quantum state diffusion localization and quantum dispersion entropy the quantum state diffusion model introduced in an earlier paper represents the evolution of an individual open quantum system by an ito diffusion equation for its quantum state the diffusion and drift terms in this equation are derived from interaction with the environment in this paper two localization theorems are proved the dispersion entropy theorem shows that under special conditions which are commonly satisfied to a good approximation the mean quantum dispersion entropy which measures the mean dispersion or delocalization of the quantum states decreases at a rate equal to a weighted sum of effective interaction rates so that the localization always increases in the mean except when the effective interaction with the environment is zero the general localization theorem provides a formula for more general conditions

a kinetic theory for quantum transport in aperiodic media transport theory is reinvestigated in a one particle model including dissipation mechanisms through a phenomenological time dependent part using the description of an aperiodic crystal in terms of noncommutative geometry it is then possible to rigorously derive kubo formulas for the electric conductivity within the framework of the relaxation time approximation a fractal analysis applied to the spectral measures entering in the kubo formula allows us to examine anomalous transport owing to quantum interferences this leads to anomalies in drude s formula

on the quantum dissipative generator weak coupling approximation and stochastic approach for a quantum open system the so called schr dinger langevin picture has been revisited in a second order perturbation it is shown that a non markovian evolution for the stochastic state vector leads to a dissipative generator which has a kossakowski lindblad form in this context it is possible to analyse the completely positive condition the equivalence of this picture with the trace out technique in the weak coupling approximation has been proved

path integral semiclassical and stochastic propagators for markovian open quantum systems we develop the path integral theory for master equations of general lindblad form positive semigroups describing markovian open quantum systems first the hamiltonian path integral expression for the propagator is derived which exhibits nicely the decoherence of pairs of phase space histories a very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional for the important class of effective hamiltonians quadratic in the momenta we derive the lagrangian version of the path integral propagator we then evaluate the path integral approximately in a stationary phase approximation leading to a van vleck type propagator valid under semiclassical conditions we also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals finally connections to the active field of stochastic pure state descriptions of open quantum systems are established here in particular to linear quantum state diffusion

additive resonances of a controlled van der pol duffing oscillator abstract the trivial equilibrium of a controlled van der pol duffing oscillator with nonlinear feedback control may lose its stability via a non resonant interaction of two hopf bifurcations when two critical time delays corresponding to two hopf bifurcations have the same value such an interaction results in a non resonant bifurcation of co dimension two in the vicinity of the non resonant hopf bifurcations the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response when the frequency of the external excitation and the frequencies of the two hopf bifurcations satisfy a certain relationship with the aid of centre manifold theorem and the method of multiple scales three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four dimensional ordinary differential equations on the centre manifold the amplitudes of the free oscillation terms are found to admit three solutions two non trivial solutions and the trivial solution of two non trivial solutions one is stable and the trivial solution is unstable a stable non trivial solution corresponds to a quasi periodic motion of the original system it is also found that the frequency response curves for three cases of additive resonances are an isolated closed curve it is shown that the forced response of the oscillator may exhibit quasi periodic motions on a three dimensional torus consisting of three frequencies the frequencies of two bifurcating solutions and the frequency of the excitation illustrative examples are given to show the quasi periodic motions

the quantum state diffusion model applied to open systems a model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation derived from a density operator equation such as the bloch equation and consistent with it the advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described the method is applied to the nonlinear absorber cascades of quantum transitions second harmonic generation and a measurement reduction process the model provides graphic illustrations of these processes with statistical fluctuations that mimic those of experiments the stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics the model is compared with the quantum jump model of dalibard 1992 carmichael and others which originated among experimenters looking for intuitive pictures and rules of computation

controlling chaos in high dimensions theory and experiment naval surface warfare center carderock division silver spring maryland 20903 received 27 november 1995 the main contribution of this work is the development of a high dimensional chaos control method that iseffective robust against noise and easy to implement in experiment assuming no knowledge of the modelequations the method achieves control by stabilizing a desired unstable periodic orbit with any number ofunstable directions using small time dependent perturbations of a single system parameter speci cally ourmajor results are as follows first we derive explicit control laws for time series produced by discrete maps second we show how to apply this control law to continuous time problems by introducing straightforwardways to extract from a continuous time series a discrete time series that measures the dynamics of somepoincare map of the original system third we illustrate our approach with two examples of high dimensionalordinary differential equations one autonomous and the other periodically driven fourth we present the resulton our successful control of chaos in a high dimensional experimental system demonstrating the viability ofthe method in practical applications pacs number s 05 45 1bi introduction

