global attractor for some partial functional differential equations with finite delay abstract in this paper we study a class of partial functional differential equations with finite delay whose linear part is not necessarily densely defined but satisfies the hille yosida condition using the classical theory about global attractors in infinite dimensional dynamical systems we establish some sufficient conditions for guaranteeing the existence of a global attractor under small delays

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

poincare husimi representation of eigenstates in quantum billiards for the representation of eigenstates on a poincare section at the boundary of a billiard different variants have been proposed we compare these poincare husimi functions discuss their properties and based on this select one particularly suited definition for the mean behavior of these poincare husimi functions an asymptotic expression is derived including a uniform approximation we establish the relation between the poincare husimi functions and the husimi function in phase space from which a direct physical interpretation follows using this a quantum ergodicity theorem for the poincare husimi functions in the case of ergodic systems is shown

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

on impulsive functional differential inclusions with hille yosida operators in banach spaces abstract we consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a banach space assuming that its linear part is a non densely defined hille yosida operator we assume that the multivalued nonlinearity of upper caratheodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness we apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results the application to an optimization problem for an impulsive feedback control system is given

global attractor for a class of functional differential inclusions with hille yosida operators abstract we study the dynamics for a class of functional differential inclusions whose linear part generates an integrated semigroup some techniques of measure of noncompactness are deployed to prove the global solvability and the existence of a compact global attractor for the m semiflow generated by our system the obtained results generalize recent ones in the same direction

universal spectral form factor for chaotic dynamics we consider the semiclassical limit of the spectral form factor k of fully chaotic dynamics starting from the gutzwiller type double sum over classical periodic orbits we set out to recover the universal behaviour predicted by random matrix theory both for dynamics with and without time reversal invariance for times smaller than half the heisenberg time th f 1 we extend the previously known expansion to include the cubic term beyond confirming the random matrix behaviour of individual spectra the virtue of that extension is that the diagrammatic rules come in sight which determine the families of orbit pairs responsible for all orders of the expansion

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

distribution and fluctuation properties of transition probabilities in a system between integrability and chaos the authors study the statistical properties of transition probabilities or generalized intensities equal to the squared matrix elements of operators having a classical limit they first generalize the shnirelman theorem to include semiclassically integrable states and use this to generalize the feingold peres formula for the average value of generalized intensities they perform an unfolding procedure to separate the smooth mean part of the intensities as a function of frequency from its fluctuating part by applying the generalized feingold peres formula this formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos the average values of the intensities as a function of frequency are excellently described by the generalized feingold peres formula

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

pole placement for linear discrete time systems by periodic output feedbacks abstract necessary and sufficient conditions for the existence of a solution to the pole placement problem by periodic output feedbacks for single input single output linear discrete time systems are established an algorithm for finding a sequence of output feedback gains is presented and illustrated by a simple numerical example

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

multi component random model of diffusion in chaotic systems we extend our recent study robnik et al 1997 j phys a math gen 30 l803 of diffusion in strongly chaotic systems the random model to systems composed of several weakly coupled ergodic components by this we mean that the system as a whole is ergodic but the typical time for the transition from one to another component is very long much longer than the ergodic time inside each individual component thus for short times the system behaves like a single component ergodic system and the random model applies neglecting the coupling to other components at times much longer than the transition time the system behaves like an ergodic system without internal structure without decomposition into several components and the random model applies again with different parameters at intermediate times there is the crossover regime which we describe in detail analytically for a two component system and test it numerically in a double billiard system butterfly billiard

general poissonian model of diffusion in chaotic components we extend our recent study of diffusion in strongly chaotic systems the random model to the general systems of mixed type dynamics including especially kam systems regarding the diffusion in chaotic components we do this by introducing a poissonian model as in our previous random model describing the strongly chaotic systems except that now we allow for different a priori probabilities in different cells of the discretized phase space surface of section thus the concept of greyness of cells denoted by g such that is introduced as is its distribution w g we derive the relationship between the dynamical property namely the normalized fraction of chaotic component as a function of discrete time j and w g we predict again the universal scaling law namely that for any w g the chaotic fraction is a function of only and not separately of j and n where n is the number of cells of equal size 1 n the random model of exponential is reproduced if all cells have g 1 i e we argue that in two dimensional systems at any finite n w g is non trivial due to the fractal dimension of the boundary of the chaotic component but is such that it goes to as whilst in systems with three or more degrees of freedom w g has a well defined limit with non zero values also at g 1 this is due to the existence of the arnold web we suggest how through our formalism one can calculate the lebesgue measure of the chaotic component at each finite discretization whose limit exists for our findings are verified and illustrated for two and three dimensional billiards

