quasistationary trajectories of the mean field xy hamiltonian model a topological perspective we employ a topological approach to investigate the nature of quasistationary states of the mean field xy hamiltonian model we focus on the quasistationary states reached when the system is initially prepared in a fully magnetized configuration by means of numerical simulations and analytical considerations we show that along the quasistationary trajectories the system evolves in a manifold of critical points of the potential energy function although these critical points are maxima the large number of directions with marginal stability may be responsible for the slow relaxation dynamics and the trapping of the system in such trajectories

self organized partially synchronous dynamics in populations of nonlinearly coupled oscillators we analyze a minimal model of a population of identical oscillators with a nonlinear coupling a generalization of the popular kuramoto model in addition to well known for the kuramoto model regimes of full synchrony full asynchrony and integrable neutral quasiperiodic states ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics self organized bunch states and self organized quasiperiodic dynamics the analysis based on the watanabe strogatz ansatz allows us to describe the self organized bunch states in any finite ensemble as a set of equilibria and the self organized quasiperiodicity as a two frequency quasiperiodic regime an analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

basic theory of f bounded quantification abstract system f bounded is a second order typed lambda calculus where the basic features of object oriented languages can be naturally modelled f bounded extends the better known system f in a way that provides an immediate solution for the treatment of the so called binary methods although more powerful than f and also quite natural system f bounded has only been superficially studied from a foundational perspective and many of its essential properties have been conjectured but never proved in the literature the aim of this paper is to give a solid foundation to f bounded by addressing and proving the key properties of the system in particular transitivity elimination completeness of the type checking semi algorithm the subject reduction property for reduction conservativity with respect to system f and antisymmetry of a full subsystem are considered and various possible formulations for system f bounded are compared finally a semantic interpretation of system f bounded is presented based on partial equivalence relations

on the limiting zeros of sampled multivariable systems for time invariant linear systems with single input single output it has been shown 3 that as the sampling period goes to 0 the zeros of the corresponding discrete time system obtained by sampling converge to some specific constant values determined only be the degrees of the finite and the infinite zeros of the original continuous time system it is natural to inquire the extent to which this result may be carried over to the case of multivariable systems in this note it is shown that the result is true also for multivariable systems if the difference among the degrees of infinite elementary divisors of the pencil corresponding to the original continuous system is less than two when the last condition is not satisfied the limiting values of the zeros of the discrete system depend in general not only on the integers cited above but also on system parameters

retractions on dl domains as a model for type type abstract we introduce an extension with the type type assumption for a higher order functional language in the framework of p the extension of the type free lambda calculus introduced in r amadio and g longo 1986b in ifip conference formal description of programming concepts ebberup dk august 1986 then we find using the stable functions and berry s di domains a model for p this settles the consistency problem for p up to now open and gives very simple mathematical models for higher order calculi with type type such models are based on the relevant property that the range of a stable retraction on a di domain is still a di domain finally we give the categorical description of the models of p based on moggi asperti s notion of internalization in a asperti 1988 models of higher order calculi internal categories carnegie mellon report in preparation

minimal positive realizations of transfer functions with nonnegative multiple poles this note concerns a particular case of the minimality problem in positive system theory a standard result in linear system theory states that any nth order rational transfer function of a discrete time invariant linear single input single output siso system admits a realization of order n in some applications however one is restricted to realizations with nonnegative entries i e a positive system and it is known that this restriction may force the order n of realizations to be strictly larger than n a general solution to the minimality problem i e determining the smallest possible value of n is not known in this note we consider the case of transfer functions with nonnegative multiple poles and give sufficient conditions for the existence of positive realizations of order n n with the help of our results we also give an improvement of an existing result in positive system theory

recursion over realizability structures abstract realizability structures play a major role in the metamathematics of intuitionistic systems and they are a basic tool in the extraction of the computational content of constructive proofs besides their rich categorical structure and effectiveness properties provide a privileged mathematical setting for the semantics of data types of programming languages in this paper we emphasize the modelling of recursive definitions of programs and types a realizability model for a language including girard s system f and an operator of recursion on types is given and some of its local properties are studied

universal retractions on di domains we prove that every di domain which contains an infinite set of pairwise incompatible elements in particular every reflexive di domain has an infinite set of universal retractions which moreover forms a di domain when ordered by the stable order isomorphic to n where is the divisibility relation

