the geometry of mixing in time periodic chaotic flows i asymptotic directionality in physically realizable flows and global invariant properties abstract this paper demonstrates that the geometry and topology of material lines in 2d time periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality this property implies the existence of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the jacobian matrix of the n th iterative of the poincare map associated with the flow asymptotic directionality also determines the geometry of the invariant unstable manifolds which are everywhere tangent to the field of asymptotic eigendirections this fact is used to derive simple non perturbative methods for reconstructing the global invariant manifolds to any desired level of detail the geometric approach associated with the existence of a field of invariant unstable subspaces permits us to introduce the concept of a geometric global unstable manifold as an intrinsic property of a poincare map of the flow defined as a class of equivalence of integral manifolds belonging to the invariant unstable foliation the connection between the geometric global unstable manifold and the global unstable manifold of hyperbolic periodic points is also addressed since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the poincare map in a sense that will be made clear in the article the reconstruction of unstable integral manifolds can be used to obtain a quantitative characterization of the topological and statistical properties of partially mixed structures two physically realizable systems are analyzed closed cavity flow and flow between eccentric cylinders asymptotic directionality provides evidence of a global self organizing structure characterizing chaotic flow which is analogous to that of anosov diffeomorphisms which turns out to be the basic prototype of mixing systems in this framework we show how partially mixed structures can be quantitatively characterized by a nonuniform stationary measure different from the ergodic measure associated with the dynamical system generated by the field of asymptotic unstable eigenvectors

derivatives and constraints in chaotic flows asymptotic behaviour and a numerical method in a smooth flow the leading order response of trajectories to infinitesimal perturbations in their initial conditions is described by the finite time lyapunov exponents and associated characteristic directions of stretching we give a description of the second order response to perturbations in terms of lagrangian derivatives of the exponents and characteristic directions these derivatives are related to generalised lyapunov exponents which describe deformations of phase space elements beyond ellipsoidal when the flow is chaotic care must be taken in evaluating the derivatives because of the exponential discrepancy in scale along the different characteristic directions two matrix decomposition methods are used to isolate the directions of stretching the first appropriate in finding the asymptotic behaviour of the derivatives analytically the second better suited to numerical evaluation the derivatives are shown to satisfy differential constraints that are realised with exponential accuracy in time with a suitable reinterpretation the results of the paper are shown to apply to the eulerian framework as well

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

non uniform stationary measure properties of chaotic area preserving dynamical systems abstract this article shows the existence of a non uniform stationary measure referred to as the w invariant measure associated with the space filling properties of the unstable manifold and characterizing some statistical properties of chaotic two dimensional area preserving systems the w invariant measure which differs from the ergodic measure and is non uniform in general plays a central role in the statistical characterization of chaotic fluid mixing systems since several properties of partially mixed structures can be expressed as ensemble averages over the w invariant measure a closed form expression for the w invariant density is obtained for a class of mixing systems topologically conjugate with the linear toral automorphism the physical implications in the theory of fluid mixing and in the statistical characterization of chaotic hamiltonian systems are discussed

a generalized problem of optimal selection and assignment we consider the infinite version of the secretary problem from an infinite stream of applicants we are to select m choices and assign them in order to m positions to be filled we receive a reward ckm for 1 k m0 c1m cmm 1 if the best k applicants are selected and correctly ranked i e the ith best applicant is assigned to the ith position for 1 i k we consider two versions of the problem i once an applicant is selected and assigned to some position he cannot be promoted to a better position at a later time and ii previously selected applicants can be promoted in this paper we concentrate on the double selection problem m 2 and derive for both the promotion and nonpromotion versions of the problem the optimal strategy that maximizes the expected reward

finite time lyapunov exponent and advection diffusion equation the diffusive transport of a scalar in a flowing fluid is determined by the advection diffusion equation if the flow is chaotic the properties of the transport are determined by the spatial and temporal dependence of the finite time lyapunov exponent the rapid diffusive transport occurs only along the field line of the vector s which defines the stable direction in which neighboring points asymptotically converge the geometry of the s line affects the diffusive transport through the finite time lyapunov exponent which varies smoothly along an s line for example the finite time lyapunov exponent of a 2d conservative system can always be written as t t with s 0 0 and the infinite time lyapunov exponent the spatial dependence of the finite time lyapunov exponent is related to the geometry of the s line through by s 0 0 s 0 diffusion is empeded if the finite time lyapunov exponent becomes small along the field line of the vector it is shown that the finite time lyapunov exponent has a sharp dip where the line encounters a sharp bend hence the barriers for diffusion are located at these sharp bends of the lines which are generic for chaptic flows due to nonhyperbolicity this new class of diffusive transport barrier persists even in systems that are far from integrable

passive scalars and three dimensional liouvillian maps abstract global aspects of the motion of passive scalars in time dependent incompressible fluid flows are well described by volume preserving liouvillian three dimensional maps in this paper the possible invariant structures in liouvillian maps and the two most interesting nearly integrible cases are investigated in addition the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail

