symbolic dynamics and characterization of complexity abstract symbolic dynamics deals with robust properties of dynamics without digging into numbers for one dimensional maps there have been some new results periodic window theorem generalized composition rule construction of median words without using harmonics and antiharmonics incorporation of discrete symmetry to analyze symmetry breaking and symmetry restoration etc this new understanding has been applied to ordinary differential equations in particular the systematics of stable periodic solutions in the lorenz model has been shown to be given basically by symbolic dynamics of the cubic map symbolic dynamics may be used to extract invariant characteristics from time series its relation with grammatical complexity will also be commented on

a circle map in a human heart abstract a circle is divided into two regions a black one and a white one successive iterates of an invertible nonlinear circle map generate a symbolic string indicating whether each iterate is in the black or white region a number of remarkable properties of the symbolic sequences are described these properties were previously described for a linear circle map corresponding to a rigid rotation in the gaps and steps problem these results have direct application to a cardiac arrhythmia parasystole that results from the competition between two pacemakers in the heart one in the sinus mode and the other in the ventricles the theoretical results are directly applicable to a clinical case of a young man who had frequent extra heartbeats

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

chaotic systems counting the number of periods abstract characterization of chaotic motion may proceed both at an averaged macroscopic level using such notions as lyapunov exponents dimensions and entropies and at a microscopic level in the latter case the number of periodic orbits being a topological invariant plays an important role for various one dimensional mappings the counting problem itself has many interesting facets and may be solved more or less completely in different ways recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher dimensional dissipative chaotic systems

the presentation and q deformation of special linear lie superalgebras it is shown that the naive serre presentation corresponding to the simplest cartan matrix of the special linear lie superalgebra sl m n yields for m n 2 a larger lie superalgebra s m n of which sl m n is a nontrivial quotient the missing relations are found and the q deformation of the universal enveloping algebra of sl m n is constructed some auxiliary results related to associative algebras and e lie algebras defined in terms of generators and relations are also included

phase resetting phase locking and bistability in the periodically driven saline oscillator experiment and model the saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water with a small orifice in the center of the bottom of the inner vessel there is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice we develop a very stable i e stationary version of this saline oscillator we first investigate the response of the oscillator to periodic forcing with a train of stimuli text period t mathrm p of large amplitude each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume for t mathrm p sufficiently close to the intrinsic period of the oscillator t 0 there is 1 1 synchronization or phase locking between the stimulus train and the oscillator as t mathrm p is decreased below t 0 one finds the succession of phase locking rhythms 1 1 2 2 2 1 2 2 and 1 1 as t mathrm p is increased beyond t 0 one encounters successively 1 1 1 2 2 4 2 3 2 4 and 1 2 phase locking rhythms we next investigate the phase resetting response in which injection of a single stimulus transiently changes the period of the oscillation by systematically changing the phase of the cycle at which the stimulus is delivered the old phase we construct the new phase old phase curve the phase transition curve from which we then develop a one dimensional finite difference equation map that predicts the response to periodic stimulation these predicted phase locking rhythms are close to the experimental findings in addition iteration of the map predicts the existence of bistability between two different 1 1 rhythms which was then searched for and found experimentally bistability between 1 1 and 2 2 rhythms is also encountered finally with one exception numerical modeling with a phenomenologically derived rayleigh oscillator reproduces all of the experimental behavior

an embedding of the farey web in the parameter space of simple families of circle maps we describe an embedding of the farey web an extension of the better known farey tree in the parameter space of simple families of circle maps we also discuss some consequences of this embedding on the organization of frequency locking and on topological properties of the boundary between simple and complicated dynamics

q oscillator realizations of the quantum superalgebras slq m n andospq m 2n realizations of the quantum superalgebras corresponding to thea m n b m n c n 1 andd m n series are given in terms of the creation and annihilation operators ofq deformed bose and fermi oscillators