on functions preserving levels of approximation a refined model construction for various lambda calculi abstract in this paper di domains are enriched by a family of projections which assign to each point a sequence of canonical approximations the morphisms are stable maps that preserve the levels of approximation generated by the projections for the computation of an approximation of a given level of the output they require in addition only information about the input of at most the same level of approximation it is shown that the category of these domains and maps is cartesian closed the set of morphisms between two such domains is a domain of the same kind but turns out not to be an exponent in the category domains d are constructed which are isomorphic to the exponent d d moreover it is proved that the space of retractions in an exponent d d is a retract of this both results together provide new models of amadio longo s extension p of the calculus as has been shown by amadio and longo strong type theories which incorporate the type type assumption can be syntactically interpreted in this calculus

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

bifurcation and stability analyses for a coupled brusselator model abstract this paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two brusselators of particular interest is the study of the associated hopf bifurcation and double hopf bifurcations the motion of the oscillator may either be periodic bifurcating from a hopf type critical point or quasi periodic bifurcating from a compound critical point furthermore bifurcation analysis reveals that the limit cycles associated with the first brusselator are always stable while that generated by the second brusselator may be unstable if the parameter values are chosen far from the stability boundary it is interesting to note that in the vicinity of the double hopf compound critical point there exist periodic as well as quasi periodic solutions the quasi periodic motion is stable for a small parameter region a robust gauss seidel like implicit finite difference method gs1 has been developed and used for the solution of the resulting initial value problem ivp in addition to being of comparable accuracy judging by the similarity of the profiles generated with the fourth order runge kutta method rk4 the gs1 method will be seen to have better numerical stability property than rk4 unlike the rk4 which fails when large time steps are used to integrate the ivp extensive numerical simulations with appropriate initial data suggest that the gs1 method is unconditionally convergent moreover it is more economical computationally

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

time delay effects on coupled limit cycle oscillators at hopf bifurcation abstract we present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at hopf bifurcation for a simple model consisting of just two oscillators with a time delayed coupling the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death phase locked and incoherent regions a novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators similar results are obtained for an array of n oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified some general analytic results for the n thermodynamic limit are also obtained and the implications of the time delay effects for physical applications are discussed

a variety of period doubling universality classes in multi parameter analysis of transition to chaos abstract in multi parameter analysis of the onset of chaos non feigenbaum period doubling behavior may occur at some special paths in the parameter space there are two possibilities i the dynamics at the onset of chaos remains essentially one dimensional but the one dimensional map is distorted in such a way that leaves feigenbaum s universality class ii a new mode comes to the threshold of instability and increases the effective dimension of the dynamics we submit a list of one dimensional and two dimensional maps which represent distinct classes of the period doubling universality discuss the properties of the associated types of critical behavior at the border of chaos and demonstrate pictures of the universal parameter space arrangement near the critical points

properties of synchronization in the systems of non identical coupled van der pol and van der pol duffing oscillators broadband synchronization the particular properties of dynamics are discussed for dissipatively coupled van der pol oscillators non identical in values of parameters controlling the hopf bifurcation possibility of a special synchronization regime in an infinitively long band between oscillator death and quasiperiodic areas is shown for such system features of the bifurcation picture are discussed for different values of the control parameters and for the case of additional duffing type nonlinearity analysis of the slow flow equations is presented

role of saddles in topologically driven phase transitions the case of the d dimensional spherical model analyzing the d dimensional spherical model we show that underlying saddles defined through a map in the configuration space play a central role in driving the phase transition at the phase transition point the underlying saddle energy reaches its lowest value corresponding to the trivial boundary topological singularity

complicated quasiperiodic oscillations and chaos from driven piecewise constant circuit chenciner bubbles do not necessarily occur via simple phase locking abstract we analyze the complex quasiperiodic oscillations and chaos generated by two coupled piecewise constant hysteresis oscillators driven by a rectangular wave force oscillations generate arnol d resonance webs wherein lower dimensional resonance tongues extend such as that of a web in numerous directions we provide the fundamental tools for bifurcation analysis of nonautonomous piecewise constant oscillators to optimize the outstanding simplicity of piecewise constant circuits we formulate a generalized procedure for calculating the lyapunov exponents in nonautonomous piecewise constant dynamics the lyapunov exponents in these dynamics can be calculated with a precision approximately similar to that of maps we observe two parameter lyapunov diagrams near the fundamental resonance region called chenciner bubbles wherein the oscillation frequencies of the two oscillators and the force are synchronized with a ratio of 1 1 1 inevitably the hysteresis considerably distorts the chenciner bubbles this result suggests that the chenciner bubbles do not necessarily occur due to simple phase locking of two dimensional tori that can be explained by homeomorphism on the circle furthermore we observe the farey sequence in the experimental measurements