synchronization via active control of identical and non identical φ6 chaotic oscillators with external excitation abstract using the active control technique with lyapunov stability theory and the routh hurwitz criteria control functions are designed to achieve complete synchronization between two identical 6 van der pol oscillators 6 vdpos two identical 6 duffing oscillators 6 dos and two non identical 6 oscillators comprising 6 vdpo and 6 do for the triple well configuration of the 6 potential the coefficient matrix of the error dynamics between each pair of synchronized systems is chosen such that the number of active control functions reduces from two to one thereby significantly reducing controller complexity in the design the designed controllers enable the state variables of the response system to synchronize with those of the drive system in both the identical and non identical cases the results are validated using numerical simulations

the coexistence of periodic almost periodic and chaotic attractors in the van der pol duffing oscillator in this paper are presented the computer simulation and the approximate theoretical analysis of the behaviour of the van der pol duffing forced oscillator at the passage through principal resonance at increasing and decreasing driving frequency almost periodic oscillations frequency locking transition to chaotic motion and the jump from the non resonant to the resonant state are observed and interpreted in the light of approximate analytical theory and by the use of topological concepts

scattering theory for the belavkin equation describing a quantum particle with continuously observed coordinate in the article the complete investigation is given of the large times behavior of the solutions of the belavkin quantum filtering equation describing a quantum particle with continuously observed coordinate it turns out that these solutions have an extraordinary property namely as t any solution will tend to a gaussian function for almost all realizations of innovating wiener process as a consequence it follows that the dispersion of coordinate will always tend to the same limit as t not depending on initial data

series parallel languages and the bounded width property abstract in order to model concurrency we extend automata theory from the usual languages of words to sets of labelled series parallel posets or sp languages we use the algebraic properties of the set of series parallel posets to investigate the notion of recognizability based on finite index congruences and a notion of regularity based on a new class of branching automata the main results of the paper concern the sp languages having bounded width we show that such languages are recognizable if and only if they are regular in addition the regular bounded width sp languages are exactly those which can be described by series rational expressions that is expressions using the letters the operations of sequential and parallel product and sequential iterations we also give a strictly algebraic characterization of the bounded width property in terms of depth nilpotent series parallel algebras

van der pol s oscillator under delayed feedback the effect of delayed feedback on oscillatory behaviour is investigated for the classical van der pol oscillator it is shown how the presence of delay can change the amplitude of limit cycle oscillations or suppress them altogether the result is compared to the conventional proportional and derivative feedback the derivative like effect of delay is also demonstrated in a modified equation where a delayed term provides the damping

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

on the time it takes a state vector to reduce it is pointed out in theories which give a dynamical description of the reduction of the state vector that the reduction should take place in a finite time it is shown that the reduction time is infinite in the bohm bub theory and finite in the author s theory

on the optimality of the ott grebogi yorke control scheme abstract some of the characteristics of the ott grebogi yorke ogy control technique are presented as applied to nonlinear flows as distinct from nonlinear maps specifically we consider the case where the magnitude of the control parameter varies in time within each control cycle in proportion to a given function referred to as a basis function the choice of the basis function is shown to influence the basin of convergence for a given level of parameter variation in the ogy controller an algorithm for designing the optimal basis function is presented the optimal basis function is shown to be defined by a step function with potentially several jumps thus revealing the intrinsic power of a standard ogy technique that uses a single step function as a basis function two numerical applications of the optimal design technique to a duffing oscillator are also presented to show that the standard ogy technique may be significantly improved by making an optimal choice of a basis function copyright

stability and bifurcation analysis in van der pol s oscillator with delayed feedback abstract the classical van der pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered linear stability of the equations is investigated by analyzing the associated characteristic equations it is found that there exist the stability switches when delay varies and the hopf bifurcation occurs when the delay passes through a sequence of critical values the bifurcation diagram is drawn in e k plane and the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem

