rate of type ii blowup for a semilinear heat equation a solution u of a cauchy problem for a semilinear heat equation left begin array ll u t delta u u p quad rm in bf r n times 0 t u x 0 u 0 x geq 0 quad rm in bf r n end array right is said to undergo type ii blowup at t t if lim sup t nearrow t t t 1 p 1 u t infty infty let varphi infty be the radially symmetric singular steady state suppose that u 0 in l infty is a radially symmetric function such that u 0 varphi infty and u0 t change sign at most finitely many times we determine the exact blowup rate of type ii blowup solution with initial data u0 in the case of p pl where pl is the lepin exponent

stability of steady states for one dimensional parabolic equations with nonlinear boundary conditions abstract we consider one dimensional parabolic equations with nonlinear boundary conditions u t u x x q u 2 q 1 in r 0 t u u q on 0 0 t u x 0 u 0 x 0 in r this equation admits a family of positive stationary solutions x 0 0 such that 1 x 2 x if 1 2 the main purpose of this paper is to study the stability of these stationary solutions furthermore we discuss the large time behavior of global solutions in particular we prove that every global solution is uniformly bounded and converges to one of the stationary solutions

on the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical sobolev exponent abstract this paper is concerned with a semilinear parabolic equation involving critical sobolev exponent in a ball or in r n the asymptotic behavior of unbounded radially symmetric nonnegative global solutions which do not decay to zero is given the structure of the space of initial data is also discussed

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains we consider fully nonlinear parabolic equations on bounded domains under dirichlet boundary conditions assuming that the equation and the domain satisfy certain symmetry conditions we prove that each bounded positive solution of the dirichlet problem is asymptotically symmetric compared with previous results of this type we do not assume certain crucial hypotheses such as uniform with respect to time positivity of the solution or regularity of the nonlinearity in time our method is based on estimates of solutions of linear parabolic problems in particular on a theorem on asymptotic positivity of such solutions

weak chaos and poincare recurrences for area preserving maps the spectrum f t of poincare recurrence times for the standard map exhibits two distinct limits an integrable weak coupling limit with an inverse power law and a chaotic strong coupling limit with exponential decay in the domain where chaotic regions coexist with integrable structures the spectrum f t exhibits a superposition of exponential and power law decay such a law can be proved to occur in a model of area preserving map at the boundary of the mixing and integrable components

a new approach for adaptive blind equalization of chaotic communication the optimal linearization technique together with the optimal linearization technique a blind channel equalization for the extended kalman filter based chaotic communication is proposed in this paper first the optimal linearization technique is utilized to find the exact linear models of the chaotic system at operating states of interest the proposed blind channel equalization is formulated as a mixed nonlinear parameter and state estimation problem by an autoregressive ar model the channel coefficients of a fading and multipath channel can be represented by an ar process then an extended kalman filter algorithm is utilized to reduce the effect of channel noise by using the extended kalman filter the channel coefficients and the state of the system which is the signal before going through the channel can be estimated the stability problem of the proposed blind channel equalization is also addressed numerical examples and simulations are given to show the effectiveness and speed of convergence for the proposed methodology

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

noise reduction in chaotic time series data a survey of common methods this paper surveys some of the methods that have been suggested for reducing noise in time series data whose underlying dynamical behavior can be characterized as low dimensional chaos although the procedures differ in details all of them must solve three basic problems how to reconstruct an attractor from the data how to approximate the dynamics in various regions on the attractor and how to adjust the observations to satisfy better the approximations to the dynamics all current noise reduction methods have similar limitations but the basic problems are reasonably well understood the methods are an important tool in the experimentalist s repertoire for data analysis in our view they should be used more widely particularly in studies of attractor dimension lyapunov exponents prediction and control

on the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity this paper is concerned with a cauchy problem mbox p left begin array ll u t delta u u p quad mbox in bf r n times 0 infty u x 0 lambda varphi x quad mbox in bf r n end array right where p p ast equiv n 2 n 2 lambda 0 and varphi is a nonnegative radially symmetric function in c 1 bf r n with compact support denote the solution of p by u lambda let p ast infty if 3 leq n leq 10 and p ast 1 6 n 10 if n geq 11 we show that if p ast p p ast then there is lambda varphi 0 such that i if lambda lambda varphi then u lambda exists globally in time in the classical sense and u lambda t converges to zero locally uniformly in bf r n as t to infty ii if lambda lambda varphi then u lambda blows upincompletely in finite time iii if lambda lambda varphi then u lambda blows upcompletely in finite time