the category theoretic solution of recursive metric space equations it is well known that one can use an adaptation of the inverse limit construction to solve recursive equations in the category of complete ultrametric spaces we show that this construction generalizes to a large class of categories with metric space structure on each set of morphisms the exact nature of the objects is less important in particular the construction immediately applies to categories where the objects are ultrametric spaces with extra structure and where the morphisms preserve this extra structure the generalization is inspired by classical domain theoretic work by smyth and plotkin for many of the categories we consider there is a natural subcategory in which each set of morphisms is required to be a compact metric space our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category as another application we present a construction that relates solutions of generalized domain equations in the sense of smyth and plotkin to solutions of equations in our class of categories our primary motivation for solving generalized recursive metric space equations comes from recent and ongoing work on kripke style models in which the sets of worlds must be recursively defined we show a series of examples motivated by this line of work

numerical and experimental observation of arnol d resonance webs in an electrical circuit abstract an extensive bifurcation analysis of partial and complete synchronizations of three frequency quasi periodic oscillations generated in an electric circuit is presented our model uses two coupled hysteresis oscillators and a rectangular wave forcing term the governing equation of the circuit is represented by a piecewise constant dynamics generating a three dimensional torus the lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation by analyzing this extremely simple circuit we clearly demonstrate that it generates an extremely complex bifurcation structure called arnol d resonance web inevitably chaos is observed in the neighborhood of chenciner bubbles around which regions generating three dimensional tori emanate furthermore the numerical results are experimentally verified

categorical models of polymorphism abstract we present and discuss the relations between two classes of categorical models of the second order or polymorphic lambda calculus namely those based on internal categories internal models and those based on indexed categories external models we start in part i with a detailed introduction to internal categories and their relations to indexed categories the presentation is by means of equations between arrows in an ambient category with finite limits in part ii we recall the definition of the two classes of models and we present the externalization process that given an internal model yields an external model we show how one can go back in a straightforward way and that by making a full round trip from an internal model to an internal model via an external one or vice versa one does obtain equivalent models part iii discusses three major examples of models provable retractions inside a per model per inside set pl categories inside their grothendieck completions the appendix contains an account of internal adjunctions and internal cccs

a full continuous model of polymorphism we introduce a model of the second order lambda calculus such a model is a scott domain whose elements are themselves scott domains and in it polymorphic maps are interpreted by generic continous maps

a stability condition of zeros of sampled multivariable systems this note derives a new condition for zeros of sampled multivariable systems to be stable for sufficiently small sampling periods it is a natural extension of hagiwara s result for single input single output systems 1993 to multivariable systems

zero dynamics of sampled data models for nonlinear multivariable systems in fractional order hold case the paper is concerned with the properties of approximate sampled data models and their zero dynamics as the sampling period tends to zero composed of a fractional order hold froh a continuous time multivariable plant and a sampler in cascade the emphasis of this paper is the stability of discrete zero dynamics with the generalized gain of the froh where we also present a condition to assure the stability of the sampling zero dynamics which they have no counterpart in the underlying continuous time system of the resulting model similar to the linear case the parameter is the only factor in affecting the stability of discrete zero dynamics and the appropriate is determined to obtain the froh that provides zero dynamics as stable as possible or with improved stability properties even when unstable for a given continuous time multivariable plant the study is also shown that the stability of the sampling zero dynamics is improved compared with a zero order hold zoh

coupled van der pol duffing oscillators phase dynamics and structure of synchronization tongues abstract synchronization in the system of coupled non identical non isochronous van der pol duffing oscillators with inertial and dissipative coupling is discussed generalized adler s equation is obtained and investigated in the presence of all relevant factors affecting the synchronization non isochronism of the oscillators their non identity coupling of the dissipative and inertial types characteristic symmetries are revealed for the adler s equation responsible for equivalence of some of the factors numerical study of the parameters space of the initial differential equations is carried out using the method of charts of dynamic regimes in the parameter planes results obtained by both these approaches are compared and discussed