the calculation of lyapunov spectra abstract linearization around the chaotic orbits of dynamical systems offers good prospects for understanding complex systems the naturally available information that can be obtained from such studies are the magnitudes and directions in phase space of stretching and shrinking of an initial small sphere of realizations this information is a property of a time displacement matrix that relates the initial bundle of realizations to its later configuration in this paper we extract information from this matrix through its singular values and the associated orthogonal matrices this given an algebraic foundation to the geometric methods that have been used previously the equations of the orthogonal principal stretching and contraction directions show that asymptotically after a long time they depend only on the location in phase space and not on the orbit as an example we show that these directions sketch out the attractor of the lorenz model

behavioural differential equations a coinductive calculus of streams automata and power series we present a theory of streams infinite sequences automata and languages and formal power series in terms of the notions of homomorphism and bisimulation which are the cornerstones of the theory of universal coalgebra this coalgebraic perspective leads to a unified theory in which the observation that each of the aforementioned sets carries a so called final automaton structure plays a central role finality forms the basis for both definitions and proofs by coinduction the coalgebraic counterpart of induction coinductive definitions take the shape of what we have called behavioural differential equations after brzozowski s notion of input derivative a calculus is developed for coinductive reasoning about all of the afore mentioned structures closely resembling calculus from classical analysis

invariant geometric properties of a class of 3d chaotic flows this article extends the analysis developed by giona and adrover m giona a adrover phys rev lett 81 1998 3864 for 2d area preserving diffeomorphisms to 3d volume preservingc 1 diffeomorphisms of the 3d torus topologically conjugate to a linear map the article analyzes the invariant geometric properties of vector dynamics and surface element evolution in 3d systems and provides an analytic expression for the probability measure describing pointwise statistical properties of the unstable foliations in the hyperbolic case the convergence properties of this measure are addressed starting from the dynamics of surface elements the application of the methods developed to physically realizable 3d chaotic flows such as abc flow is discussed in detail 2000 elsevier science b v all rights reserved

modular control of discrete event systems with coalgebra modular supervisory control of discrete event systems where the overall system is a synchronous parallel product of subsystems is considered the main results of this paper are formulations of sufficient conditions for the compatibility between the synchronous product and various operations stemming from supervisory control as supervised product and supremal controllable sublanguages these results are generalized to the case of modules with partial observations e g modular computation of supremal normal sublanguages is studied coalgebraic techniques based on the coinduction proof principle are used in our main results sufficient conditions are derived for modular to equal global control synthesis an algorithmic procedure for checking the new conditions is proposed and the computational benefit of the modular approach is discussed and illustrated by comparing the time complexity of modular and monolithic computation

passive scalars three dimensional volume preserving maps and chaos the dynamics of a medium sized particle passive scalar suspended in a general time periodic incompressible fluid flow can be described by three dimensional volume preserving maps in this paper these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot the former are called slow variables or actions the latter fast variables or angles the maps are classified by their number of actions for maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a kam like way on the other hand for the two action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity instead we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion at these intersections surfaces are locally broken and transitions from one to another can occur we call this process which is analogous to arnold diffusion singularity induced diffusion it is characteristic of two action maps in one action maps this diffusion is blocked by kam like surfaces

stretching and curvature of material lines in chaotic flows as a streak of dye is advected by a chaotic flow it stretches and folds and becomes indistinguishable from a one dimensional idealized material line the variation along a material line of the total stretching experienced by fluid elements is examined and it is found that it can be decomposed into an overall time dependent factor constant along the line and a smooth time independent deviation the stretching is related by a power law to the curvature of the line near sharp bends this is confirmed numerically and motivated by a simple model a conservation law for lyapunov exponents explains deviations from a power law

maximal solutions in decentralized supervisory control the decentralized supervisory control problem is to construct for a discrete event system a set of supervisors each observing only part of the system and each controlling only part of the events such that the interconnection of the system and the supervisors meets control objectives of safety and liveness definitions are provided of the concepts of a maximal solution of a nash equilibrium and of a strong nash equilibrium for a set of supervisors with as order relation the inclusion relation on the set of closed loop languages the main result is that a set of supervisors is a maximal solution if and only if it is a strong nash equilibrium a procedure to determine a nash equilibrium is described and illustrated by an example there is no guarantee that the procedure halts in finite time however in the case that it halts in finite time then it is proven that a nash equilibrium is obtained

think globally act locally decentralized supervisory control decentralized supervisory control is investigated by considering problem formulations that model systems whose specifications are given as global constraints but whose solution is described by local controllers a necessary and sufficient condition is given for the existence of a solution to the problem of finding decentralized supervisors that ensure that the behavior of the closed loop system lies in a given range where the range of behavior can be described by regular languages it can be effectively tested whether the decentralized control problem is solvable in this case a procedure is given to compute the associated supervisors

