a limit theorem in dynamic fuzzy systems with a monotone property abstract a dynamic fuzzy system defined on a compact state space is considered under a monotone property of the fuzzy relation we study the limit of a sequence of fuzzy states and obtain a convergence theorem the limit is characterized as the solution of a fuzzy relational equation a numerical example is given to comprehend our idea in this paper

cyclic classes and an ergodic theorem in dynamic fuzzy systems this paper analyses cyclic behavior of dynamic fuzzy systems defined by fuzzy relations on a metric space for the system we give an ergodic theorem with a strata structure

sharkovskii s order for the appearance of superstable cycles in one parameter families of simple real maps an elementary proof we give an elementary proof of a theorem a la sharkovskii which orders according to the period the first occurrence of superstable cycles in some one parameter families of endomorphisms of the real line

functional equations for circle homeomorphisms with golden ratio rotation number the investigation of a scaling limit for mappings of the circle to itself with golden ratio rotation number leads to a pair of functional equations with at least a formal resemblance to the functional equation using the accumulation of period doubling bifurcations we discuss the general theory of these functional equations assuming that solutions exist

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

entrainment control in a noisy neural system the open plus closed loop opcl entrainment control put forth by jackson and grosu physica d 85 1995 is applied to an effective neuron system as a way to extract stable limit cycles from a chaotic attractor analogous to the retrieval of memories from a memory searching state additive gaussian white noise representing the natural noise inherent in any real dynamical system is added to the entrainment control mechanism moderate levels of additive noise have little effect on successful entrainment as reflected in phase space plots and lyapunov exponents all three lyapunov exponents are negative which suggests parallels between opcl control and chaotic synchronization al 2

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

adaptive recognition and control of chaos abstract exploiting the information provided by the local variation rates of a dynamical system we show how an adaptive algorithm is able to recognize and stabilize the unstable periodic orbits embedded in a chaotic attractor through additive corrections to the dynamics the relative weight of each correction is adjusted upon the size of the local expansion rate application of this method to chaotic lorenz and four dimensional roessler systems is discussed

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a continuous time dynamic fuzzy system i a limit theorem abstract a continuous time dynamic fuzzy system with possibility transition is constructed and a differential of fuzzy states is given by paths a limit theorem for fuzzy states is discussed under a contractive condition on fuzzy relations

analysis of joint angle frequency estimation using esprit high resolution parameter estimation techniques have been applied to jointly estimate multiple signal parameters we consider the problem of determining the directions and center frequencies of a number of narrowband sources in a band of interest we present a joint angle frequency estimation method based on the multidimensional esprit algorithm a perturbation error analysis gives bounds on the parameter estimates and provides optimal values for the temporal and spatial smoothing parameters the analysis is shown to be consistent with simulation results

conserved quantities and adaptation to the edge of chaos certain dynamical systems such as the shift map and the logistic map have an edge of chaos in their parameter spaces on one side of this edge the dynamics are chaotic for many parameter values on the other side of the edge they are periodic we find that discrete time dynamical systems with wavelet filtered feedback from the dynamical variable to the parameters are attracted to a narrow parameter range near the edge of chaos the periodic boundary regime we show that the migration from the chaotic regime to the periodic boundary regime can be attributed to a conserved quantity and find that such adaptation to the edge of chaos is accompanied by a depopulation of the chaotic regime we use this conserved quantity to determine the location of the periodic boundary regime and show that its size is proportional to the size of the feedback further we compute the dynamics of the probability density for the parameter for a specific example

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

a numerical study of the likelihood of phase locking abstract we report the results of numerical computation of the size of the set of s for which the hemeomorphism of the circle f k x x k 2 sin 2 x mod 1 is phased locked for values of k ranging from 0 1 to 1 the results suggest that for moderate values of k phase locking is unlikely except for small periods but that for k 1 almost all s give phase locking with large periods accounting for a surprisingly large portion of the total

the set of maps f_ ɑ b x x ɑ b 2π sin 2πx with any given rotation interval is contractible consider the two parameter family of real analytic maps f b x x b 2 sin 2 x which are lifts of degree one endomorphisms of the circle the purpose of this paper is to provide a proof that for any closed interval i the set of maps f b whose rotation interval is i form a contractible set

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

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

learning to control dynamical behaviour abstract control of dynamical behaviour requires the transformation of one dynamical sequence into another the ability of a neural network to perform this task depends on the complexity of a time series involved and on the connectivity of the network used scaling estimates of the information extracted during learning and the resulting control error achieved are compared with simulations of period doubling maps