berry robnik level statistics in a smooth billiard system the berry robnik level spacing distribution is demonstrated clearly in a generic quantized plane billiard for the first time however this ultimate semiclassical distribution is found to only be valid for extremely small semiclassical parameters effective planck s constant where the assumption of statistical independence of regular and irregular levels is achieved for sufficiently larger semiclassical parameters we find fractional power law level repulsion with a phenomenological brody distribution providing an adequate global fit

pole assignment for linear time invariant systems by periodic memoryless output feedback it is well known that the poles of a linear time invariant controllable and observable system can be assigned arbitrarily by state feedback when only the output is available pole assignment is still possible by means of dynamic output feedback in this paper the potential of time varying memoryless output feedback is considered it is shown that up to some technical conditions it is indeed possible to allocate the poles of a linear time invariant discrete time system by memoryless output feedback with periodic gains the period of the gains is n 1 with n the order of the system the power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

maximum norms of chaotic quantum eigenstates and random waves abstract the growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated the maximum norms provide a measure for localization effects in eigenfunctions an upper bound for the maxima of random superpositions of random functions is derived for the random wave model this gives the bound c ln e in the semiclassical limit e the growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic the maximum norms of these systems are numerically shown to be conform with the random wave model furthermore the distribution of the locations of the maximum norms is discussed

fractal weyl law for chaotic microcavities fresnel s laws imply multifractal scattering we demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities as an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium we show that the conjectured fractal weyl law for open chaotic systems lu et al phys rev lett 91 154101 2003 is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating fresnel s laws

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

amplitude distribution of eigenfunctions in mixed systems we study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space for an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and a good agreement with numerical computations for the family of limacon billiards is found the natural extension of our result to more general systems e g with a potential is also discussed

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

spectral fluctuations of billiards with mixed dynamics from time series to superstatistics a statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards one with mushroomlike shape and the other from the family of the limacon billiards is presented these billiards have mixed regular chaotic dynamics but different structures in their classical phase spaces the spectrum of each billiard is represented as a time series where the level order plays the role of time two most important findings follow from the time series analysis first the spectra can be characterized by two distinct relaxation lengths this is a prerequisite for the validity of the superstatistical approach which is based on the folding of two distribution functions second the shape of the resulting probability density function of the so called superstatistical parameter is reasonably approximated by an inverse chi2 distribution this distribution is used to compute nearest neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics the obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions including the one from berry and robnik however in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the porter thomas distribution indeed the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the limacon billiards within the framework of superstatistics

symbolic dynamics and the discrete variational principle we show how to construct symbolic dynamics for the class of 2d dimensional twist mappings generated by piecewise strictly convex concave generating functions the method is constructive and gives an efficient way to find all periodic orbits of these high dimensional symplectic mappings it is illustrated with the cardioid and the stadium billiard

dynamical localization in chaotic systems spectral statistics and localization measure in the kicked rotator as a paradigm for time dependent and time independent systems we study the kicked rotator in the classically fully chaotic regime using izrailev s n dimensional model for various n 4000 which in the limit n tends to the quantized kicked rotator we do treat not only the case k 5 as studied previously but also many different values of the classical kick parameter 5 k 35 and many different values of the quantum parameter ke 5 60 we describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time periodic and time independent autonomous fully chaotic or and mixed type hamilton systems we generalize the scaling variable l n to the case of anomalous diffusion in the classical phase space by deriving the localization length l for the case of generalized classical diffusion we greatly improve the accuracy and statistical significance of the numerical calculations giving rise to the following conclusions 1 the level spacing distribution of the eigenphases or quasienergies is very well described by the brody distribution systematically better than by other proposed models for various brody exponents br 2 we study the eigenfunctions of the floquet operator and characterize their localization properties using the information entropy measure which after normalization is given by loc in the interval 0 1 the level repulsion parameters br and loc are almost linearly related close to the identity line 3 we show the existence of a scaling law between loc and the relative localization length now including the regimes of anomalous diffusion the above findings are important also for chaotic eigenstates in time independent systems batistic and robnik j phys a math gen 43 215101 2010 arxiv 1302 7174 2013 where the brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates even in the mixed type systems after separation of regular and chaotic eigenstates