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

partial morphisms in categories of effective objects abstract this paper is divided in two parts in the first we analyse in great generality data types in relation to partial morphisms we introduce partial function spaces partial cartesian closed categories and complete objects motivate their introduction and show some of their properties in the second part we define the partial cartesian closed category gen of generalized numbered sets prove that it is a good extension of the category of numbered sets and show how it is related to the recursive topos

stability of the limiting zeros of sampled data systems with zero and first order holds this paper is concerned with the zeros of sampled data systems resulting from continuous time systems preceded by a hold and followed by a sampler the holds we shall consider are a zero order hold and a first order hold for sufficiently small or large sampling periods such zeros are called limiting zeros for sufficiently small sampling periods they are known to consist of two different types of zeros the zeros of the first type correspond to the zeros of the original continuous time system while those of the second type have no continuous time counterparts we first show basic properties of the zeros of sample data systems for sufficiently small sampling periods next we clarify in more detail the correspondence between the former type zeros of the sampled data systems and the zeros of the original continuous time system including the stability property of these zeros we also study stability properties of the latter type zeros in addition we study limiting zeros for sufficiently large sampli

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

exploring the energy landscape of xy models we investigate the energy landscape of two and three dimensional xy models with nearest neighbor interactions by analytically constructing several classes of stationary points of the hamiltonian these classes are analyzed in particular with respect to possible signatures of the thermodynamic phase transitions of the models we find that even after explicitly breaking the global o 2 symmetry of the xy spins an exponentially large class of stationary points are singular and occur in continuous one parameter families this property may complicate the use of theoretical tools developed for the investigation of phase transitions based on stationary points of the energy landscape and we discuss strategies to avoid these difficulties

synchronization and multi frequency oscillations in the low dimensional chain of the self oscillators abstract the problem of growing complexity of the dynamics of the coupled phase oscillators as the number of oscillators in the chain increases is considered the organization of the parameter space parameter of the frequency detuning between the second and the first oscillators versus parameter of dissipative coupling is discussed the regions of complete synchronization quasi periodic regimes of different dimensions and chaos are identified we discuss transformation of the domains of different dynamics as the number of oscillators grows we use the method of charts of lyapunov exponents and modification of the method of the chart of dynamical regimes to visualize two frequency regimes of different type limits of applicability of the quasi harmonic approximation and the features of the dynamics of the original system which are not described by the approximate phase equations are discussed for the case of three coupled oscillators

phase transitions and topology changes in configuration space the relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant the euler characteristic of certain submanifolds in the configuration space of two physical models the models are the mean field xy model and the one dimensional xy model with nearest neighbor interactions the former model undergoes a second order phase transition at a finite critical temperature while the latter has no phase transitions the computation of this topologic invariant is performed within the framework of morse theory in both models topology changes in configuration space are present as the potential energy is varied however in the mean field model there is a particularly strong topology change corresponding to a big jump in the euler characteristic connected with the phase transition which is absent in the one dimensional model with no phase transition the comparison between the two models has two major consequences i it lends new and strong support to a recently proposed topological approach to the study of phase transitions ii it allows us to conjecture which particular topology changes could entail a phase transition in general we also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two dimensional surfaces and a possible direct connection between topological invariants and thermodynamic quantities

extensional models for polymorphism we present a general method for constructing extensional models for the polymorphic lambda calculus the polymorphic extensional collapse the method yields models that satisfy additional computationally motivated constraints like having only two polymorphic booleans and having only the numerals as polymorphic integers moreover the method yields models that prove that the polymorphic lambda calculus can be conservatively added to arbitrary algebraic data type specifications even with complete transfer of the computational power to the added data types

phase space geometry and slow dynamics we describe a non arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space we show that such a mechanism is at work both in a family of mean field spin glass models without any domain structure and in the case of ferromagnetic domain growth the marginality of spin glass dynamics as well as the existence of a quasi equilibrium regime can be understood within this scenario we discuss the question of ergodicity in an out of equilibrium situation

amplitude death through a hopf bifurcation in coupled electrochemical oscillators experiments and simulations amplitude death was observed in experiments with two coupled periodic electrochemical oscillators without time delay simulation results confirmed that the observed amplitude death was obtained via a hopf bifurcation the two oscillators must have a minimum discrepancy and both be near their individual hopf bifurcations for amplitude death to occur phase drift coexisting with unstable asymmetric phase locked solutions amplitude death and in phase synchronization were observed in both the experiments and simulations as coupling strength was increased in addition the simulations showed that a stable asymmetric phase locked solution exists between the phase drift and amplitude death regions