continuous quantum jumps and infinite dimensional stochastic equations from a mathematical point of view a class of infinite dimensional stochastic differential equations describing continuous spontaneous localization in quantum dynamics will be studied existence and uniqueness of weak and strong solutions of respective equations are proven via cameron martin girsanov transformation the case of gaussian initial states is explicitly solved

a perturbation method for computing the simplest normal forms of dynamical systems abstract a previously developed perturbation method is generalized for computing the simplest normal form at each level of computation the minimum number of terms are retained of general n dimensional differential equations this direct approach combining the normal form theory with center manifold theory in one unified procedure can be used to systematically compute the simplest or unique normal form two particular singularities of the jacobian of the system are considered in this paper the first one is associated with one pair of purely imaginary eigenvalues hopf type singularity and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues hopf zero type singularity the approach can be easily formulated and implemented using a computer algebra system maple programs have been developed in this paper which can be automatically executed by a user without the knowledge of computer algebra a physical oscillator model is studied in detail to show the computational efficiency of the direct method and the advantage of using the simplest normal form which greatly simplifies the analysis on dynamical systems in particular for bifurcations and stability

bifurcation and chaos in the double well duffing van der pol oscillator numerical and analytical studies the behavior of a driven double well duffing char21 van der pol oscillator for a specific parametric choice ensuremath alpha ensuremath beta is studied the existence of different attractors in the system parameters f ensuremath omega domain is examined and a detailed account of various steady states for fixed damping is presented the transition from quasiperiodic to periodic motion through chaotic oscillations is reported the intervening chaotic regime is further shown to possess islands of phase locked states and periodic windows including period doubling regions boundary crisis three classes of intermittencies and transient chaos we also observe the existence of local global bifurcation of intermittent catastrophe type and global bifurcation of blue sky catastrophe type during the transition from quasiperiodic to periodic solutions using a perturbative periodic solution an investigation of the various forms of instabilities allows one to predict neimark instability in the f ensuremath omega plane and eventually results in the approximate predictive criteria for the chaotic region

large time behavior and convergence rate for quantum filters under standard non demolition conditions a quantum system mathcal s undergoing continuous time measurement is usually described by a jump diffusion stochastic differential equation such an equation is called a quantum filtering equation or quantum stochastic master equation and its solution is called a quantum filter or quantum trajectory this solution describes the evolution of the state of mathcal s in the context of quantum non demolition measurement we investigate the large time behavior of this solution it is rigorously shown that for large time this solution behaves as if a direct von neumann measurement has been performed at time 0 in particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate using the theory of girsanov transformation we obtain the explicit rate of convergence towards this random state the problem of state estimation used in experiment is also investigated

a continuous counting observation and posterior quantum dynamics a stochastic model for a continuous photon counting measurement is proposed an equation for the generating functional of the statistics of counts is found a stochastic linear equation for the unnormalised posterior state vector of atoms continuously collapsing without mixing is derived by using quantum stochastic calculus methods

hilbert space path integral representation for the reduced dynamics of matter in thermal radiation fields a hilbert space path integral for the dissipative dynamics of matter in thermal radiation fields is derived from the hamiltonian of quantum electrodynamics this path integral represents the conditional transition probability of a stochastic markov process as a sum over sample trajectories in hilbert space the realizations of the process are piecewise deterministic paths broken by instantaneous quantum jumps it is shown that the operators which define the possible quantum jumps form a continuous family parametrized by the polarization vector of the emitted or absorbed photons the stochastic process is shown to be representation independent and to be invariant with respect to space rotations the precise physical interpretation of the stochastic process is given in particular the expansion of the density matrix in terms of quantum jumps is derived for finite temperatures from the hilbert space path integral

jump diffusion unravelling of a non markovian generalized lindblad master equation the correlated projection technique has been successfully applied to derive a large class of highly non markovian dynamics the so called non markovian generalized lindblad type equations or lindblad rate equations in this article general unravelings are presented for these equations described in terms of jump diffusion stochastic differential equations for wave functions we show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions the main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not such restrictions can be seen as the effect of some kind of superselection rule finally we develop a concrete example and discuss possible effects on the heterodyne spectrum of a two level system due to a structured thermal like bath with memory