systematic template extraction from chaotic attractors i genus one attractors with an inversion symmetry describing the topological properties by a template is a powerful technique to classify chaotic attractors most of the time reduced templates are used but direct templates in which all mechanisms torsions branch permutation etc identified in the attractor are explicitly described without any simplification are of great interest for a better description of the subtleties of the dynamics we introduce here two additional conventions for representing in a unique way the reduced template from a given linking matrix we then propose an addition law for linking matrices which allows us to manipulate combine different mechanisms a direct template can thus be described by using a series of linking matrices on the other hand we show how to analytically build the linking matrix associated with an attractor which is the image under an inversion symmetry of an attractor whose linking matrix is known

splitting words and lorenz braids abstract a canonical way to split lorenz periodic orbits is presented this allows one to recognize the braids on the lorenz template which can occur in non chaotic semi flows on this surface we first present a combinatorial version of our main result and then merely reinterpret it in topological terms

symbolic dynamics of the cubic map abstract a symmetric cubic map on the interval describing isolated stable or bistable orbits and different kind of bifurcation was analyzed stable orbits were presented by itineraries permutation parentheses and matrices itineraries were used to order the cycles with respect to the parameter and also to order the coordinates of the cycle in the interval linear symbolic dynamics was introduced to study the periodic unstable points coexisting with stable orbits by means of a matrix of intervals the non zero entries of these matrices have an n shape corresponding to the n shape of the cubic the characteristic polynomial of the matrix of intervals was in one to one correspondence with the itinerary equations to calculate the number of different kind of orbits were explicitly written and used to calculate those numbers for a small number of periods

quantized enveloping algebras for borcherds superalgebras the rst author was supported in part by nsf grant dms 9300523 the second author was supported in part by the nondirected research fund korea research foundation 1996 the third author was supported in part by a faculty research grant from st lawrence university

a realistic minimal middle layer for neural networks abstract to demonstrate some dynamical consequences of making the formal characteristic of neuronal elements and their connectivities more realistic than those of modern neural computer algorithms we study two multiplicatively coupled logistic maps each with the quadratic response characteristics of neurons with autoreceptor mechanisms which iteratively exchange their outputs the dynamic phenomenology of this two neuron system though noninvertible demonstrates characteristics of both automorphisms of the real line and diffeomorphisms of the plane with smooth parametric transitions between them we portray a unified parameter space containing the negative positive and mixed coupled regimes categorize the phase space structures using characteristic exponent inequalities explore representative parametric paths demonstrate generic bifurcation sequences and report number theoretic ordering in parameter space we conjecture that most if not all middle layer like brain systems in contrast to primary sensory and motor information transport systems are dominated by intrinsic dynamics of the sort demonstrated here external input serves as a perturbation of these already ongoing complex systems the intrinsic instabilities of these middle layer dynamics return the unpredictability required by theoretical studies of real brain function to models of neural networks we demonstrate a variety of parameter dependent intrinsically ordered attractor sequences and number theoretic series which emerge from the global dynamics and exemplify the rich dynamical machine language potential for specific coding by brain like information transport and learning algorithms

genealogy of periodic points of maps of the interval we describe the behavior of families of periodic points in one parameter families of maps of the interval which feature a transition from simple dynamics with finitely many periodic points to chaotic mappings in particular we give topological criteria for the appearance and disappearance of these families our results apply specifically to quadratic maps of the form f a x x l x 0 introduction in recent years much attention has been paid to the dynamical properties of smooth mappings of the real line to itself the logistic function g x ux i x in particular has been studied by many authors as the parameter l for this family is varied through positive values the associated dynamical systems become increasingly more complex for example when 0 4 the dynamical system is much more complicated yet still well understood all points tend to oo with the exception of a subset a of the unit interval which is mapped into itself a is known to be homeomorphic to a cantor set see 1 below and moreover the restriction of g to a is topologically conjugate to a one sided shift automorphism on two symbols in particular there are 2 fixed points for g s so the dynamics here are quite a bit different from those for lower values of p a major question in bifurcation theory is how does one progress from the simple dynamical systems for p b the precise hypotheses are given in 1 for these mappings we show that a particular point arises via a finite sequence of simple bifurcations we catalogue these bifurcations in the sense that we present an algorithm that decides which lower period points give rise via a sequence of bifurcations to a given received by the editors october 9 1979 and in revised form april 18 1980 1980 mathematics subject classification primary 58f14 58f22 secondary 28d99 research partially supported by nsf grant mcs 78 00430 o 1981 american mathematical society 0002 9947 81 0000 0208 03 50

