from any pre existing recurrent points we discuss the following in a generic one parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur 1 a saddle node bifurcation isolated from other recurrent points 2 a saddle node bifurcation embedded in the set of recurrent points 3 outer homoclinic tangencies and 4 outer heteroclinic tangencies the term outer tangency refers to a particular configuration of the stable and unstable manifolds at tangency in particular we examine different types of tangencies of stable and unstable manifolds from orbits of pre existing invariant sets this leads to a general theory that unites phenomena such as crises basin boundary metamorphoses explosions of chaotic saddles etc we illustrate this theory with numerical examples

some dynamical properties of a classical dissipative bouncing ball model with two nonlinearities abstract some dynamical properties for a bouncing ball model are studied we show that when dissipation is introduced the structure of the phase space is changed and attractors appear increasing the amount of dissipation the edges of the basins of attraction of an attracting fixed point touch the chaotic attractor consequently the chaotic attractor and its basin of attraction are destroyed given place to a transient described by a power law with exponent 2 the parameter space is also studied and we show that it presents a rich structure with infinite self similar structures of shrimp shape

on retarded integral inequalities and their applications abstract the main results of j math anal appl 285 2003 436 443 are here generalized to the following retarded integral inequalities u m t c m m n m m n 0 t f s u n s w u s g s u n s d s and u m t c m m n m m n 0 t f s u n s w u s d s m m n 0 t g s u n s w u s d s where m n 0 are constants and t r 0 the results given here can be applied to the global existence of solutions to differential equations with time delay

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

large time step maximum norm regularity of l stable difference methods for parabolic equations we establish almost sharp maximum norm regularity properties with large time steps for l stable finite difference methods for linear second order parabolic equations with spatially variable coefficients the regularity properties for first and second spatial differences of the numerical solution mimic those of the continuous problem with logarithmic factors in second differences the regularity results for the inhomogeneous problem imply that the uniform rate of convergence of the numerical solution and its differences is controlled only by the maximum norm of the local truncation error

codimension two bifurcations of nonlinear systems driven by fuzzy noise abstract by means of a fuzzy generalized cell mapping method a duffing van der pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period one attractors grow and merge as the intensity of the fuzzy noise is increased by introducing a small symmetry breaking parameter to break the symmetry the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations considering both the intensity of the fuzzy noise and the symmetry breaking parameter together as controls a codimension two bifurcation of fuzzy attractors is defined and examples of additive and multiplicative fuzzy noise are given such a codimension two bifurcation is fuzzy noise induced effects which cannot be seen in the deterministic systems

amplitude modulated and chaotic dynamics in resonant motion of strings abstract the weakly non linear resonant motions of a stretched string to a harmonic planar excitation are investigated one mode truncation of the in plane and out of plane equation results in a two degree to freedom system which is examined by using the method of averaging and the method of integral manifolds it is shown that the averaged equations for small enough damping possess non planar constant solutions which become unstable by a hopf bifurcation the resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string there are two limit cycle branches for the averaged equations one arising due to the hopf bifurcation and the other due to a global saddle node bifurcation the two branches merge as the damping is further reduced also with variation in detuning the isolated branch exhibits period doubling bifurcations chaotic attractors and merging of attractors giving rise to rossler as well as lorentz type solutions the truncated string equations are also directly integrated it is shown that there are non planar periodic responses that bifurcate into amplitude modulated motions on a 2 torus changes in detuning result in torus doubling merging of tori and then destruction of the torus leading to chaotic amplitude modulations the results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response