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

periodic entrainment of chaotic logistic map dynamics consider the map dynamics xn 1 f xn c with a control parameter c let the governing set gn n 0 1 2 be a desired periodic dynamic set gn n gn it is noted that the non autonomous system xn 1 f xn c gn has such a solution xn gn if gn gn 1 f gn c in particular the set of values gn38 might be obtained from the periodic solutions of gn 1 f gn c using suitable values of c this study explores the values of c which yield entrained solutions their basin of entrainment x0 limn xn gn 0 and more generally the basins of bounded solutio ns and their character when f x c cx 1 x the logistic map of particular interest is the entrainment of chaotic dynamics c 4 generally the basin of entrainment for period one governing g0 is x0 1 g0 1 c g0 1 c 2c with the above b asin or if c 1 2c g0 c 4 2c with the basin x 0 g0 x0 1 g0 the complimentary x0 region yields unbounded dynamics period two governing values have two basins depending on the initial value used for g0 these basins are investigated for the sets gn 1 f gn c other periodic sets gn in gn do not become disjoint i n space period four entrainment for suitable c can have even larger basins the unwanted occurrence of other disjoint basins of attraction which interlace the basins of entrainment as a function of both x0 and g0 is discussed other periodic and non periodic responses to these periodic gn are also studied

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

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

dynamics control of chaotic systems by parametric destochastization it is shown that deterministic noise chaos appearing via destruction of the quasiperiodic motion may be easily suppressed by weak parametric perturbation of the system

dynamics of chaos order interface in coupled map lattices abstract we study a coupled map lattice model with two states a simple fixed point and spatio temporal chaos preparing properly initial conditions we investigate the dynamics of the interface between order and chaos in the one dimensional lattice regimes of irregular and regular front propagation behavior are observed and analyzed by introducing a local front map and a front lyapunov exponent corresponding to these different regimes of front propagation we can characterize different types of transitions from laminar state to chaos using comoving lyapunov exponents in the two dimensional lattice these types of front motion are related to regimes of roughening and flattening of the interface

entrainment and migration controls of two dimensional maps a general approach to several types of controls of complex dynamic systems formulated by jackson is applied to two classic two dimensional maps xk 1 e xk x r2 the henon and the ikeda maps the convergent regions of their phase spaces ck are shown to respectively possess translational and rotatational symmetries several topologically distinct entrainment goals e goals gk ck are illustrated for the henon map for both maps the basins of entrainment be g0 x0 limk xk g0 0 of the controlled dynamics for fixed point goals in each convergent region g0 ck are illustrated the be g0 are invariant relative to g0 when g0 is varied by the above symmetries it is proved that for the ikeda map be g0 0 r2 for experimental purposes spheres of entrainment are introduced se r0 x0 x0 g0 r g0 where r g0 maxr r x0 g0 r limk xk g0 0 the ikeda map has two attractors in a bounded region so that migration controls can be illustrated this is done using several migration goals m goals which transfer this system from one attractor to the basin of attraction of the other attractor a1 ba2 or a2 ba1 these require controls only for finite times the influence of noise on the entrainment process is analyzed and compared with numerical examples

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

a limit theorem in some dynamic fuzzy systems abstract in this paper using a fuzzy relation we define a dynamic fuzzy system with a compact state space by introducing a contraction property we study the limit of a sequence of fuzzy states and obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation as an application we deal with a dynamic fuzzy system with a terminal fuzzy gain and give some characterizations for the fuzzy expected gain a numerical example is given to comprehend our idea in this paper

applications of cumulants to array processing i aperture extension and array calibration an interpretation for the use of cumulants in narrowband array processing problems is proposed it is shown how fourth order cumulants of multichannel observations increase the directional information compared with second order statistics based on the interpretation it is shown how cumulants can be used to increase the effective aperture of an arbitrary antenna array the amount of partial information necessary to jointly calibrate an arbitrary array and estimate the directions of far field sources is also investigated it is proven that the presence of a doublet and use of fourth order cumulants is sufficient to accomplish this task the proposed approach is computationally efficient and more general than covariance based algorithms that have addressed the calibration problem under constraints a class of beamforming techniques is proposed to recover the source waveforms proposed estimation procedures are based on cumulants which bring insensitivity to the spatial correlation structure of additive gaussian measurement noise simulations are provided to illustrate the use of the proposed algorithms