counting the number of periods in one dimensional maps with multiple critical points abstract the problem of counting the number of different periodic orbits in continuous maps of an interval is solved a map with m monotone pieces laps and m 1 critical points may have at most m 1 independent kneading sequences which provide the most convenient parameters for the map when one or more kneading sequences are kept constant or bound to vary simultaneously various degenerated cases of the map arise the number of period n orbits of the general m lap map as various degenerated cases is given by combinations of nink n with k m where nm n is the number of period n orbits of a particular kind of degenerated m lap map with only one kneading sequence the quantity nm n may be calculated or enumerated by many different methods which are all discussed in this paper we also give tabulated values for nm n for m 2 to 7

a nine dimensional lorenz system to study high dimensional chaos we examine the dynamics of three dimensional cells with square planform in dissipative rayleigh benard convection by applying a triple fourier series ansatz up to second order we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations depending on two control parameters namely the rayleigh number and the prandtl number the asymptotic behaviour can be stationary periodic quasiperiodic or chaotic a period doubling cascade is identified as a route to chaos hereafter the asymptotic behaviour progressively evolves towards a hyperchaotic attractor for given values of control parameters beyond the accumulation point we observe a low dimensional chaotic attractor as is currently done for dissipative systems although the correlation dimension strongly suggests that this attractor could be embedded in a three dimensional space a topological characterization reveals that a higher dimensional space must be used thus we reconstruct a four dimensional model which is found to be in agreement with the properties of the original dynamics the nine dimensional lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3

decision feedback differential detection based on linear prediction for 16dapsk signals transmitted over flat ricean fading channels in this article prediction based decision feedback differential detection df dd for 16 level differentially encoded amplitude phase shift keying is proposed unlike previously reported df dd schemes this scheme provides a performance gain over conventional differential detection under general ricean fading conditions a further important advantage of the novel scheme is that it is able to compensate a small carrier frequency offset the linear predictor coefficients may be updated using the recursive least squares algorithm which can start blind i e without a priori knowledge about the channel statistics and without a training sequence this makes the scheme attractive for application in mobile communications since the statistics of a nonstationary mobile channel can be tracked

m ary phase coding for the noncoherent awgn channel this work considers coded m ary phase shift keying mpsk schemes with noncoherent detection a class of block codes called module phase codes is described the algebraic framework used for describing these codes relies on elements from module theory which are discussed along with a method for constructing such codes for noncoherent detection it is shown that differential encoding may be viewed as a specific code from a particular class of module phase codes two classes of codes that achieve significant coding gain with respect to coherent detection of uncoded mpsk are presented in the first class of module phase codes the coding gain is achieved at the expense of bandwidth expansion in the second class however the coding gain is achieved at the expense of signal constellation expansion without expanding bandwidth finally an integrated demodulation decoding technique based on a modification of information set decoding is presented it is shown that this reduced complexity suboptimal decoding strategy performs nearly as well as maximum likelihood decoding

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

some flesh on the skeleton the bifurcation structure of bimodal maps abstract one of the main tools in the numerical study of two parameter families of one dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits we use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points then we explain how the bifurcation structure hangs on this skeleton

attractors and symmetries of vector fields the inverse problem abstract the inverse problem of constructing dynamical systems x with a prescribed compact attracting set k r n and non trivial dynamics on it is solved the boundaries of the attracting basins corresponding to different connected components of k and the relationship between the symmetries of k x and are also studied

