complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

monopoles and rational maps a note on a theorem of donaldson in a recent paper donaldson gave a description of the moduli space of su 2 monopoles in terms of rational maps this was done indirectly via the associated solution of nahm s equations we give here an interpretation of these rational maps in terms of the monopole s spectral curve and then as scattering data for the monopole itself

reachability analysis of dynamical systems having piecewise constant derivatives abstract in this paper we consider a class of hybrid systems namely dynamical systems with piecewise constant derivatives pcd systems such systems consist of a partition of the euclidean space into a finite set of polyhedral sets regions within each region the dynamics is defined by a constant vector field hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction with respect to such systems we investigate the reachability question given an effective description of the systems and of two polyhedral subsets p and q of the state space is there a trajectory starting at some x p and reaching some point in q our main results are a decision procedure for two dimensional systems and an undecidability result for three or more dimensions

slow scale split step tau leap method for stiff stochastic chemical systems abstract tau leaping is a family of algorithms for the approximate simulation of the discrete state continuous time markov chains a motivation for the development of such methods can be found for instance in the fields of chemical kinetics and systems biology it is known that the dynamical behavior of biochemical systems is often intrinsically stiff representing a serious challenge for their numerical approximation the naive extension of stiff deterministic solvers to stochastic integration often yields numerical solutions with either impractically large relaxation times or incorrectly resolved covariance in this paper we propose a splitting heuristic which helps to resolve some of these issues the proposed integrator contains a number of unknown parameters which are estimated for each particular problem from the moment equations of the corresponding linearized system we show that this method is able to reproduce the exact mean and variance of the linear scalar test equation and demonstrates a good accuracy for the arbitrarily stiff systems at least in the linear case the numerical examples for both linear and nonlinear systems are also provided and the obtained results confirm the efficiency of the considered splitting approach

a new conceptual framework for analog computation in this paper we show how to explore the classical theory of computability using the tools of analysis a differential scheme is substituted for the classical recurrence scheme and a limit operator is substituted for the classical minimization we show that most relevant problems of computability over the non negative integers can be dealt with over the reals elementary functions are computable turing machines can be simulated the hierarchy of non computable functions can be represented the classical halting problem being solvable at some level the most typical concepts in analysis become natural in this framework the most relevant question is posed can we solve open problems of classical computability and computational complexity using as popper says the toolbox of analysis

min max theory for the yang mills higgs equations in each monopole sector there exist an infinite number of finite energy solutions to the prasad sommerfield limit of the su 2 yang mills higgs equations on 3 whose energy is greater than any finite number

differential automata and their discrete simulators on presente la notion d automate differentiel pour la formulation d une classe de problemes aux valeurs initiales sur des champs vectoriels hysteretiques

monopoles and their spectral data a new definition of spectral data of a monopole is given for any compact lie or kac moody group it is shown that the spectral data determines the irreducible monopole in the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the dynkin diagram of the group the structure of solutions to nahm s equations corresponding to the monopole is discussed

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

the classification of monopoles for the classical groups by studying a construction of nahm we compute the moduli spaces of monopoles with maximal symmetry breaking at infinity forsu n so n andsp n these are found to be equivalent to spaces of holomorphic maps from 1 into flag manifolds

the existence of multi monopole solutions to the non abelian yang mills higgs equations for arbitrary simple gauge groups we prove that for arbitrary simple gauge groups the non abelian yang mills higgs equations on 3 in the prasad sommerfield limit have at least a countably infinite set of distinct solutions these solutions may be interpreted physically as configurations of widely spaced non interacting fundamental monopoles the solutions are generically not spherically symmetric

the asymptotic higgs field of a monopole a simple formula is computed for the asymptotic higgs field of an su 2 monopole this formula is derived from the twistor description of monopoles and is applied to the study of boundary behaviour it is found to be harmonic and to have as its natural domain of definition a branched covering of 3 explicit formulae are given in simple cases

on the construction of monopoles for the classical groups forg a classical group an equivalence is exhibited between a g monopoles over 3 with maximal symmetry breaking at infinity b families of rank g algebraic curves int 1 along with divisors on those curves satisfying certain constraints c solutions of nahm s equations over rank g intervals satisfying the appropriate boundary conditions a and b are linked by twistor techniques b and c via the krichever method for solving non linear differential equations and a and c via the adhmn construction providing a unified picture of techniques for solution amongst other things an asymptotic formula for the higgs field of the monopole is computed