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

a potential of fuzzy relations with a linear structure the unbounded case this paper is a sequel to yoshida et al 1993 in which the potential theory for linear fuzzy relations on the positive orthant r n is considered in the class of fuzzy sets with a compact support under the contractive assumption in this paper potential treatment for unbounded fuzzy sets is developed without the assumption of contraction and compactness the objective of this paper is to give the existence and the characterization of potentials for linear fuzzy relations under some reasonable conditions also introducing a partial order in fuzzy sets we prove riesz decomposition theorem in the fuzzy potential theory the proofs are shown by using only the linear structure and the monotonicity of fuzzy relations in the one dimensional case the potential and its cuts are explicitly calculated numerical examples are given to comprehend further discussions

high order synchronization transitions and competition among arnold tongues in a rotator under harmonic forcing we consider a rotator whose equation of motion for the angle theta consists of the zeroth and first fourier modes numerical analysis based on the trailing of saddle node bifurcations is used to locate the n 1 arnold tongues where synchronization occurs several of them are wide enough for high order synchronization to be seen in passive observations by sweeping the system parameters within a certain range we find that the stronger the dependence of theta over on theta the wider the regions of synchronization use of a synchronization index reveals a vast number of very narrow n m arnold tongues a competition phenomenon among the tongues is observed in that they push and squeeze one another as some tongues widen others narrow two mechanisms for transitions between different n m synchronization states are considered slow variation of the driving frequency and the influence of low frequency noise on the rotator

nonlinear analysis of data sampled nonuniformly in time abstract we demonstrate how some of thee common methods in nonlinear data analysis can be modified to account for nonuniform time sampling by using fuzzy delay coordinate reconstructions the effects of measurement noise and time sampling errors are compared we have also developed a learning algorithm to find optimal fuzzy delay coordinate reconstructions of the data to suit one s goals this can be employed to find representations optimized for measuring topological invariants forecasting control or other goals

nonlinear transformation on the transfer entropy of financial time series abstract transfer entropy te now is widely used in the data mining and economic field however te itself demands that time series intend to be stationary and meet markov condition naturally we are interested in investigating the effect of the nonlinear transformation of the two series on the te therefore the paper is designed to study the te of five nonlinear volatile transformations based on the data which are generated by the linear modeling and the logistic maps modeling as well as the dataset that come from financial markets with only one of the te of nonlinear transformations fluctuating around the te of original series the te of others all have increased with different degrees

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

a potential of fuzzy relations with a linear structure the contractive case abstract in this paper we develop a potential theory of fuzzy relations on the positive orthant in a euclidean space by introducing a linear structure for fuzzy relations the existence of a potential and its characterization by fuzzy relational equation are derived under the assumption of contraction and compactness in the one dimensional unimodal case a potential is given explicity also a numerical example is shown to illustrate our approaches

rhythm synchronization and chaotic modulation of coupled van der pol oscillators in a model for the heartbeat we investigate a phenomenological model for the heartbeat consisting of two coupled van der pol oscillators the coupling between these nodes can be both unidirectional or bidirectional and an external driving produced by a pacemaker is also included in this model in order to warrant a robust operation it is desirable that both units oscillate in a synchronized way even though in the presence of external in0uences or parameter mismatches which are unavoidable in a physiological setting we study the synchronization properties of such an association with respect to the nature andintensity of coupling we analyze in particular the generalized synchronization of rhythms characterizedby a chaotic mod ulation of the oscillator frequencies we also investigate the shadowing breakdown of numerically generated chaotic trajectories of the coupled oscillator system via unstable dimension variability in its chaotic invariant set c

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

a fuzzy relational equation in dynamic fuzzy systems for a dynamic fuzzy system the fundamental method is to analyze its recursive relation of the fuzzy states it is similar in that the bellman equation is an important tool in the dynamic programming here we will consider the existence and the uniqueness of solution of a fuzzy relational equation two examples which satisfies our conditions are given to illustrate the results

computing arnol d tongue scenarios a famous phenomenon in circle maps and synchronisation problems leads to a two parameter bifurcation diagram commonly referred to as the arnol d tongue scenario one considers a perturbation of a rigid rotation of a circle or a system of coupled oscillators in both cases we have two natural parameters the coupling strength and a detuning parameter that controls the rotation number frequency ratio the typical parameter plane of such systems has arnol d tongues with their tips on the decoupling line opening up into the region where coupling is enabled and in between these arnol d tongues quasi periodic arcs in this paper we present unified algorithms for computing both arnol d tongues and quasi periodic arcs for both maps and odes the algorithms generalise and improve on the standard methods for computing these objects we illustrate our methods by numerically investigating the arnol d tongue scenario for representative examples including the well known arnol d circle map family a periodically forced oscillator caricature and a system of coupled van der pol oscillators