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

noncoherent block coded tapsk for coherent detection block coded modulation is a bandwidth efficient scheme in this paper we propose theorems about the error performance of block coded modulation using m ary phase shift keying mpsk for noncoherent detection based on these theorems we propose a novel block coded modulation scheme for noncoherent detection called noncoherent block coded mpsk the proposed scheme provides flexible designs of noncoherent block codes with different code rate block length and error performance good noncoherent block codes can be easily obtained by properly choosing binary linear block codes as the component codes moreover noncoherent block codes of this new scheme can be decoded by multistage decoding which has the advantage of low complexity and satisfactory error performance in this paper two algorithms of multistage decoding for noncoherent detection are proposed as well the error performance of some designed codes and decoding algorithms is verified by computer simulation

parametric adaptive control and parameter identification of low dimensional chaotic systems abstract the presented method of adaptive control corrects perturbations in the parameters of a system based on observations of its variables after a transient during which the desired parameter setting is reached exponentially quickly the system exhibits the goal behavior which can be a periodic quasiperiodic or chaotic solution of the equations of motion the length of the control transient scales inversely with the control strength and logarithmically with the magnitude of the original parameter perturbation it is shown that time series based models are capable of substituting for detailed knowledge of the system s nonlinear dynamics the method can be generalized to control more than one parameters to rely on observations of a single variable or to utilize a variety of control functions finally a method of parameter identification dual to the adaptive control is also discussed

unitary space time modulation for multiple antenna communications in rayleigh flat fading motivated by information theoretic considerations we propose a signaling scheme unitary space time modulation for multiple antenna communication links this modulation is ideally suited for rayleigh fast fading environments since it does not require the receiver to know or learn the propagation coefficients unitary space time modulation uses constellations of t spl times m space time signals spl phi sub i l 1 l where t represents the coherence interval during which the fading is approximately constant and m t is the number of transmitter antennas the columns of each spl phi sub i are orthonormal when the receiver does not know the propagation coefficients which between pairs of transmitter and receiver antennas are modeled as statistically independent this modulation performs very well either when the signal to noise ratio snr is high or when t spl gt m we design some multiple antenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding we demonstrate that two antennas have a 6 db diversity gain over one antenna at 15 db snr

quasiperiodic resonance and bifurcations of tori in the weakly nonlinear duffing oscillator abstract we study analytically and numerically the weakly nonlinear quasiconservative duffing oscillator under 2 periodic forcing utilizing the secondary averaging approach we derive approximate analytical expressions for the quasiperiodic response of the oscillator and for conditions of torus bifurcations the theoretical results are in good agreement with the numerical ones the structure of the bifurcation sets in the oscillator parameter space is similar to that of a one dimensional map with two extrema

dissecting shrimps results for some one dimensional physical models this paper describes how certain shrimp like clusters of stability organize themselves in the parameter space of dynamical systems clusters are composed of an infinite affine similar repetition of a basic elementary cell containing two primary noble points a head and a tail defining an axis of approximate symmetry knowledge of the axis and the skewness of the k periodic main cell of a k 2 cluster is enough to define the orientation of the whole cluster in space peculiarly simple directions along which shrimp like clusters align are formed by the locus of doubly degenerate saddle zero multipliers corresponding to the main shrimp head in addition we report a family of models having the boundaries of all isoperiodic domains of stability totally degenerate and describe different aspects of their mathematical arrangement and some of their consequences for example that shrimps are diffeomorphic copies of shrimps

topological analysis and synthesis of chaotic time series abstract we have developed a topological procedure for analyzing chaotic time series which identifies the stretching and squeezing mechanisms responsible for chaotic behavior in low dimensional dynamical systems these mechanisms quantitatively described by a template or knot holder can then be used to model the processes which generate the original chaotic data set

a genealogy for finite kneading sequences of bimodal maps on the interval we generate all the finite kneading sequences of one of the two kinds of bimodal map on the interval building each sequence uniquely from a pair of shorter ones there is a single pair at generation 0 with members of length 1 concomitant with this genealogy of kneading sequences is a unified genealogy of all the periodic orbits see figure 0