convergence analysis for initial condition estimation in coupled map lattice systems in this correspondence we focus on studying the problem of initial condition estimation for chaotic signals within the coupled map lattice cml systems to investigate the effectiveness of a cml initial condition estimation method with different maps and coupling coefficients the convergence and divergence properties of the inverse cml systems are analyzed an inverse largest lyapunov exponent ille is proposed to investigate the strength of convergence and divergence in the inverse cml systems and it can determine if the cml initial condition estimation method is effective computer simulations are included to verify the relationship between the effectiveness of the cml initial condition estimation method and its corresponding ille

asymptotic behavior of type i blowup solutions to a parabolic elliptic system of drift diffusion type we consider a cauchy problem for a parabolic elliptic system of drift diffusion type the problem is formally of the form u t nabla cdot nabla u u nabla delta 1 u this system describes a mass conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis our concern is the asymptotic behavior of blowup solutions when the blowup is type i in the sense that its blowup rate is the same as the corresponding ordinary differential equation y t y 2 up to a multiple constant it is shown that all type i blowup is asymptotically backward self similar provided that the solution is radial nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three

sign changing stationary solutions and blowup for the nonlinear heat equation in a ball in this paper we prove that if phi is a radially symmetric sign changing stationary solution of the nonlinear heat equation u t delta u u alpha u quad quad quad quad quad in the unit ball of mathbb r n n 3 with dirichlet boundary conditions then the solution of nlh with initial value lambda phi blows up in finite time if 1 0 is sufficiently small and if is subcritical and sufficiently close to 4 n 2

comments on an aperiodic phenomenon of the extended kalman filter in filtering noisy chaotic signals in the above paper an aperiodic behavior of the extended kalman filter ekf when filtering chaotic signals is reported in this comment we point out that this erratic behavior should be expected when filtering an aperiodic time varying signal more interestingly the convergence to a fixed point that is observed in some systems is due to the special form of the minimum bayesian mean square error

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

methods for chaotic signal estimation a dynamic programming algorithm and a suboptimal but computationally efficient method for estimation of a chaotic signal in white gaussian noise are proposed the nonlinear map is assumed known so that only the initial condition need be estimated computer simulations confirm that both approaches produce efficient estimates at high signal to noise ratios

nonexistence of type ii blowup solution for a semilinear heat equation a solution u of a cauchy problem for a semilinear heat equation ut u u p 1uin rn 0 t u x 0 u0 x in rn is said to undergo type ii blowup at t t if lim supt t t t 1 p 1 u t let ps and pjl be the exponents of sobolev and of joseph and lundgren respectively we prove that when ps p pjl a radial solution u does not exhibit type ii blowup if u does not blow up at infinity let be the positive singular stationary solution with radial symmetry it was shown in matano and merle 2009 12 that for ps p pjl if the number of intersections with is at most finite then the radial solution does not undergo type ii blowup we do not impose an assumption on the number of intersections with for example when a radial initial data u0 is nonnegative and nonincreasing in r x the result in matano and merle 2009 12 does not exclude type ii blowup for p in the range while our result does it

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

an aperiodic phenomenon of the extended kalman filter in filtering noisy chaotic signals in this correspondence we report an interesting behavior of the extended kalman filter ekf when it is used to filter a chaotic system we show both theoretically and experimentally that the gain of the ekf does not converge or diverge but oscillates aperiodically more precisely when a nonlinear system is periodic the kalman gain and error covariance of the ekf converge to zero however when the system is chaotic they either converge to a fixed point with magnitude larger than zero or oscillate our theoretical analyses are verified using monte carlo simulations based on some popular chaotic systems