system identification for the ott grebogi yorke controller design a technique to compute the linearized poincar e map and the sensitivity vector required for the implementation of an ott grebogi yorke ogy controller from experimental data is presented unlike previous methods the linearized map and the sensitivity vector are computed using only data collected over a single period of the limit cycle the main advantage of the present method is the fact that it eliminates the long waiting period required to design an ogy controller using previously known methods numerical examples are presented using the duffing oscillator with random perturbations to simulate noisy experimental data

combined analytical and numerical approaches in dynamic stability analyses of engineering systems abstract dynamic stability is a widely studied area that has attracted many researchers from various disciplines although dynamic stability is usually associated with mechanics theoretical physics or other natural and technical disciplines it is also relevant to social economic and philosophical areas of our lives therefore it is useful to occasionally highlight the general aspects of this amazing area to present some relevant examples and to evaluate its position among the various branches of rational mechanics from this perspective the aim of this study is to present a brief review concerning the dynamic stability problem its basic definitions and principles important phenomena research motivations and applications in engineering the relationships with relevant systems that are prone to stability loss encountered in other areas such as physics other natural sciences and engineering are also noted the theoretical background which is applicable to many disciplines is presented in this paper the most frequently used dynamic stability analysis methods are presented in relation to individual dynamic systems that are widely discussed in various engineering branches in particular the lyapunov function and exponent procedures routh hurwitz lienard and other theorems are outlined together with demonstrations the possibilities for analytical and numerical procedures are mentioned together with possible feedback from experimental research and testing the strengths and shortcomings of these approaches are evaluated together with examples of their effective complementing of each other the systems that are widely encountered in engineering are presented in the form of mathematical models the analyses of their dynamic stability and post critical behaviour are also presented the stability limits bifurcation points quasi periodic response processes and chaotic regimes are discussed the limit cycle existence and stability are examined together with their separating roles as attractors and repulsers two levels of stability loss recovery of the system is possible or final collapse is inevitable as can be observed in softening systems are noted time limited excitation and relevant transition effects e g seismic excitation are also discussed together with the evaluation of possible system reliability improvement the dynamic stability investigation of two degrees of freedom aero elastic systems in a linear formulation using several approaches is briefly highlighted further systems modelling problems that arise in transport engineering are also outlined a few hints for applications are given some open problems and possible future research strategies are outlined

analytical conditions for strange chaotic and nonchaotic attractors of the quasiperiodically forced van der pol equation we propose an analytical method for predicting the boundaries of the region in parameter space within which a strange nonchaotic attractor exists for a quasiperiodically forced van der pol equation one boundary is obtained from the condition for existence of a simple quasiperiodic solution to the equation the other is shown to be well approximated by the condition for hopf bifurcation in a suitably averaged system of equation good agreement with numerical calculations of lyapunov exponents is demonstrated

stability and hopf bifurcation of a magnetic bearing system with time delays the effect of time delays occurring in the feedback control loop on the linear stability of a simple magnetic bearing system is investigated by analyzing the associated characteristic transcendental equation it is found that a hopf bifurcation can take place when time delays pass certain values the direction and stability of the hopf bifurcation are determined by constructing a center manifold and by applying the normal form method it is also found that a codimension two bifurcation can occur through a hopf and a steady state bifurcation interaction finally numerical simulations are performed to verify the analytical predictions

two families of super harmonic resonances in a time delayed nonlinear oscillator abstract two stable bifurcating periodic solutions are numerically found to coexist in a time delayed nonlinear oscillator by using different initial conditions after the trivial equilibrium loses its stability via two to one resonant hopf bifurcations these two coexisting solutions have different amplitudes and frequency components with one having the frequencies of hopf bifurcations while the other containing different frequencies from those of hopf bifurcations the dynamic interaction of the periodic excitation and the two coexisting solutions can induce two families of super harmonic resonances when the forcing frequency is approximately at half the lower frequency component of the stable bifurcating solutions it is found that the forced response under two families of super harmonic resonances exhibits qualitatively different dynamic behaviour in addition one family of super harmonic resonances may suddenly disappear when the excitation magnitude reaches a certain value and then the forced response becomes non resonant response the other family of super harmonic resonances can be established by adjusting the forcing frequency accordingly time trajectories phase portraits frequency spectra basin of attraction and bifurcation diagrams are given to characterise the different dynamic behaviours of the time delayed nonlinear oscillator