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

complete devil s staircase fractal dimension and universality of mode locking structure in the circle map it is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil s staircase at the onset of chaos the complementary set to the stability intervals is a cantor set of fractal dimension d 0 87 this exponent is found to be universal for a large class of functions

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

tori and klein bottles in four dimensional chaotic flows abstract we have analyzed the local structure of the flow in the neighborhood of the unstable periodic orbits within the chaotic solution of a four dimensional vector field surfaces within the invariant manifolds of the orbits are found and characterized they are either klein bottles or tori

symbolic dynamics and skeletons of circle maps a circle map maps the circumference of a circle into itself such maps are of physical and biological interest because they arise naturally in many circumstances as for example the periodic forcing of limit cycle oscillators the canonical example of circle maps is x tx a sin2 x mod 1 where a and are two constants t is an integer that gives the topological degree of cycles that contain extremal points of are stable and are called superstable cycles the locus of superstable cycles in a parameter space is called the skeleton the critical points of x 0 and the discontinuous points of x 1 break into monotonic branches in this paper we propose a novel notation for symbolically representing the branches of x and discuss the symbolic sequences for the skeleton simple rules based on continuity of periodic orbits as parameters change allow us to construct the skeleton associated with all periodic orbits for the circle maps with t 0 0 a 1 0 and t 1 0 a 1 0 recursion formulae to compute the numbers of periodic orbits for t 0 and t 1 at 0 are given

on the topology of the lu attractor and related systems we use well established methods of knot theory to study the topological structure of the set of periodic orbits of the l u attractor we show that for a specific set of parameters the l u attractor is topologically different from the classical lorenz attractor whose dynamics is formed by a double cover of the simple horseshoe this argues against the similarity between the l and lorenz attractors claimed for these parameter values by some authors on the basis of non topological observations however we show that the l system belongs to the lorenz like family since by changing the values of the parameters the behaviour of the system follows the behaviour of all members

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

generalized redundancies for time series analysis abstract extensions to various information theoretic quantities such as entropy redundancy and mutual information are discussed in the context of their role in nonlinear time series analysis we also discuss linearized versions of these quantities and their use as benchmarks in tests for nonlinearity many of these quantities can be expressed in terms of the generalized correlation integral and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics such as the bds and green savit statistics further numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation finally we consider several local versions of these quantities including a local kolmogorov sinai entropy which gives an estimate of variability of the short term predictability

differential amplitude phase modulation for correlated rayleigh fading channels performance analysis and labeling design in this paper we investigate differential amplitude phase modulation apm schemes for correlated rayleigh fading channels considering the noncoherent maximum likelihood receiver a theoretical pairwise error probability pep formula for differential signal vectors with arbitrary power values is derived the design criterion for the signal set of differential vectors is then proposed based on the asymptotic analysis of the pep for both uncoded and coded apm schemes the design criteria and optimization procedures for bit labeling rules are investigated the designed differential apm schemes are shown to provide significant improvement in bit error rate performance particularly in the error floor regions

noncoherent amplitude phase modulated transmission schemes for rayleigh block fading channels in this paper we investigate noncoherent transmission schemes using amplitude phase modulation for uncoded transmission schemes we derive the pairwise error probability pep and its high snr approximation for signal vectors with unequal power values using a noncoherent maximum likelihood detector under rayleigh block fading channels for coded transmission schemes we consider differential amplitude phase modulated signals and introduce a codeword interleaving strategy to improve the error performance the design criteria for the signal labelling and bit interleaving are also derived via the high snr approximation of the pep at the receiver we devise an iterative channel amplitude estimator to enhance the detection reliability with the help of a specially designed iterative receiving algorithm the designed noncoherent coded scheme can provide a significant improvement either in the error floor or the waterfall region

symbolic dynamics and systematics of periodic windows abstract elementary symbolic dynamics of two or more letters developed for one dimensional mappings may be of much help in studying ode s at least three periodically driven systems including one studied in great detail by us are known to be described in the large by symbolic dynamics of two letters the skeleton of periodic windows in the lorenz model is described by symbolic dynamics of three letters in close connection with the one dimensional antisymmetric cubic map the understanding of transitions from quasiperiodicity to chaos may also be deepened looking at the global picture of the interrelation between the u sequences and the farey sequences