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

paper uniformly optimal control of linear feedback systems the problem of uniformly optimal control is posed in the context of linear time varying discrete time systems the controller is optimal uniformly over a pre specified set of exogenous signals existence of an optimal controller is proved and a formula for the minimum cost is derived the time invariant case is treated in the frequency domain it is shown that for time invariant systems an optimal time varying controller is no better than an optimal time invariant controller

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

distribution of resonance strengths in microwave billiards of mixed and chaotic dynamics a new measure for statistical properties of the wave function components of quantum systems the distribution of the product of two partial widths is introduced it is tested with data obtained in analog experiments with microwave billiards where the product of two partial widths equals the resonance strengths in the microwave spectra the billiards are from the family of the limacons one with chaotic and two with mixed classical dynamics for completely chaotic systems the partial widths generically obey a porter thomas distribution we show that in this case the distribution of their product equals a k0 distribution while we find deviations of the experimental strength distribution from the k0 distribution for the billiards with mixed dynamics the distributions agree perfectly for the chaotic billiard when taking into account the experimental threshold of detection in the theoretical description hence the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory

pullback attractor for differential evolution inclusions with infinite delays we analyze the existence of pullback attractor for non autonomous differential inclusions with infinite delays in banach spaces by using measures of noncompactness the obtained results can be applied to control systems driven by semilinear partial differential equations and multivalued feedbacks

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

dynamical localization of chaotic eigenstates in the mixed type systems spectral statistics in a billiard system after separation of regular and chaotic eigenstates we study the quantum mechanics of a billiard robnik 1983 j phys a math gen 16 3971 in the regime of mixed type classical phase space the shape parameter 0 15 at very high lying eigenstates starting at about 1 000 000th eigenstate and including the consecutive 587654 eigenstates by calculating the normalized poincare husimi functions of the eigenstates and comparing them with the classical phase space structure we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates and the corresponding energies the chaotic eigenstates appear all to be dynamically localized meaning that they do not uniformly occupy the entire available chaotic classical phase space component but are localized on a proper subset we find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the poisson statistics and the chaotic ones obey the brody statistics as anticipated in a recent paper by batistic and robnik 2010 j phys a math theor 43 215101 where the entire spectrum was found to obey the berry robnik brody statistics there are no effects of dynamical tunneling in this regime due to the high energies as they decay exponentially with the inverse effective planck constant which is proportional to the square root of the energy

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

global attractor for the m semiflow generated by a quasilinear degenerate parabolic equation using theory of global attractors for multi valued semiflows we prove the existence of a global attractor for the m semiflow generated by a parabolic equation involving the nonlinear degenerate operator in a bounded domain

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

two properties of l sub 1 optimal controllers the solution of the l sub 1 sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better known h infinity sensitivity minimization problem or the lqg linear quadratic gaussian problem is given of a one parameter family of first order plants where the order of the l sub 1 optimal compensator can be arbitrarily large and thus it is impossible to bound the order of an l sub 1 optimal compensator in terms of the order of the plant a plant is considered which has a continuous one parameter family of l sub 1 optimal compensators and thus l sub 1 optimal compensators need not be unique the author s two examples are considered to answer two questions left open by m a daleh and j b pearson ibid vol 32 p 314 23 1987

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

a tradeoff between period and order of linear periodic stabilizing compensators here we consider the problem of stabilizing a discrete time linear time invariant plant by a linear periodic compensator we show that we can trade off compensator order for an increased period in particular with n the plant order it is shown that for every l 1 n we can do almost arbitrary pole placement using an lth order n 2 l th period linear periodic controller for most plants the period can be reduced by one we also show that first order linear periodic controllers can be used to solve the strong and simultaneous stabilization problems

