upper bounds in phase synchronous weak coherent chaotic attractors an approach is presented for coupled chaotic systems with weak coherent motion from which we estimate the upper bound value for the absolute phase difference in phase synchronous states this approach shows that synchronicity in phase implies synchronicity in the time of events a characteristic explored to derive an equation to detect phase synchronization based on the absolute difference between the time of these events we demonstrate the potential use of this approach for the phase coherent and the funnel attractor of the rossler system as well as for the spiking bursting rulkov map

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

general framework for phase synchronization through localized sets we present an approach which enables one to identify phase synchronization in coupled chaotic oscillators without having to explicitly measure the phase we show that if one defines a typical event in one oscillator and then observes another one whenever this event occurs these observations give rise to a localized set our result provides a general and easy way to identify ps which can also be used to oscillators that possess multiple time scales we illustrate our approach in networks of chemically coupled neurons we show that clusters of phase synchronous neurons may emerge before the onset of phase synchronization in the whole network producing a suitable environment for information exchanging furthermore we show the relation between the localized sets and the amount of information that coupled chaotic oscillator can exchange

transmission of information in active networks shannon s capacity theorem is the main concept behind the theory of communication it says that if the amount of information contained in a signal is smaller than the channel capacity of a physical media of communication it can be transmitted with arbitrarily small probability of error this theorem is usually applicable to ideal channels of communication in which the information to be transmitted does not alter the passive characteristics of the channel that basically tries to reproduce the source of information for an active channel a network formed by elements that are dynamical systems such as neurons chaotic or periodic oscillators it is unclear if such theorem is applicable once an active channel can adapt to the input of a signal altering its capacity to shed light into this matter we show among other results how to calculate the information capacity of an active channel of communication then we show that the channel capacity depends on whether the active channel is self excitable or not and that contrary to a current belief desynchronization can provide an environment in which large amounts of information can be transmitted in a channel that is self excitable an interesting case of a self excitable active channel is a network of electrically connected hindmarsh rose chaotic neurons

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

reduced order controller synthesis with regional pole constraint abstracta reduced order output feedback controller is designed for a linear time invariant system which guarantees that the closed loop poles are placed within some pre specified stability region in the complex plane a convex approximation of the non convex constraints is used to pose a sequence of semi definite programs which provide the lowest order proper controller satisfying the approximate constraints the proposed method is demonstrated on two practical controller design applications

partial pole placement with controller optimization an arbitrary subset n m of the n closed loop eigenvalues of an nth order continuous time single input linear time invariant system is to be placed using full state feedback at pre specified locations in the complex plane the remaining m closed loop eigenvalues can be placed anywhere inside a pre defined region in the complex plane this region constraint on the unspecified poles is translated into a linear matrix inequality constraint on the feedback gains through a convex inner approximation of the polynomial stability region the closed loop locations for these m eigenvalues are optimized to obtain a minimum norm feedback gain vector this reduces the controller effort leading to less expensive actuators required to be installed in the control system the proposed algorithm is illustrated on a linearized model of a 4 machine 2 area power system example

existence of global solutions and invariant measures for stochastic differential equations driven by poisson type noise with non lipschitz coefficients the purpose of this paper is twofold firstly we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi martingales secondly we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

globally convergent algorithms for robust pole assignment by state feedback it is observed that an algorithm proposed in 1985 by kautsky nichols and van dooren knv amounts to maximizing at each iteration the determinant of the candidate closed loop eigenvector matrix x with respect to one of its columns with unit length constraint subject to the constraint that it remains an achievable closed loop eigenvector matrix this interpretation is used to prove convergence of the knv algorithm it is then shown that a more efficient algorithm is obtained if det x is concurrently maximized with respect to two columns of x and such a scheme is easily extended to the case where the eigenvalues to be assigned include complex conjugate pairs variations exploiting the availability of multiple processors are suggested convergence properties of the proposed algorithms are established their superiority is demonstrated numerically