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

spectrally efficient noncoherent communication this paper considers noncoherent communication over a frequency nonselective channel in which the time varying channel gain is unknown a priori but is approximately constant over a coherence interval unless the coherence interval is large coherent communication which requires explicit channel estimation and tracking prior to detection incurs training overhead which may be excessive especially for multiple antenna communication in contrast noncoherent detection may be viewed as a generalized likelihood ratio test glrt which jointly estimates the channel and the data and hence does not require separate training the main results in this paper are as follows 1 we develop a signal space criterion for signal and code design for noncoherent communication in terms of the distances of signal points from the decision boundaries 2 the noncoherent metric thus obtained is used to guide the design of signals for noncoherent communication that are based on amplitude phase constellations these are significantly more efficient than conventional differential phase shift keying psk especially at high signal to noise ratio snr also known results on the high snr performance of multiple symbol demodulation of differential psk are easily inferred from the noncoherent metric 3 the glrt interpretation is used to obtain near optimal low complexity implementations of noncoherent block demodulation in particular this gives an implementation of multiple symbol demodulation of differential psk which is of linear complexity in the block length and whose degradation from the exact exponential complexity implementation can be made as small as desired

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

decision feedback differential detection of mdpsk for flat rayleigh fading channels in this paper decision feedback differential detection df dd of m ary differential phase shift keying mdpsk signals which has been introduced previously for the additive white gaussian noise awgn channel by leib et al 1988 and edbauer 1992 is extended to flat rayleigh fading channels the corresponding df dd metric is derived from the multiple symbol detection msd metric and for genie aided df dd an exact expression for the bit error rate ber of qdpsk m 4 is calculated furthermore the dependence of ber on the power spectrum of the fading process is investigated for feedback filters of infinite order it is shown that in this case for ideally bandlimited fading processes the error floor of conventional differential detection dd can be removed entirely simulation results confirm that both msd and df dd with feedback filters of finite order can reduce the error floor of conventional dd significantly df dd thereby causes considerably less computational load

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

on the llr metrics for dpsk modulations over two symbol observation intervals for the flat rician fading channel we consider the differential phase shift keying modulation over the flat rician fading channel with perfect channel model information pcmi and derive the two symbol observation interval log likelihood ratio llr metric from first principles the work generalizes the previous llr derivations in 16 19 and demonstrates the necessity of the knowledge of the statistical channel model in the exact llr computation assuming the channel is slowly time varying we also propose a simple approximate llr metric based on the generalized likelihood ratio test glrt which requires only the information of the noise spectral density the advantage of the newly derived pcmi llr metric over the approximate metric and the other llr metrics in the literature is explained from the information theoretic perspective via the generalized mutual information computer simulations are also provided to demonstrate the superiority of the pcmi llr metric in both error rate performance and convergence speed for iterative decoding of turbo and low density parity check codes in practical systems this paper aims to present the fundamental principles of llr computation emphasizing the importance of selecting the appropriate metric for iterative decoding over fading channels

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

bifurcation and chaos in a periodically stimulated cardiac oscillator abstract periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking period doubling bifurcations and aperiodic chaotic dynamics at different values of the stimulation parameters this behavior is analyzed by considering an experimentally determined one dimensional poincare or first return map a simplified version of the experimentally determined poincare map is proposed and several features of the bifurcations of this map are described