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

topology and phase transitions the case of the short range spherical model we characterize the topology of the phase space of the berlin kac spherical model in the context of the so called topological hypothesis for spins lying in hypercubic lattices of dimension d for zero external field we are able to characterize the topology exactly up to homology we find that even though there is a continuum of changes in the topology of the corresponding manifolds for d 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level we show however that these changes do not coincide with the phase transitions and conversely that no topological discontinuity can be associated to the points where the phase transitions take place at variance with what happens in the mean field version of this same model we show that these abrupt topological changes are accessible thermodynamically we conclude that even in short range systems the topological mechanism does not seem to be responsible for the triggering of a phase transition we also analyze the case of spins connected to a macroscopic number of but not all neighbors and find that similar to the results found for the fully connected version in this case the topological hypothesis seems to hold the phase transition coincides with an accumulation point of the topological changes present in configuration space the question of the ensemble equivalence in the short range spherical model is also considered

amplitude response of coupled oscillators abstract we investigate the interaction of a pair of weakly nonlinear oscillators eg each near a hopf bifurcation when the coupling strength is comparable to the attraction of the limit cycles changes in amplitude cannot then be ignored and there are new phenomena we show that even for simple forms of these equations there are parameter regimes in which the interaction causes the system to stop oscillating and the rest state at zero stabilized by the interaction is the only stable solution when the uncoupled oscillators have a local frequency that is independent of amplitude zero shear and the coupling is scalar we give a complete description of the behavior we then analyze more complicated equations to show the effects of amplitude dependent frequency in the uncoupled equations nonzero shear and nonscalar coupling for example when there is shear there can be bistability between a phase locked solution and an unlocked drift solution in which the oscillators go at different average frequencies there can also be bistability between a phase locked solution and an equilibrium nonoscillatory solution the equations for the zero shear case have a pair of degenerate bifurcation points and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities when the coupling is nonscalar there are a variety of new behaviors including bistability of an in phase and an anti phase solution for two identical oscillators parameters for which only the anti phase solution is stable and phase trapping i e nonlocked solutions in which the oscillators still have the same average frequency it is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system

trees and semantics abstract several types of trees are proposed in the literature to model the behaviour of nondeterministic processes the purpose of this paper is to clarify the distinction between the various types of trees and to discuss their use as semantic domains trees in the sense of milner s derivation trees and bisimulation equivalence classes of trees are considered in three different settings in the category of sets in the metric setting and in the partial order setting each type of tree is characterized by a domain equation special attention is given to the use of trees as models for processes which allow for unbounded nondeterminism as for instance milners ccs processes in this case one has to deal with infinitely branching trees which cause problems to give denotational semantics that are fully abstract w r t a transition system based operational semantics

polymorphic type inference and containment abstract type expressions may be used to describe the functional behavior of untyped lambda terms we present a general semantics of polymorphic type expressions over models of untyped lambda calculus and give complete rules for inferring types for terms some simplified typing theories are studied in more detail and containments between types are investigated

dynamics of two van der pol oscillators coupled via a bath abstract in this work we study a system of two van der pol oscillators x and y coupled via a bath z x 1 x 2 x x k z x y 1 y 2 y y k z y z k x z k y z we investigate the existence and stability of the in phase and out of phase modes for parameters 0 and k 0 to this end we use floquet theory and numerical integration surprisingly our results show that the out of phase mode exists and is stable for a wider range of parameters than the in phase mode this behavior is compared to that of two directly coupled van der pol oscillators and it is shown that the effect of the bath is to reduce the stability of the in phase mode we also investigate the occurrence of other periodic motions by using bifurcation theory and the auto bifurcation and continuation software package our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes we present a simplified model of a circadian oscillator which shows that it can be modeled as a van der pol oscillator in our model although there is no direct connection between the two eyes they can influence each other by affecting the concentration of melatonin in the bloodstream which is represented by the bath

inverse stable sampled low pass systems it is shown that for low pass systems the fractional order hold and the pulse amplitude modulation signal reconstruction methods produce an inverse stable sampled transfer function h z in some cases when zero order hold fails to do so therefore they may be desirable alternatives to the zero order hold reconstruction in certain control problems the problem of how well h z models the continuous g s is also discussed