quantum stochastic calculus operation valued stochastic processes and continual measurements in quantum mechanics the physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process ovsp in this article it is shown how the formalism of quantum stochastic calculus of hudson and parthasarathy allows in a simple way for constructing a large class of ovsp s that in particular contains the quantum counting processes of davies and srinivas and continual gaussian measurements this result is obtained by means of a stochastic dilation of the ovsp s at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time dependent commuting self adjoint operators

non resonant double hopf bifurcations the complex case the analysis of the unfolding of a non resonant double hopf singularity is considered using the frequency domain and the normal form methodologies a higher order harmonic balance and the evaluation of floquet multipliers are used to obtain accurate approximations of the limit cycles and its local bifurcations this type of hybrid methodology using harmonic balance and normal forms gives a complementary view in the vicinity of the singularity more specifically some of the bifurcation curves arising in the unfolding of the double hopf bifurcation are computed with great accuracy using the harmonic balance method compared to the classical normal form however this last method is able to predict the appearance of complex quasiperiodic behavior such as three dimensional 3d torus numerical simulations corroborate this observation

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

two equational theories of partial words abstract partial words are in our terminology isomorphism classes of labeled partial orders we consider two structures starting with the partial orders reduced to a unique element the first one is defined as the closure under the three operations of series product parallel product and power series product of the same partial word and the second one as the closure under the three operations of series product parallel product and product series product of possibly different partial words we show that the two equational theories can be axiomatized by an infinite collection of simple identities and that no finite axiomatization exists we characterize the free algebras in the corresponding varieties as algebras of certain partial words subject to order and graph theoretic conditions we also show that the first equational theory is decidable

non markovian quantum trajectories instruments and time continuous measurements the linear and nonlinear non markovian quantum state diffusion nmqsd equations are well known tools for the description of certain non markovian open quantum systems in this work we systematically investigate whether the normalized linear nmqsd or the nonlinear nmqsd solutions can be generated by means of a time continuous measurement by considering any conceivable measurement scheme we derive a necessary criterion for the measurement interpretation of both equations concrete examples show that the normalized linear nmqsd solutions are realizable only in the markovian limit in general the application of the presented criterion to the nonlinear nmqsd remains an open issue furthermore this criterion is a necessary one not only in the context of the nmqsds but also for a much broader class of stochastic schrodinger equations

series parallel languages on scattered and countable posets in this paper we investigate the recognition by finite automata of languages of countable labelled posets we unify and generalize several previous results from two different directions the theory of finite or wn free posets and automata over countable and scattered linear orderings first we establish that the smallest class of posets obtained from the empty set and the singleton and closed under finite parallel operation and sequential concatenation indexed by all linear orderings corresponds precisely to the class of scattered and countable n free posets without infinite antichains next we prove a kleene like theorem we define automata and rational expressions for languages of countable scattered n free labelled posets without infinite antichains and show that both formalisms have the same expressive power

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

advancement of estimation fidelity in continuous quantum measurement we estimate an unknown qubit from the long sequence of n random polarization measurements of precision using the standard ito stochastic equations of the a posteriori state in the continuous measurement limit we calculate the advancement of fidelity we show that the standard optimum value 2 3 is achieved asymptotically for n 2 96 1 we append a brief derivation of novel ito equations for the estimate state

optimal multi dimensional ogy controller abstract a technique which generalizes the standard ogy control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented the proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with ogy type controllers an optimal ogy control scheme designed for larger dimensional systems having improved convergence properties compared to the standard ogy controller is also presented numerical examples for the well known van der pol duffing oscillator are presented to illustrate the proposed control schemes

the equational theory of pomsets abstract pomsets have been introduced as a model of concurrency since a pomset is a string in which the total order has been relaxed to be a partial order in this paper we view them as a generalization of strings and investigate their algebraic properties in particular we investigate the axiomatic properties of pomsets sets of pomsets and ideals of pomsets under such operations as concatenation parallel composition union and their associated closure operations we find that the equational theory of sets pomsets under concatenation parallel composition and union is finitely axiomatizable whereas the theory of languages under the analogous operations is not a similar result is obtained for ideals of pomsets which incorporate the notion of subsumption which is also known as augmentation finally we show that the addition of any closure operation parallel or serial leads to nonfinite axiomatizability of the resulting equational theory