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

deformations of enveloping algebra of lie superalgebra sl m n theq deformations of the universal enveloping algebra ofsl m n are considered a poincare birkhoff witt type theorem is proved for these deformations and the extra relations which are needed to definesl m n as a contragredient algebra in addition to the serre type relations are identified with proof

bifurcation analysis of a periodically forced relaxation oscillator differential model versus phase resetting map we compare the dynamics of the periodically forced fitzhugh nagumo oscillator in its relaxation regime to that of a one dimensional discrete map of the circle derived from the phase resetting response of this oscillator the phase resetting map the forcing is a periodic train of gaussian shaped pulses with the width of the pulses much shorter than the intrinsic period of the oscillator using numerical continuation techniques we compute bifurcation diagrams for the periodic solutions of the full differential equations with the stimulation period being the bifurcation parameter the period 1 solutions which belong either to isolated loops or to an everywhere unstable branch in the bifurcation diagram at sufficiently small stimulation amplitudes merge together to form a single branch at larger stimulation amplitudes as a consequence of the fast slow nature of the oscillator this merging occurs at virtually the same stimulation amplitude for all the period 1 loops again using continuation we show that this stimulation amplitude corresponds in the circle map to a change of topological degree from one to zero we explain the origin of this coincidence and also discuss the translational symmetry properties of the bifurcation diagram

a genealogy for the periodic orbits of a class of 1d maps abstract we describe a genealogy for the kneading sequences of one of the two kinds of bimodal map of the interval it provides a unified genealogy for all the periodic orbits and a complete road map for the periodic dynamics in the parameter plane

construction of phenomenological models from numerical scalar time series abstract the problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed techniques are exemplified for rossler and lorenz chaotic attractors with emphasis on improved algorithms with respect to those previously published the steps which are still required in order to investigate experimental noisy data are discussed

problems in estimating dynamics from data abstract this paper discusses some difficulties in estimating dynamics from time delay embeddings of experimental data that can be characterized as low dimensional a new procedure is described to reduce noise by exploiting the properties of saddle periodic orbits on the reconstructed attractor

nonlinear dynamics chaos and complex cardiac arrhythmias periodic stimulation of a nonlinear cardiac oscillator in vitro gives rise to complex dynamics that is well described by one dimensional finite difference equations as stimulation parameters are varied a large number of different phase locked and chaotic rhythms is observed similar rhythms can be observed in the intact human heart when there is interaction between two pacemaker sites simplified models are analysed which show some correspondence to clinical observations the normal adult human heart at rest usually beats at a rate of between 50 and 100 times per minute in many circumstances some of which are lifethreatening but most of which are not the normal rhythmicity is altered resulting in abnormal rhythms called cardiac arrhythmias the point of this paper is to show that a branch of mathematics called nonlinear dynamics may be useful in the analysis of physiological processes believed to underlie normal heart rate regulation and some cardiac arrhythmias the idea that mathematical analysis can play a role in understanding cardiac arrhythmias is not novel indeed in the 1920s it was demonstrated that as parameters in mathematical models for the heart were varied several different rhythms that resembled clinically observed arrhythmias could be generated mobitz 1924 van de pol van der mark 1928 in nonlinear mathematics these changes in the qualitative features of the rhythms that are observed as parameters vary are called bifurcations thus the problem of understanding cardiac arrhythmias in the human heart is identified with understanding the bifurcations and complex dynamics in mathematical models of the human heart one type of dynamic behaviour that is the object of intensive analysis in mathematics is chaos loosely chaos is defined as aperiodic dynamics in deterministic systems in which there is sensitive dependence to the initial conditions this means that although in principle one could determine precisely the future evolution of the system starting from some initial condition for chaotic dynamics any difference in the initial condition no matter how small will eventually lead to marked differences in the future evolution of the system although the existence of chaos was known to poincare and others since the end of the last century in the