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

threshold and generic type i behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume that n 3 and p ps n 2n 2 our first goal is to analyze a threshold behavior for solutions with initial data u0 v where v c h1 and v 0 v 0 it is known that there exists 0 such that the solution converges to 0 as t if 0 with j j 1 2 k the blow up is complete and of type i with a flat local profile our method is based on a combination of the zero number principle and energy estimates in the second part of the paper we employ the very same idea to show that the constant solution attains the smallest rescaled energy among all non zero stationary solutions of the rescaled equation using this result we derive a sharp criterion for no blow up

existence of type ii blowup solutions for a semilinear heat equation with critical nonlinearity abstract we consider the blowup rate of solutions for a semilinear heat equation u t u u p 1 u x r n t 0 with critical power nonlinearity p n 2 n 2 and n 3 first we investigate the profiles of backward self similar solutions by making use of the variational method and then by employing the intersection comparison argument with a particular self similar solution we derive the criteria of the blowup rate of solutions assuming the positivity of solutions in backward space time parabola in particular we show the existence of the so called type ii blowup solutions for the cauchy dirichlet problems on suitable shrinking domains in the case n 3

growup of solutions for a semilinear heat equation with supercritical nonlinearity abstract we consider a cauchy problem for a semilinear heat equation p u t u u p in r n 0 u x 0 u 0 x 0 in r n let v be the radially symmetric singular steady state of p it is proved that if p n 2 n 1 n 4 2 n 1 and n 11 then for each nonnegative even integer n there exists a radially symmetric global solution u n of p with n intersections with v such that t a n u n t 1 as t for some a n 0 depending on n the exact value of a n is also given we show that a 0 is the optimal upper bound of growup rate for solutions below v

parameter estimation in chaotic noise the problem of parameter estimation in chaotic noise is considered in this paper since a chaotic signal is inherently deterministic a new complexity measure called the phase space volume psv is introduced for estimation instead of using the conventional probabilistic measures we show that the unknown parameters of a signal embedded in chaotic noise ran be obtained by minimizing the psv mpsv of the output of an inverse filter of the received signal in a reconstructed phase space monte carlo simulations are carried out to analyze the efficiency of the mpsv method for parameter estimation in chaotic noise to illustrate the usefulness of the mpsv technique in solving real life problems the problem of sinusoidal frequency estimation in real radar clutter unwanted radar backscatters is considered modeling radar clutter as a chaotic process we apply the mpsv technique to estimate the sinusoidal frequencies by estimating the coefficients of an autoregressive ar spectrum the results show that the frequency estimates generated by the mpsv method are more accurate than those obtained by the standard least square ls technique

asymptotic maximum likelihood estimator performance for chaotic signals in noise the performance of the maximum likelihood estimator for a 1 d chaotic signal in white gaussian noise is derived it is found that the estimator is inconsistent and therefore the usual asymptotic distribution large data record length is invalid however for high signal to noise ratios snrs the maximum likelihood estimator is asymptotically unbiased and attains the cramer rao lower bound

boundedness of global solutions of a supercritical parabolic equation we study radially symmetric classical solutions of the dirichlet problem for a heat equation with a supercritical nonlinear source we give a sufficient condition under which blow up in infinite time cannot occur this condition involves only the growth rate of the source term at infinity we do not need the homogeneity property which played a key role in previous proofs of similar results we also establish the blow up rate for a class of solutions which blow up in finite time

an ergodic approach for chaotic signal estimation at low snr with application to ultra wide band communication this paper proposes a novel method to estimate the parameters of chaotic signals corrupted by noise by exploiting the ergodic property of chaotic signals it is shown here that signal parameters can be estimated accurately from the noisy chaotic signal the proposed estimator is proved to be consistent the asymptotic variance of the proposed method and the cramer rao lower bound crlb are derived analytically to assess the estimation performance computer simulations confirm that the proposed ergodic estimation approach provides good parameter estimates even at low signal to noise ratio snr and its performance is superior to conventional estimation techniques this method is then applied to ultra wide band uwb communication by proposing a novel ergodic chaotic parameter modulation based uwb ecpm uwb scheme the ecpm uwb scheme is analog and noncoherent the communication performance of ecpm uwb is found to be superior through theoretical mean square error mse analysis and computer simulations the ecpm uwb scheme is shown to have good spectral characteristics in addition the proposed scheme is shown to be robust against channel estimation error and multipath fading using both theoretical and numerical analyses