the algebraic quantisation of the birkhoff gustavson normal form the author develops an algebraic quantisation method for the birkhoff gustavson normal form for this purpose the weyl quantisation rule is used the method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities the author gives a brief review of the normal form derives some of its general properties and finds a general analytic solution for the fourth degree normal form for hamiltonians of two degrees of freedom in particular this includes the henon heiles system the author compares the results of specific examples with other works the question of canonically invariant quantisation the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed the author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals if the normal form accidentally terminates so that the classical system is integrable then this implies quantum integrability of the normal form hamiltonian

local existence and stability for some partial functional differential equations with infinite delay we study the existence regularity and stability of solutions for some partial functional dierential equations with infinite delay we assume that the linear part is not necessarily densely defined and satisfies the hille yosida condition on a banach space x the nonlinear term takes its values in space larger than x namely the extrapolated favard class of the extrapolated semigroup corresponding to the linear part our approach is based on the theory of the extrapolation spaces

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

beyond the berry robnik regime a random matrix study of tunneling effects we consider the energy spectra of quantum hamilton systems whose classical dynamics is of mixed type i e regular for some initial conditions in the classical phase space and chaotic for the complementary initial conditions in the strict semiclassical limit when the effective planck constant eff is sufficiently small the berry robnik br statistics applies while at larger values of eff or smaller energies one sees deviations from br due to localization and tunneling effects we derive a two level random matrix model which describes these effects and can be treated analytically in a closed form the coupling between the regular and chaotic levels due to tunneling is assumed to be gaussian distributed this two level model describes most of the features of matrices of large dimensions here n 1000 which we treat numerically and is predicted to apply in mixed type systems at low energies the proposed analytical level spacing distribution function has two parameters the br parameter characterizing the classical phase space and the coupling antenna distortion or tunneling parameter between states localization effects so far are not included in our analysis except in the special case where can be replaced by some effective eff

existence for nonlinear evolution inclusions with nonlocal retarded initial conditions abstract in this paper we prove a sufficient condition for the global existence of bounded c 0 solutions for a class of nonlinear functional differential evolution equation of the form u t a u t f t t 0 f t f t u t u t 1 u t n t 0 u t g u t t 0 where x is a real banach space a is the infinitesimal generator of a nonlinear compact semigroup f r d a n 1 x is a nonempty convex weakly compact valued and almost strongly weakly u s c multi function and g c b d a c 0 d a is nonexpansive

on the discontinuities of the boundary in billiards abstract this work concerns billiards in domains of r 2 the set s of pre images of discontinuities of t the bouncing map induced by the flow is studied particularly those discontinuities due to points where the boundary is not c1 we prove that if the angle at the discontinuity is a divisor of there exists a power of t which is continuous around such a point and that under conditions of symmetry this power is locally differentiable it is assumed that continuity may often be extended to certain areas of phase space this property is used to show by means of numerical simulations of two billiards how s fashions the phase space in general if the angle is not a divisor of s separates the distinct areas of quasi interagrable motion one of the billiards is isomorphic to a mechanical model of three interacting particles on a line

abstract impulsive differential equations without predefined time impulses abstract we introduce and study a general class of abstract impulsive differential equations with non instantaneous impulses at non predefined times we study the existence uniqueness and qualitative properties of different types of solutions in the last section some examples are presented

intermediate statistics in the regime of mixed classical dynamics we explain the arguments in support of the berry robnik 1984 17 2413 picture of the energy level statistics in the asymptotic strict far semiclassical limit of sufficiently small where the entire energy spectrum can be represented as a statistically independent superposition of regular and irregular level sequences the regular ones obeying the poissonian statistics and the irregular ones the rmt statistics goe or gue we generalize the results to describe not only the level spacing distribution the number variance and the delta statistics but also arbitrary statistics the probability to find k levels inside an interval of length l after unfolding very useful and effective approximations for are described we demonstrate very clearly that this regime is excellently described by this picture for k as high as the outer energy scale even when taking into account the regular component and only one the dominant chaotic component this we show numerically for the compactified standard map and for the quartic 2d billiard

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically

intermediate statistics of the quasi energy spectrum and quantum localisation of classical chaos the statistical properties of the quasi energy spectrum in a simple quantum model are investigated for the case when the corresponding classical system is fully chaotic while quantum chaos is restricted by the localisation effects it is shown that the level spacing distribution depends effectively on some parameter which is the ratio of the dimension of the eigenfunctions mean localisation length to the total number of the quasi energy levels numerical data for a wide range of parameters of the system are given