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

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

coded unitary space time modulation with iterative decoding error performance and mapping design this paper studies the bit error probability of coded unitary space time modulation with iterative decoding where neither the transmitter nor the receiver knows the channel fading coefficients the tight error bound with respect to the asymptotic performance is first analytically derived for any given unitary constellation and mapping rule design criteria regarding the choice of unitary constellation and mapping are then established furthermore using the unitary constellation obtained from orthogonal design with quadrature phase shift keying qpsk or 4 psk and 8 psk two different mapping rules are proposed the first mapping rule gives the most suitable mapping for systems that do not implement iterative processing which is similar to a gray mapping in coherent channels the second mapping rule yields the best mapping for systems with iterative decoding in particular analytical and simulation results show that with the proposed mappings of the unitary constellations obtained from orthogonal designs the asymptotic error performance of the iterative systems can closely approach a lower bound which is applicable to any unitary constellation and mapping

further results on differential encoding by a table recently we proposed a novel differential encoder by a look up table in this paper we present further research results on this subject we first propose a bit assigning algorithm and construct some new tables for various constellations that maximize the minimum noncoherent distance then differential encoding for multiple symbol differential detection is proposed after that we indicate that differential encoding by a table is equivalent to differential encoding by a trellis so we propose a novel differential detector which uses the viterbi algorithm on the trellis a theorem about the sufficient number of states for various modulations is proposed in addition we further enhance the error performance of the proposed differential encoder by concatenating it with trellis coding for this noncoherent trellis coding scheme we propose augmented state viterbi decoding finally we extend the proposed differential encoding and detection to differential space time modulation dstm we propose a new definition of dstm by which the set of transmitted blocks is determined first the differential encoding is implemented by a look up table which can be optimized the proposed dstm has satisfactory error performance without constellation expansion

fractal bifurcation sets renormalization strange sets and their universal invariants the transition structure of the most common routes to chaos are organized by fractal bifurcation sets examples include the quasiperiodic transitions to chaos and the period doubling structure found in arnol d tongues in this paper i discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations an important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic this implies that they have the same fractal structure and in particular the same hausdorff dimension and scaling spectra some other invariants are introduced

star transformations and their genealogical varieties in symbolic dynamics of four letters the composition rules of period tripling and period doubling in symbolic dynamics of four letters are proposed the methods to generate various singly and doubly superstable kneading sequences from triply superstable kneading sequences are given these composition rules and methods will present all the essential elements in the three dimensional parameter space including the joint bones membrane surfaces on the skeleton and many fractal objects constructed by period doubling and period tripling bifurcations it is shown that the generalization from unimodal and bimodal to trimodal maps is not trivial and the promptly exponential growth in the kinds of star transformations is even seen when the number of critical points or parameters of the map increases

maximum likelihood differential detection of uncoded and trellis coded amplitude phase modulation over awgn and fading channels spl minus metrics and performance this paper derives metrics for maximum likelihood differential detection of uncoded and trellis coded mpsk and qam transmitted over rayleigh and rician fading channels receiver structures based on these metrics are proposed and their error probability performance analyzed and or simulated the results represent a generalization of the notion of multiple symbol differential detection previously introduced by the authors for mpsk over an awgn to the fading channel and other modulations for the coded cases ideal interleaving deinterleaving is assumed and furthermore the presence or absence of channel state information an interesting side result is that for a constant envelope modulation transmitted over a fading channel with unknown but rapidly varying phase error the other extreme to the slowly varying phase error case normally assumed for differential detection under certain practical assumptions it is shown that the optimum receiver is of the limiter discriminator type

differential encoding by a look up table for quadrature amplitude modulation in this paper we propose a novel differential encoder for qam quadrature amplitude modulation by a look up table instead of a rule using a table for differential encoding has the advantages of both low complexity and good error performance we propose algorithms to construct the table for maximizing the minimum noncoherent distance and minimizing the number of different data bits between two codewords with small noncoherent distance for 16qam and 32qam we show that the differential encoders obtained by the proposed algorithms are optimal in terms of minimum noncoherent distance differential detectors for qam including a reduced complexity two symbol detector and multiple symbol detectors are proposed as well simulation results of 16qam show that the proposed differential encoder provides significant gain over existent differential encoders as expected by distance analysis and the proposed low complexity multiple symbol differential detector can further improve the error performance

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