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

limiting zeros of decouplable mimo systems for sufficiently rapid zero order hold zoh sampling it is known that the zeros of single input single output discrete time systems expressed in the forward shift operator converge to values determined only by the degrees of the finite and infinite zeros of the underlying continuous time system in this paper it is shown how this result can be generalized to multi input multi output mimo systems decouplable by static state feedback equivalently having a diagonal interactor matrix in the fast sampling limit the authors show how invariant and infinite zero structure is mapped under zoh sampling for discrete time systems expressed in either the delta forward difference or forward shift operators

a quick construction of a retraction of all retractions for stable bifinites in the framework of stable domain theory we show that the space of retractions over a bifinite is a retract of the functional space

on sampled data models for nonlinear systems models for deterministic continuous time nonlinear systems typically take the form of ordinary differential equations to utilize these models in practice invariably requires discretization in this paper we show how an approximate sampled data model can be obtained for deterministic nonlinear systems such that the local truncation error between the output of this model and the true system is of order spl delta sup r 1 where spl delta is the sampling period and r is the system relative degree the resulting model includes extra zero dynamics which have no counterpart in the underlying continuous time system the ideas presented here generalize well known results for the linear case we also explore the implications of these results in nonlinear system identification

bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators abstract we study the bifurcation and dynamical behaviour of the system of n globally coupled identical phase oscillators introduced by hansel mato and meunier in the cases n 3 and n 4 this model has been found to exhibit robust slow switching oscillations that are caused by the presence of robust heteroclinic attractors this paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors we consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function these bifurcations preserve the permutation symmetry of the system we then investigate the implications of these bifurcations for the sensitivity to detuning i e the size of the smallest perturbations that give rise to loss of frequency locking for n 3 we find three types of heteroclinic bifurcation that are codimension one with symmetry on varying two parameters in the coupling function we find three curves giving a an s 3 transcritical homoclinic bifurcation b a saddle node heteroclinic bifurcation and c a z 3 heteroclinic bifurcation we also identify several global bifurcations with symmetry that organize the bifurcation diagram these are codimension two with symmetry for n 4 oscillators we determine many but not all codimension one bifurcations with symmetry including those that lead to a robust heteroclinic cycle a robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region furthermore we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system while for other parts of the phase plane it can coexist with stable limit cycles we finish with a discussion of bifurcations that appear for this coupling function and general n as well as for more general coupling functions

synchronization of two non scalar coupled limit cycle oscillators being one of the fundamental phenomena in nonlinear science synchronization of oscillations has permanently remained an object of intensive research development of many asymptotic methods and numerical simulations has allowed an understanding and explanation of various phenomena of self synchronization but even in the classical case of coupled van der pol oscillators a full description of all possible dynamical regimes their mutual transitions and characteristics is still lacking we present here a study of the phenomenon of mutual synchronization for two non scalar coupled non identical limit cycle oscillators and analyze phase frequency and amplitude characteristics of synchronization regimes a series of bifurcation diagrams that we obtain exhibit various regions of qualitatively different behavior among them we find mono bi and multistability regions beating and oscillation death ones also a region where one of the oscillators dominates the other one is observed the frequency characteristics that we obtain reveal three qualitatively different types of synchronization i on the mean frequency the in phase synchronization ii with a shift from the mean frequency caused by a conservative coupling term the anti phase synchronization and iii on the frequency of one of the oscillators when one oscillator dominates the other 2003 elsevier b v all rights reserved

domain theoretic models of polymorphism abstract we give an illustration of a construction useful in producing and describing models of girard and reynolds polymorphic calculus the key unifying ideas are that of a grothendieck fibration and the category of continuous sections associated with it constructions used in indexed category theory the universal types of the calculus are interpreted as the category of continuous sections of the fibration as a major example a new model for the polymorphic calculus is presented in it a type is interpreted as a scott domain in fact understanding universal types of the polymorphic calculus as categories of continuous sections appears to be useful generally for example the technique also applies to the finitary projection model of bruce and longo and a recent model of girard indeed the work here was inspired by girard s and arose through trying to extend the construction of his model to scott domains it is hoped that by pin pointing a key construction this paper will help towards a deeper understanding of models for the polymorphic calculus and the relations between them