achilles and the tortoise climbing up the hyper arithmetical hierarchy abstract in this paper we characterize the computational power of dynamical systems with piecewise constant derivatives pcd considered as computational machines working on a continuous real space with a continuous real time we prove that piecewise constant derivative systems recognize precisely the languages of the kth respectively k 1 th level of the hyper arithmetical hierarchy in dimension d 2k 3 respectively d 2k 4 k 0 hence we prove that the reachability problem for pcd systems of dimension d 2k 3 resp d 2k 4 k 1 is hyper arithmetical and is k complete resp k complete

on the construction of monopoles we show that any self dual su 2 monopole may be constructed either by ward s twistor method or nahm s use of the adhm construction the common factor in both approaches is an algebraic curve whose jacobian is used to linearize the non linear ordinary differential equations which arise in nahm s method we derive the non singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions

spectral curves and the adhm method for a general monopole the algebraic curves defined by nahm are shown to be the same as the spectral curves

universal computation and other capabilities of hybrid and continuous dynamical systems we explore the simulation and computational capabilities of hybrid and continuous dynamical systems the continuous dynamical systems considered are ordinary differential equations odes for hybrid systems we concentrate on models that combine odes and discrete dynamics e g finite automata we review and compare four such models from the literature notions of simulation of a discrete dynamical system by a continuous one are developed we show that hybrid systems whose equations can describe a precise binary timing pulse exact clock can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of r n the simulations require continuous odes in ir2n with the exact clock as input all four hybrid systems models studied here can implement exact clocks we also prove that any discrete dynamical system in rn can be simulated by continuous odes in rt2n 1 we use this to show that smooth odes in ri3 can simulate arbitrary turing machines and hence possess the power of universal computation we use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems we prove that one cannot build an arbiter with devices described by a system of lipschitz odes on the other hand all four hybrid systems models considered can implement arbiters even if their odes are lipschitz

continuous time computation with restricted integration capabilities recursion theory on the reals the analog counterpart of recursive function theory is an approach to continuous time computation inspired by the models of classical physics in recursion theory on the reals the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals linear differential equations and more general cauchy problems we define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity we prove both upper and lower bounds for several classes of real recursive functions which lie inside the elementary functions and can be characterized in terms of space complexity in particular we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy the results in this paper combined with earlier results suggest that there is a close connection between analog complexity classes and subrecursive classes at least in the region between flinspace and the primitive recursive functions

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

characterization of well posedness of piecewise linear systems one of the basic issues in the study of hybrid systems is the well posedness existence and uniqueness of solutions problem of discontinuous dynamical systems the paper addresses this problem for a class of piecewise linear discontinuous systems under the definition of solutions of caratheodory the concepts of jump solutions or of sliding modes are not considered here in this sense the problem to be discussed is one of the most basic problems in the study of well posedness for discontinuous dynamical systems first we derive necessary and sufficient conditions for bimodal systems to be well posed in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions next its extensions to the multimodal case are discussed as an application to switching control in the case that two state feedback gains are switched according to a criterion depending on the state we give a characterization of all admissible state feedback gains for which the closed loop system remains well posed

low dimensional hybrid systems decidable undecidable don t know even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems the affair is far from being resolved more and more low dimensional systems are being shown to be undecidable with respect to reachability and many open problems in between are being discovered in this paper we present various two dimensional hybrid systems for which the reachability problem is undecidable we show their undecidability by simulating minsky machines their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable we also show that for other two dimensional systems the reachability question remains unanswered by proving that it is as hard as the reachability problem for piecewise affine maps on the real line which is a well known open problem

computational bounds on polynomial differential equations in this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations we consider elementary in the sense of analysis discrete time dynamical systems satisfying certain criteria of robustness we show that those systems can be simulated with elementary and robust continuous time dynamical systems which can be expanded into fully polynomial ordinary differential equations in q p this sets a computational lower bound on polynomial odes since the former class is large enough to include the dynamics of arbitrary turing machines we also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial value problem defined with polynomial odes is bounded or not is in general undecidable even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56 combined with earlier results on the computability of solutions of polynomial odes one can conclude that there is from a computational point of view a close connection between these systems and turing machines

nahm s equations and the classification of monopoles solutions of nahm s system of ordinary differential equations are produced by variational methods this leads to an explicit parametrisation of the solutions to the bogomolny equation over 3