full characterization of a strange attractor chaotic dynamics in low dimensional replicator system abstract two chaotic attractors observed in lotka volterra equations of dimension n 3 are shown to represent two different cross sections of one and the same chaotic regime the strange attractor is studied in the equivalent four dimensional catalytic replicator network analytical expressions are derived for the lyapunov exponents of the flow in the centre of the chaotic regime the strange attractor was characterized by the lyapunov dimension dl 2 06 0 02 and the renyi fractal dimensions the set corresponding to the attractor represents a multifractal its singularity spectrum is computed one route in parameter space leading into the chaotic regime and crossing it was studied in detail it leads to a feigenbaum type cascade of bifurcations before the chaos is fully developed the dynamic system passes an internal crisis through a period two tangent bifurcation then a second sequence of period doubling bifurcations leading to another chaotic regime is observed which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type a series of one dimensional maps is derived from a properly chosen poincare cross section which illustrates structural changes in the attractor mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on perturbation theory the most striking result is the gradual disappearance of complex dynamics with increasing mutation rates the parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet

chaotic saddles at the onset of intermittent spatiotemporal chaos in a recent study rempel and chian phys rev lett 98 014101 2007 it has been shown that nonattracting chaotic sets chaotic saddles are responsible for intermittency in the regularized long wave equation that undergoes a transition to spatiotemporal chaos stc via quasiperiodicity and temporal chaos in the present paper it is demonstrated that a similar mechanism is present in the damped kuramoto sivashinsky equation prior to the onset of stc a spatiotemporally chaotic saddle coexists with a spatially regular attractor after the transition to stc the chaotic saddle merges with the attractor generating intermittent bursts of stc that dominate the post transition dynamics

route to high dimensional chaos we present a route to high dimensional chaos that is chaos with more than one positive lyapunov exponent in this route as a system parameter changes a subsystem becomes chaotic through say a cascade of period doubling bifurcations after which the complementary subsystem becomes chaotic leading to an additional positive lyapunov exponent for the whole system a characteristic feature of this route as suggested by numerical evidence is that the second largest lyapunov exponent passes through zero continuously three examples are presented a discrete time map a continuous time flow and a population model for species dispersal in evolutionary ecology

discrete maximal regularity of time stepping schemes for fractional evolution equations in this work we establish the maximal ell p regularity for several time stepping schemes for a fractional evolution model which involves a fractional derivative of order alpha in 0 2 alpha ne 1 in time these schemes include convolution quadratures generated by backward euler method and second order backward difference formula the l1 scheme explicit euler method and a fractional variant of the crank nicolson method the main tools for the analysis include operator valued fourier multiplier theorem due to weis math ann 319 735 758 2001 doi 10 1007 pl00004457 and its discrete analogue due to blunck stud math 146 157 176 2001 doi 10 4064 sm146 2 3 these results generalize the corresponding results for parabolic problems

on certain integral inequalities in two independent variables for retarded equations in this paper we establish some new retarded integral inequalities in two independent variables which can be used as tools in the theory of integral equations with time delays applications are given to illustrate the usefulness of the inequalities

flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus we present evidence for a sequence of bifurcations in simulations of the differentially heated baroclinic rotating annulus similar to bifurcations of the logistic map the met office oxford rotating annulus laboratory simulation morals code is used to construct a detailed numerical regime diagram for the annulus and the distribution of regimes in parameter space is discussed the bifurcations are observed in a sequence of runs at high temperature forcing identified by poincare sections of the dominant temperature mode amplitude time series higher order return maps and predictions using quadratic fits to the data are used to verify this result and lyapunov exponents are calculated to identify and quantify the chaotic parts of the sequence

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

boundary crisis in quasiperiodically forced systems abstract we study boundary crisis in quasiperiodically forced dissipative systems using the henon map as a characteristic example the crisis is due to a homoclinic tangency of the stable and unstable manifolds of an accessible invariant circle of saddle type on the basin boundary of the attractor numerical evidence shows that the type of boundary crisis can change as the saddle circle loses its accessibility due to the quasiperiodic forcing this implies the existence of codimension two double crisis vertices where a curve of boundary crisis and a curve of interior crisis meet we argue that bifurcation points of higher codimension must exist in the full parameter space

on retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

on some new integral inequalities and their applications the aim of the present paper is to establish some new integral inequalities in two independent variables which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain partial differential equations