estimating initial conditions of noisy chaotic signals generated by piecewise linear markov maps using itineraries estimating a one dimensional 1 d chaotic signal in noise is an important problem in chaotic communications and information processing this problem is theoretically equivalent to the estimation of the initial condition of a chaotic signal a few studies on this initial condition estimation problem have been carried out for certain specific maps such as the tent map and the logistic map this problem is investigated for the piecewise linear markov maps as well as maps that are topologically conjugate to piecewise linear markov maps by using the one to one correspondence between the initial conditions of a chaotic map and its space of itineraries several algorithms extending the halving method are developed to estimate the initial condition of a 1 d chaotic signal embedded in additive noise performance of these estimators is evaluated using monte carlo simulations at high snr the variance of these estimators is found to approach the cramer rao bound

asymptotic behavior of some nonlinear heat equations abstract in this paper a semilinear heat equation with a convex nonlinearity is considered the asymptotic behavior of the solutions is completely determined and this gives in particular a very precise description of the global stability of stationary solutions author

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

return time entropies for a class of circle homeomorphisms poincare recurrence for a class of circle maps is used to study the properties of the corresponding invariant measures in the subcritical case when the map is a diffeomorphism the return time measure is smooth and in the critical case when the map is only a homeomorphism the measure is only continuous furthermore in the considered class of critical maps the behaviour of the return time entropy depends only on the tail in the continued fraction expansion of the rotation number

ergodic attractors and almost everywhere asymptotics of scalar semilinear parabolic differential equations abstract we consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions and on the entire real line with a general nonlinearity g t x u u x either not depending on t or periodic in t while the topological and geometric structure of their attractors has been investigated in depth we focus here on ergodic theoretical properties the main result is that the union of supports of all the invariant measures projects one to one to r 2 we rely on a novel application of the zero number techniques with respect to evolution of measures on the phase space and on properties of the flux of zeroes and the dissipation of zeroes as an example of an application we prove uniqueness of an invariant measure for a large family of considered equations which conserve a certain quantity mass thus generalizing the results by sinai for the scalar viscous burgers equation

consistent estimation of symmetric tent chaotic sequences with coded itineraries maximum likelihood ml estimation of long chaotic sequences is generally statistically inefficient therefore as the length of the sequences increase we do not obtain the usual ml behavior of consistency and normality we discuss here a specific class of chaotic sequences for which consistency is preserved the itineraries of the chaotic sequences in this class are derived from a set of binary code words as a result we are able to guarantee a minimal hamming distance between them which increases linearly with the sequences length in other words pairs of sequences from this class have a non vanishing normalized hamming distance between their itineraries as their length goes to infinity we derive expressions related to the associated euclidean distance between these chaotic sequences using these expressions we show that the condition for consistent estimation of a sequence from the class is satisfied with probability 1 throughout the paper we use the discrete time symmetric tent chaotic sequences as an example and the expressions are derived for this specific case we argue however that our results apply for a wider class of chaotic sequences we mention the applicability of the work to the field of error correcting codes for analog signals however it may be of interest for readers working on other aspects of chaotic maps as well

hausdorff dimension of measures via poincare recurrence we study the quantitative behavior of poincare recurrence in particular for an equilibrium measure on a locally maximal hyperbolic set of a c1 diffeomorphism f we show that the recurrence rate to each point coincides almost everywhere with the hausdorff dimension d of the measure that is inf k 0 fkx b x r r d this result is a non trivial generalization of work of boshernitzan concerning the quantitative behavior of recurrence and is a dimensional version of work of ornstein and weiss for the entropy we stress that our approach uses different techniques furthermore our results motivate the introduction of a new method to compute the hausdorff dimension of measures

various behaviors of solutions for a semilinear heat equation after blowup abstract the present paper is concerned with a cauchy problem for a semilinear heat equation p u t u u p in r n 0 u x 0 u 0 x 0 in r n we show that if p n 2 n 1 n 4 2 n 1 and n 11 then there exists a solution u i i 1 2 3 of p which blows up at t t i becomes a regular solution for all t t i and behaves as follows i lim t u 1 t 0 ii 0 lim inf t u 2 t lim sup t u 2 t iii lim t u 3 t where denotes the supremum norm in r n