the w sequence for circle maps and misbehaved itineraries the w sequence an ordering of maximal itineraries for circle maps is constructed by means of the farey transformation and the gluing operation the similarity in the dynamical behaviour of the maps is explicitly described in terms of itineraries a class of badly ordered itineraries termed misbehaved is defined with the language of symbolic sequences a misbehaved itinerary is shown to imply a rotation interval

explicit renormalisation in piecewise linear bimodal maps abstract we discuss 2 two parameter families of piecewise linear bimodal two hump maps the first example arises naturally in the study of bimodal maps any bimodal map is semi conjugate to a member of this family and topological entropy is preserved the second example is a symmetric subcase of a map derived from a simple pulse modulated feedback system in each case we define an explicit map of parameter space which governs the effect of renormalisation and use it to describe the non wandering set

chaotic firing in the sinusoidally forced leaky integrate and fire model with threshold fatigue the leaky integrate and fire lif model is one of the elementary neuronal models that has been widely used to gain understanding of the behavior of many excitable systems the sinusoidally forced standard leaky integrate and fire model reproduces the quasiperiodic and phase locked discharge trains observed experimentally in neurons however this basic model fails to generate chaotic firing whereas this form of behavior has been observed experimentally we modify the standard lif through the introduction of threshold fatigue responsible for progressive decrease of excitability during high frequency firing as observed experimentally we show that the dynamics of this neuron model under sinusoidal forcing are governed by the iterates of an annulus map and derive expressions for its two characteristic lyapunov exponents using these exponents it is shown that chaotic dynamics are possible for this model unlike the standard leaky integrate and fire model chaotic dynamics occur when memory effects are strong and only under certain forms of threshold fatigue

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

suppression of chaos by cyclic parametric excitation in two dimensional maps we study the qualitative change of the dynamics of a generalized two dimensional quadratic map under the influence of parametric perturbations which operate in the chaotic parameter set it is shown that such perturbations can lead to the suppression of chaos and appearance of a regular periodic behaviour numerically we can argue that the suppression of chaos due to the parametric excitation is caused by a shift of the windows of periodic behaviour in the bifurcation diagram

a conjugate augmented approach to direction of arrival estimation in this paper we propose a new direction of arrival doa estimator called conjugate augmented music cam the basic idea of cam is to use the second order statistics of the received signals to get the conjugate steering matrix this together with the steering matrix is used to find the fourth order cumulants from that the source directions are obtained using the music like algorithm cam can resolve two times the number of directions when compared to music like estimator moreover simulation results show that the estimation capacity angle resolution immunity to noise and the number of required snapshots are all better than music like algorithm

direction finding algorithms based on high order statistics two direction finding algorithms are presented for nongaussian signals which are based on the fourth order cumulants of the data received by the array the first algorithm is similar to music while the second is asymptotically minimum variance in a certain sense the first algorithm requires singular value decomposition of the cumulant matrix while the second is based on nonlinear minimization of a certain cost function the performance of the minimum variance algorithm can be assessed by analytical means at least for the case of discrete probability distributions of the source signals and spatially uncorrelated gaussian noise the numerical experiments performed seem to confirm the insensitivity of these algorithms to the gaussian noise parameters

mode locking in a periodically forced resonate and fire neuron model the resonate and fire rf model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system impact oscillator we analyze the response of the rf neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one dimensional time map based on such a firing map we describe mode locked solutions and their stability leading to the so called arnol d tongues the boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow despite the fact that the periodically driven rf system shows periodic firing its behavior may become chaotic when the forcing frequency is near the resonant frequency we compare these results to numerical simulations of the model undergoing sinusoidal forcing furthermore upon varying a system parameter the rf system can be reduced to the integrate and fire system and in this case we show the consistency of the results on mode locked solutions