peculiarities of synchronization of a resonant limit cycle on a two dimensional torus the peculiarities of external synchronization of a resonant limit cycle on a torus are studied in an autonomous oscillator of quasiperiodic oscillations with two basic frequencies we show numerically and experimentally that in the resonance conditions the synchronization effect takes place only at one of the two basic frequencies of the system while the oscillations at the second basic frequency remain unsynchronized our results convincingly indicate a principal difference between synchronization of the resonant limit cycle on the torus and of a typical limit cycle this is in contrast to the well established theory of synchronization of a limit cycle this finding opens new strategies for controlling systems with multiple time scales

two parameter scaling of correlation functions near continuous phase transitions we discuss the order parameter correlation function in the vicinity of continuous phase transitions using a two parameter scaling form g k k 2 c g k k k c where k is the wave vector and is the correlation length and the interaction dependent nonuniversal momentum scale k c remains finite at the critical fixed point the correlation function describes the entire critical regime and captures the classical to critical crossover one parameter scaling is recovered only in the limit k k c 0 we present an approximate calculation of g x y for the ising universality class using the functional renormalization group

rotation sets and phase locking in an electronic three oscillator system abstract the parameter space of an electronic three oscillator system is investigated and various codimension one and two bifurcations predicted by baesens guckenheimer kim and mackay are identified sampled time series from the experimental systems are recorded and analysed for partial mode locking or resonance one or two independent rational relations between the average rates of change of the angles describing the system using knowledge of where the invariant torus lies and the torus unfolding scheme of ashwin and swift examples of toroidal and annular chaos are investigated by finding bounds on the size and shape of the rotation set

an automata theoretical characterization of the oi hierarchy this paper gives an automata theoretical characterization of the oi hierarchy damm 1982 engelfriet and schmidt 1977 wand 1975 this hierarchy is generated by so called level n grammars which are natural generalizations from context free and macro grammars in that their nonterminals are treated as functionals of higher type i e they are allowed to carry up to n levels of parameters the automata model used for this characterization is the n iterated pushdown automaton its characteristic feature is the storage structure which consists of a nesting of pushdowns up to nesting depth n the equivalence proof is given constructively its method is illustrated using examples by viewing level n grammars as modeling recursive procedures on higher types the iterated pushdown automation thus provides an operational model for the run time behavior of procedures defined by recursion on higher types which makes the results of this paper interesting not only from a language theoretical point of view

improvement of stability of zeros in discrete time multivariable systems using fractional order hold this paper is concerned with the stability of zeros of the discrete time multivariable systems composed of a fractional order hold froh a continuous time plant and a sampler in cascade the properties of the limiting zeros are studied and a condition for obtaining stable zeros for sufficiently small sampling periods is derived an approximate fractional order hold afroh is proposed as an implementation of froh it is shown that the afroh as well as the froh can locate the zeros of discrete time multivariable systems inside the stable region and improve stability properties when the zero order hold zoh cannot experimental results of adaptive control with afroh demonstrate a significant improvement of control performance compared with the use of a zoh

on the stabilization of discrete system zeros the stabilization of discrete system zeros is commonly achieved by allowing the input pattern to deviate from zero order hold zoh in this paper we show that this design technique may not be practical at small sampling times when the continuous system has unstable zero s deviation of the input pattern from zoh may also cause undesirable fluctuation in the inter sampling time output in order to reduce this effect we present a method which uses linear quadratic optimization for the determination of suitable zero locations

the mean field φ 4 model entropy analyticity and configuration space topology a large deviation technique is applied to the mean field model phi4 providing an exact expression for the configurational entropy s v m as a function of the potential energy v and the magnetization m although a continuous phase transition occurs at some critical energy vc the entropy is found to be a real analytic function in both arguments and it is only the maximization over m which gives rise to a nonanalyticity in s v supm s v m this mechanism of nonanalyticity generation by maximization over one variable of a real analytic entropy function is restricted to systems with long range interactions and has for continuous phase transitions the generic occurrence of classical critical exponents as an immediate consequence furthermore this mechanism can provide an explanation why contradictory to the so called topological hypothesis the phase transition in the mean field model need not be accompanied by a topology change in the family of constant energy submanifolds