exponential mean square stability of the theta approximations for neutral stochastic differential delay equations in this paper a split step theta sst method is introduced and analyzed for neutral stochastic differential delay equations nsddes it is proved that the sst method with 0 1 2 can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient but for 1 2 1 the sst can reproduce the exponential mean square stability unconditionally then based on the stability results of sst scheme we examine the exponential mean square stability of the stochastic linear theta slt approximation for nsddes and obtain the similar stability results to that of the sst method moreover for sufficiently small stepsize we show that the decay rate as measured by the lyapunov exponent can be reproduced arbitrarily accurately these results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for nsddes

chaotic phase synchronization in bursting neuron models driven by a weak periodic force we investigate the entrainment of a neuron model exhibiting a chaotic spiking bursting behavior in response to a weak periodic force this model exhibits two types of oscillations with different characteristic time scales namely long and short time scales several types of phase synchronization are observed such as 1 1 phase locking between a single spike and one period of the force and 1 l phase locking between the period of slow oscillation underlying bursts and l periods of the force moreover spiking bursting oscillations with chaotic firing patterns can be synchronized with the periodic force such a type of phase synchronization is detected from the position of a set of points on a unit circle which is determined by the phase of the periodic force at each spiking time we show that this detection method is effective for a system with multiple time scales owing to the existence of both the short and the long time scales two characteristic phenomena are found around the transition point to chaotic phase synchronization one phenomenon shows that the average time interval between successive phase slips exhibits a power law scaling against the driving force strength and that the scaling exponent has an unsmooth dependence on the changes in the driving force strength the other phenomenon shows that kuramoto s order parameter before the transition exhibits stepwise behavior as a function of the driving force strength contrary to the smooth transition in a model with a single time scale

feedback norm minimisation with regional pole placement this article proposes a convex algorithm for minimising an upper bound of the state feedback gain matrix norm with regional pole placement for linear time invariant multi input systems the inherent non convexity in this optimisation is resolved by a combination of two separate approaches 1 an inner convex approximation of the polynomial matrix stability region due to henrion and 2 a novel convex parameterisation of column reduced matrix fraction system representations using a sequence of approximations enabled by the above two methods it is shown that the constraints on closed loop poles both pre specified exact locations and regional placement define linear matrix inequalities finally the effectiveness of the proposed algorithm is compared with similar pole placement algorithms through numerical examples

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

low order controller design for siso systems using coprime factors and lmi this paper develops a low order controller design method for linear continuous time invariant single input single output systems requiring only the solution of a convex optimization problem the technique integrates several well known results in control theory an important step is the use of coprime factors so that based on strictly positive real functions feedback stabilization using low order controllers becomes a zero placement problem which is convex from this result we develop algorithms to solve two optimal control problems

strong convergence of compensated split step theta methods for sdes with jumps under monotone condition abstract this paper is concerned with strong convergence of the compensated split step theta csst method for autonomous stochastic differential equations sdes with jumps under weaker conditions more precisely the diffusion coefficient and the drift coefficient are both locally lipschitz and the jump diffusion coefficient is globally lipschitz while they all satisfy the monotone condition it is proved that the csst method is strongly convergent of order 1 2 finally the obtained results are supported by numerical experiments

non transitive maps in phase synchronization abstract concepts from ergodic theory are used to describe the existence of special non transitive maps in attractors of phase synchronous chaotic oscillators in particular it is shown that for a class of phase coherent oscillators these special maps imply phase synchronization we illustrate these ideas in the sinusoidally forced chua s circuit and two coupled rossler oscillators furthermore these results are extended to other coupled chaotic systems in addition a phase for a chaotic attractor is defined from the tangent vector of the flow finally it is discussed how these maps can be used for the real time detection of phase synchronization in experimental systems

robust exact pole placement via an lmi based algorithm this technical note deals with the robust exact pole placement problem pole placement algorithms that guarantee a small variation of the assigned poles against possible perturbations the solution to this problem is related to the solvability of a sylvester like equation thus the main issue is to compute a well conditioned solution to this equation also the best candidate solution must possess the minimal condition number to reduce sensitivity to perturbation this problem is cast as a global optimization under linear matrix inequality constraints for which a numerical convergent algorithm is provided and compared with the most attractive methods in the literature