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

itineraries of rigid rotations and diffeomorphisms of the circle we examine the itinerary of email protected s 1 r z under the rotation by email protected email protected q the motivating question is if we are given only the itinerary of 0 relative to email protected s 1 a finite union of closed intervals can we recover a and i we prove that the itineraries do determine a and i up to certain equivalences then we present elementary methods for finding a and i moreover if g s 1 s 1 is a c 2 orientation preserving diffeomorphism with an irrational rotation number then we can use the orbit itinerary to recover the rotation number up to certain equivalences

on the control of complex dynamic systems a method is described for the limited control of the dynamics of systems which generally have several dynamic attractors associated either with maps or first order ordinary differential equations ode in rn the control is based on the existence of convergent regions cc k 1 2 in the phase space of such systems where there is local convergence of all nearby orbits the character of the control involves the entrainment and subsequent possible migration of the experimental system from one attractor to another entrainment means that limt x t g t 0 where x t rn is the system s controlled dynamics and the goal dynamics g t gk has any topological form but is limited dynamically and to regions of phase space gk contained in some ck gk ck the control process is initiated only when the system enters a basin of entrainment bek gk associated with the goal region gk aside from this macroscopic initial state information about the system no further feedback of dynamic information concerning the response of the system is required the experimental reliability of the control requires that the regions bek be convex regions in the phase space which can apparently be assured if gk ck simple illustrations of these concepts are given using a general linear and a piecewise linear ode in rn in addition to these entrainment goals migration goal dynamics is introduced which intersects two convergent regions g cj g cj i j and permits transferring the dynamics of a system from one attractor to another or from one convergent region to another in the present study these concepts are illustrated with various one dimensional maps involving one or more attractors and convergent regions several theorems concerning entrainment are derived for very general continuous one dimensional maps sufficient conditions are also established which ensure near entrainment for a system when the dynamic model of the system is not exactly known the applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies

dynamics of the complex standard family the complexification of the standard family of circle maps sin mod 2 whose parameter space contains the well known arnold tongues is given byf ei e 2 1 a holomorphic map of c with essential singularities at 0 and for real values of the parameters we study the dynamical plane of the familyf near the essential singularities we prove the existence ofhairsin the julia set an invariant set of curves organized by some symbolic dynamics whose points that are not endpoints tend exponentially fast to 0 or under iteration for 1 we prove that under certain conditions all hairs end up attached to the unit circle as we increase the parameter

dynamics near a periodically perturbed robust heteroclinic cycle abstract robust heteroclinic cycles rhcs arise naturally in collections of symmetric differential equations derived as dynamical models in many fields including fluid mechanics game theory and population dynamics in this paper we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle rhc we give a detailed derivation of the poincare map for trajectories near the rhc asymptotically correct in the limit of small amplitude perturbations this reduces the nonautonomous system in r 3 to a 2d map we identify three distinct dynamical regimes the distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem the first regime corresponds to the rhc being only weakly attracting here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque in the second regime the periodically perturbed rhc is more strongly attracting and the system dynamics corresponds to that of a non invertible or invertible circle map in the third regime of yet stronger attraction the dynamics of the return map is chaotic and no longer reducible to a one dimensional map this third regime has been noted previously our analysis in this paper focusses on providing quantitative results in the first two regimes

existence d orbites quasi periodiques dans les attracteurs de birkhoff for each number between the lower and the upper rotation number of the birkhoff attractor of a dissipative monotone twist map there is a periodic or quasi periodic orbit with rotation number

symbolic dynamics of order preserving orbits abstract the maps we consider are roughly those that can be obtained by truncating non invertible maps to weakly monotonic maps they have a flat piece the binary sequences that correspond to order preserving orbits are shown to satisfy a minimax principle which was already known for order preserving orbits with rational rotation number the converse is also proven all minimax orbits are order preserving with respect to some rotation number for certain families of such circle maps one can solve exactly for the parameter values for which the map has a specified rotation number rho for rho rational we obtain the endpoints of the resonance intervals recursively these parameter values can be organized in a natural way as the nodes of a farey tree we give some applications of the ideas discussed

onset of stochasticity for some one dimensional systems we study the appearance of stochastic dynamics in two situations 1 i for one parameter families of maps of the interval f undergoing an infinite sequence of period doubling bifurcations we consider 0 0 e beyond feigenbaum s parameter value 0 2 ii for a two parameter family of circle maps fb x x b2pisin2 x we consider b 1 1 e 0 1 in both cases we estimate the relative measure of parameter values corresponding to stochastic behavior

supercritical behavior of an ordered trajectory the behavior of a particle undergoing a discrete stepped motion around a circle is investigated for the case in which the average number of steps per revolution is the golden mean the new features of this study are that the map in question is not invertible and that consequently the orbit is bunched into intervals within the circle because of this bunching one can calculate the motion in some detail