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

a general efficient method for chaotic signal estimation this article presents a general computationally efficient method for chaotic signal estimation based on the connection between the symbolic sequence and the initial condition of a chaotic system the performance of the method in white gaussian noise is evaluated the new method is asymptotically unbiased and attains the cramer rao lower bound at high signal to noise ratios

secure communication with chaotic systems of difference equations the paper presents chaotic systems of difference equations that can effectively encrypt information two classes of systems are presented the first one class 1 is optimized for secure communications over reliable channels while the second class 2 tolerates transmission noise at the expense of reduced parameter space size the nonlinearity of these systems is achieved by designing proper piecewise linear functions and by using module operations the utilization of additional nonlinear terms can improve the enciphering efficiency the encrypting performance of the algorithms is evaluated analytically and by simulation experiments also the case of an imperfect transmission channel that inserts noise in the transmitted signal is addressed and the design is modified in order to offer reliable secure transmission over channels with very small signal to noise ratios

blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity abstract this paper is concerned with blowup phenomena of solutions for the cauchy and the cauchy dirichlet problem of p u t u u p with p in the supercritical range in the sense of sobolev s embedding we first show that if p 1 7 n 11 and n 12 then there are no radially symmetric bounded positive solutions of w y 2 w 1 p 1 w w p 0 in r n which intersect the radially symmetric singular solution at least twice using the above result the existence of a blowup solution of type ii for the cauchy dirichlet problem for p in a ball is proved where a solution u is said to exhibit the type ii blowup at t t if lim sup t t t t 1 p 1 u t

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

sign changing stationary solutions and blowup for the two power nonlinear heat equation in a ball abstract consider the nonlinear heat equation 0 1 u t u u p 1 u u q 1 u where t 0 and x the unit ball of r n n 3 with dirichlet boundary conditions let h be a radially symmetric sign changing stationary solution of 0 1 we prove that the solution of 0 1 with initial value h blows up in finite time if 1 0 is sufficiently small and if 1 q p p s n 2 n 2 and p sufficiently close to p s this proves that the set of initial data for which the solution is global is not star shaped around 0

on the structure of global solutions of the nonlinear heat equation in a ball abstract in this paper we consider the nonlinear heat equation nlh u t u u u in the unit ball of r n with dirichlet boundary conditions in the subcritical case more precisely we study the set g of initial values in c 0 for which the resulting solution of nlh is global we obtain very precise information about a specific two dimensional slice of g which necessarily contains sign changing initial values as a consequence of our study we show that g is not convex this contrasts with the case of nonnegative initial values where the analogous set g is known to be convex

type ii blow up mechanisms in a semilinear heat equation with critical joseph lundgren exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant it is well known that type ii blow up does occur if n 11 and p p jl where p jl stands for the joseph lundgren exponent p jl n 10 1 4 n 4 2 n 1 n 11 on the other hand it is also known that if p p jl type ii blow up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned it has long remained open whether or not type ii blow up occurs for the borderline case p p jl n 11 in the present article we prove constructively the existence of type ii blow up solutions for the joseph lundgren critical case thus giving an answer to this question

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

universal scaling of critical quasiperiodic orbits in a class of twist maps recently we have shown that the fractal properties of the critical invariant circles of the standard map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in buric n mudrinic m and todorovic k 1997 30 l161 in the present paper this result is extended on the whole class of sufficiently smooth area preserving twist maps of cylinders we present numerical evidence that the and d q are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail

a sufficient condition for type i blowup in a parabolic elliptic system abstract this paper is concerned with blowup of positive solutions to a cauchy problem for a parabolic elliptic system u t u u v in r n 0 t 0 v u in r n 0 t we say that a solution u v blows up at t t if lim sup t t u t with l norm in r n when a solution u v blows up at t t the blowup is called of type i if lim sup t t t t u t and of type ii otherwise it was shown in herrero and velazquez 1996 12 and mizoguchi and senba 2007 20 that there exist radial type ii blowup solutions for n 2 or n 11 when 3 n 9 type ii blowup solutions were given by formal analysis in brenner et al 1999 2 and herrero et al 1997 10 we give a sufficient condition for a solution to exhibit type i blowup in the case of 3 n 9