dynamics of a piecewise linear map with a gap in this paper we consider periodic solutions of discontinuous non smooth maps we show how the fixed points of a general piecewise linear map with a discontinuity a map with a gap behave under parameter variation we show in detail all the possible behaviours of period 1 and period 2 solutions for positive gaps we find that period 2 solutions can exist independently of period 1 solutions conversely for negative gaps period 1 and period 2 solutions can coexist higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation finally we compare our results with those of jain banerjee jain banerjee 2003 int j bifurcat chaos 13 3341 3351 and banerjee et al banerjee et al 2004 ieee trans circ syst ii 51 649 654 and explain their numerical simulations

a hierarchy of transitions to mixed mode oscillation in an electrochemical system abstract the dynamic behavior of the electrocatalytic oxidation of hydrogen on a platinum anode in the presence of cu 2 and cl ions under galvanostatic conditions has been studied experimentally in two control parameters current density and copper ion concentration this parameter plane can roughly be divided into three broad regions a steady state is found at low values of the current density at medium values the system shows small amplitude oscillations whereas at high current densities typical mixed mode oscillations mmos are found depending on the cu 2 concentration the transition from the small amplitude to the mixed mode regime takes place from a simple periodic period doubled or chaotic attractor arising from a feigenbaum route in this paper we focus on the characteristics of these different transitions to mmos and try to construct a bifurcation diagram the bifurcation from the small chaotic attractor to mmos very likely constitutes an interior crisis a bifurcation at which the small chaotic attractor is destroyed by colliding with a stable manifold of a saddle type limit set at lower copper ion concentrations the transition to mmos moved through the period double cascade we compile different properties of the bifurcation diagram and discuss a scenario which fits the observations in a consistent way

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

codimension three bifurcations explanation of the complex one two and three dimensional bifurcation structures in nonsmooth maps many physical and engineering systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied such systems have been found to yield piecewise smooth maps and in some cases the obtained map is discontinuous by investigating the normal form of such maps we have detected a type of codimension three bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space the results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering

the bandcount increment scenario iii deformed structures bifurcation structures in two dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely in a series of three papers we investigated the chaotic domain without periodic inclusions for a map which is considered by many authors as some kind of one dimensional canonical form for discontinuous maps in part i the basic structures in the chaotic region are explained by the bandcount increment scenario in part ii fine self similar substructures nested into the bandcount increment scenario are explained by the bandcount adding and doubling scenarios nested into each other ad infinitum hereby we fixed in both previous parts one of the parameters to a non generic value and studied the remaining two dimensional parameter subspace in this part iii finally we investigated the structures under variation of this third parameter remarkably this step is the most important with respect to practical applications since it cannot be expected that these operate exactly at the previously investigated specific value

complex balancing motions of an inverted pendulum subject to delayed feedback control we show that an inverted pendulum that is balanced on a cart by linear delayed feedback control may exhibit small chaotic motion about the upside down position in periodic windows associated with this chaotic regime we find periodic orbits of arbitrarily high period that correspond to complex balancing motion of the pendulum with bounded velocity of the cart this result shows that complex balancing is possible in a controlled mechanical system with a geometric nonlinearity even when the control law is only linear this is in contrast to other proposed models that require a nonlinearity of the controller such as round off due to digitization we find the complex motion by studying homoclinic bifurcations of a reduced three dimensional vector field near a triple zero eigenvalue singularity more generally the dynamics we describe must be expected in any system with a triple zero singularity and reflection symmetry

bifurcation behavior of the generalized lorenz equations at large rotation numbers the bifurcation structure and periodic orbits of the lorenz stenflo equations at large rotation numbers are given it is shown that rotation can lead to a much richer dynamical behavior than that of the original lorenz system and can be used to control or modify the latter s chaos behavior orbits with new topology arising from the merging and splitting of different periodic windows are observed abrupt changes in the one dimensional map are pointed out and studied in terms of the interaction of the interior and exterior boundaries