saddle index properties singular topology and its relation to thermodynamic singularities for a ϕ 4 mean field model we investigate the potential energy surface of a ensuremath phi 4 model with infinite range interactions all stationary points can be uniquely characterized by three real numbers ensuremath alpha ensuremath alpha 0 ensuremath alpha ensuremath with ensuremath alpha ensuremath alpha 0 ensuremath alpha ensuremath 1 provided that the interaction strength ensuremath mu is smaller than a critical value the saddle index n s is equal to ensuremath alpha 0 and its distribution function has a maximum at n s mathrm max 1 3 the density p e of stationary points with energy per particle e as well as the euler characteristic ensuremath chi e are singular at a critical energy e c ensuremath mu if the external field h is zero however e c ensuremath mu ensuremath ne ensuremath upsilon c ensuremath mu where ensuremath upsilon c ensuremath mu is the mean potential energy per particle at the thermodynamic phase transition point t c this proves that previous claims that the topological and thermodynamic transition points coincide is not valid in general both types of singularities disappear for h ensuremath ne 0 the average saddle index overline n s as function of e decreases monotonically with e and vanishes at the ground state energy only in contrast the saddle index n s as function of the average energy overline e n s is given by n s overline e 1 4 overline e for h 0 that vanishes at overline e ensuremath 1 4g ensuremath upsilon 0 the ground state energy

a novel approach to stable zero dynamics of sampled data models for nonlinear systems in backward triangle sample and hold case abstract it is well known that a good approximate sampled data model is very important in sampled data controller design of nonlinear continuous time system because the exact sampled data model of nonlinear continuous time system is often unavailable it is also a truth that unstable zero dynamics of sampled data model greatly limit the control performance of the system this paper investigates the properties of approximate sampled data model and its zero dynamics which generated during the process of discretization with backward triangle sample and hold btsh further a more accurate sampled data model is obtained for nonlinear systems in btsh case the zero dynamics form and stable conditions are also derived finally numerical examples are provided to better show the results developed in this paper

building continuous webbed models for system f we present here a large family of concrete models for girard and reynolds polymorphism system f in a noncategorical setting the family generalizes the construction of the model of barbanera and berardi tech report university of turin 1997 hence it contains complete models for f a complete model for system f preprint june 1998 and we conjecture that it contains models which are complete for f it also contains simpler models the simplest of them e2 being a second order variant of the engeler plotkin model e all the models here belong to the continuous semantics and have underlying prime algebraic domains all have the maximum number of polymorphic maps the class contains models which can be viewed as two intertwined compatible webbed models of untyped calculus in the sense of berline from computations to foundations the calculus and its webbed models revised version theoret comput sci 86 pp to appear but it is much larger than this finally many of its models might he read as two intertwined strict intersection type systems

zero dynamics of sampled nonlinear systems abstract the behaviour of the zero dynamics of a discrete time system obtained when sampling a continuous time nonlinear system is studied we give a result which generalizes the known behaviour of the zeros of a sampled linear system for small time intervals

evaluation of the watson integral and associated logarithmic integral for the d dimensional hypercubic lattice the watson integral for the d dimensional hypercubic lattice and the associated logarithmic integral are investigated in particular a new method is developed which enables one to calculate the numerical values of ld d 1 2 and wd d 3 4 with extremely high precision the asymptotic behaviour of ld and wd as d is also determined finally some generalizations of the results are briefly discussed

topological hypothesis on phase transitions the simplest case we critically analyze the possibility of finding signatures of a phase transition by looking exclusively at static quantities of statistical systems like e g the topology of potential energy submanifolds pes s this topological hypothesis has been successfully tested in a few statistical models but up to now there has been no rigorous proof of its general validity we make a new test of it analyzing the probably simplest example of a nontrivial system undergoing a continuous phase transition the completely connected version of the spherical model going through the topological properties of its pes it is shown that as expected the phase transition is correlated with a change in their topology nevertheless this change as reflected in the behavior of a particular topological invariant the euler characteristic is small at variance with the strong singularity observed in other systems furthermore it is shown that in the presence of an external field when the phase transition is destroyed a similar topology change in the submanifolds is still observed at the maximum value of the potential energy manifold a level which nevertheless is thermodynamically inaccessible this suggests that static properties of the pes s are not enough in order to decide whether a phase transition will take place some input from dynamics seems necessary