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

generalized lyapunov exponents corresponding to higher derivatives abstract we generalize the notion of lyapunov exponents to higher order derivatives for fixed points and periodic orbits we derive a relationship for the higher order lyapunov spectrum in terms of the usual first order lyapunov spectrum based on numerical experiments as well as general arguments we conjecture that this relationship also holds for chaotic orbits this work is relevant to a priori error estimates for time series forecasting

synthesis of controllable and normal sublanguages for discrete event systems using a coordinator synthesis of normal or controllable and normal sublanguages of global specification languages without computation of the global modular plant is a difficult problem in this paper these sublanguages are computed using a coordinator we recall the notion of conditional controllability introduce a notion of conditional normality and prove necessary and sufficient conditions where such a computation is possible specifically we show that conditionally controllable and conditionally normal languages computed by our method are controllable and normal with respect to the global plant the optimality supremality of the resulting languages is also discussed

measure theoretical properties of the unstable foliation of two dimensional differentiable area preserving systems this article analyzes in detail the statistical and measure theoretical properties of the nonuniform stationary measure referred to as the w invariant measure associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two dimensional differentiable area preserving systems the analysis is developed starting from a sequence of analytical approximations for the associated density these approximations are related to the properties of the jacobian matrix of the n mathrm th iteration of a poincar e map the w invariant measure plays a fundamental role in the study of transport phenomena in laminar chaotic fluid mixing systems for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids the w invariant measure turns out to be singular and exhibits multifractal features its associated density displays local self similarity in an ensuremath varepsilon neighborhood of hyperbolic periodic points the cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed the only case for which the w invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism the connections with the statistical properties and in particular with the stretching dynamics are addressed in detail

symmetric product measures binomial processes and invariant manifold intersections in dynamical systems abstract this article proposes the concept of symmetric product measures spm for addressing the geometric characterization of the distribution of the intersections of the stable and unstable manifolds in hyperbolic dynamical systems prototypical examples of spm associated with multiplicative binomial processes are addressed and the application to manifold characterization in non linear area preserving systems henon map is developed moreover the application of spm to the characterization of measure theoretical properties of the periodic orbit distribution in hyperbolic dynamical systems is addressed

on controllability and normality of discrete event dynamical systems abstract this paper studies the supervisor synthesis problem of discrete event dynamical systems deds s through the use of synchronous composition of the plant and the supervisor and discusses some of the simplifications achieved by using the synchronous composition to model supervisory control an alternate definition of controllability is presented this definition of controllability is then used to derive an algorithm that is computationally more efficient than previously existing ones for the construction of the supremal controllable sublanguage the algorithm is also shown to be optimal the observability and normality issues arising due to the partial observation of the system dynamics under an arbitrary mask are then investigated closed form representations of the supremal normal and supremal closd and normal sublanguages are derived in the more general setting of arbitrary mask and nonclosed languages thus extending earlier results of the authors and others

the computational complexity of decentralized discrete event control problems computational complexity results are obtained for decentralized discrete event control problems these results generalize the earlier work of tsitsiklis 1989 who showed that for a special class of centralized supervisory control problems under partial observation there is an algorithm for determining in polynomial time whether or not a solution exists the negative complexity results associated with tsitsiklis work also carry over to the decentralized case so that solution existence for the more general class is not decidable in polynomial time nor does there exist a polynomial time algorithm for producing supervisor solutions when such solutions exist

the dynamic and stochastic knapsack problem the dynamic and stochastic knapsack problem dskp is defined as follows items arrive according to a poisson process in time each item has a demand size for a limited resource the knapsack and an associated reward the resource requirements and rewards are jointly distributed according to a known probability distribution and become known at the time of the item s arrival items can be either accepted or rejected if an item is accepted the item s reward is received and if an item is rejected a penalty is paid the problem can be stopped at any time at which time a terminal value is received which may depend on the amount of resource remaining given the waiting cost and the time horizon of the problem the objective is to determine the optim al policy that maximizes the expected value rewards minus costs accumulated assuming that all items have equal sizes but random rewards optimal solutions are derived for a variety of cost structures and time horizons and recursive algorithms for computing them are developed optimal closed form solutions are obtained for special cases the dskp has applications in freight transportation in scheduling of batch processors in selling of assets and in selection of investment projects

the dynamic and stochastic knapsack problem with deadlines in this paper a dynamic and stochastic model of the well known knapsack problem is developed and analyzed the problem is motivated by a wide variety of real world applications objects of random weight and reward arrive according to a stochastic process in time the weights and rewards associated with the objects are distributed according to a known probability distribution each object can either be accepted to be loaded into the knapsack of known weight capacity or be rejected the objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward the optimal decision rules are derived and are shown to exhibit surprising behavior in some cases it is also shown that if the distribution of the weights is concave then the decision rules behave according to intuition

formulas for calculating supremal controllable and normal sublanguages abstract supremal controllable and normal sublanguages have been shown to play an important role in supervisor synthesis in this paper we discuss the computation of supremal controllable and normal sublanguages we derive formulas for both supremal controllable sublanguages and supremal normal sublanguages when the languages involved are closed as a result those languages can be computed without applying recursive algorithms we also discuss those aspects of these formulas