higher dimensional targeting this paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations such a procedure is called targeting previous work on targeting for chaotic attractors has been in the context of one and two dimensional maps here it is shown that targeting can also be done in higher dimensional cases the method is demonstrated with a mechanical system described by a four dimensional mapping whose attractor has two positive lyapunov exponents and a lyapunov dimension of 2 8 the target is reached by making very small successive changes in a single control parameter in one typical case 35 iterates on average are required to reach a target region of diameter 10 sup minus 4 as compared to roughly 10 sup 11 iterates without the use of the targeting procedure

steady state self oscillations and chaotic behavior of a controlled electromechanical device by using the first lyapunov value and the melnikov theory abstract in this paper regular and chaotic oscillations in a controlled electromechanical transducer are investigated the nonlinear control laws are defined by an electric tension excitation and an external force applied to the mobile piece of the transducer the paper shows that an andronov poincare hopf bifurcation appears as long as adequate parameters are chosen for the nonlinear control laws the stability of the weak focuses associated to such bifurcation is examined according to the sign of the first lyapunov value showing that chaotic behavior can arise when the first lyapunov value is varied harmonically the appearance of a homoclinic orbit is investigated assuming an approximated model for the device on the basis of the parametric equations of the homoclinic orbit and the presence of harmonic disturbances on the platform it is demonstrated that chaotic oscillations can also appear and they have been examined by means of the melnikov theory chaotic behavior is corroborated by means of the sensitive dependence lyapunov exponents and power spectral density and it is applied to drive the transducer mobile piece to a predefined set point assuming that noise due to the measurement process can appear in the control signals the steady state error associated to such random noise is eliminated by adding a pi linear controller to the control force numerical simulations are used to corroborate the analytical results

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

oscillation sliding in a modified van der pol duffing electronic oscillator abstract some periodic behaviour exhibited by a modified van der pol duffing electronic oscillator near a degenerate hopf pitchfork bifurcation have been studied numerical continuation of these periodic orbits leads to the appearance of interesting phenomena after varying one of the characteristics of the oscillator oscillation sliding between two periodic regimes is detected in the region where oscillation sliding is present quasiperiodic oscillations invariant torus breakdown of the torus and the corresponding resonant periodic orbits are also found

robust stabilization to non linear delayed systems via delayed state feedback the averaging method the problem of robust stabilization to dynamical systems via delayed feedback is important in applications due to the fact that the characteristic quasi polynomial of a delayed system is difficult to analyze when uncertain parameters are involved this problem has been most frequently solved on the basis of the method of lyapunov functional by solving riccati equations or by solving linear matrix inequalities by applying the averaging method that reduces the delay differential equation of infinite dimensional to an ordinary differential equation this paper presents a simple method to stabilize the trivial solution or periodic solutions of a type of non linear delayed vibration systems via delayed state feedback in particular this method is applied to the robust stabilization of those systems when the system parameters are uncertain but fall into given intervals respectively in addition an extension is made to this problem for a general class of delayed systems that result from a small perturbation of a linear delay system with characteristic roots of non positive real parts only this can serve as a straightforward application to the hopf bifurcation control of delayed systems with weak non linearity

parametrization and optimization of gaussian non markovian unravelings for open quantum dynamics we derive a family of gaussian non markovian stochastic schr odinger equations for the dynamics of open quantum systems the different unravelings correspond to different choices of squeezed coherent states reflecting different measurement schemes on the environment consequently we are able to give a single shot measurement interpretation for the stochastic states and microscopic expressions for the noise correlations of the gaussian process by construction the reduced dynamics of the open system does not depend on the squeezing parameters they determine the non hermitian gaussian correlation a wide range of which are compatible with the markov limit we demonstrate the versatility of our results for quantum information tasks in the non markovian regime in particular by optimizing the squeezing parameters we can tailor unravelings for improving entanglement bounds or for environment assisted entanglement protection

long words the theory of concatenation and o power it is shown that for any set a the algebra of ordinal words on the alphabet a equipped with the operations of concatenation and power is axiomatized by the equationsx y z x y z x y x y x xn x n 1 indeed the algebra freely generated by a in the variety determined by these equations is the algebra of tail finite ordinal words of length on the alphabet a it is further shown that this collection of identities cannot be replaced by any finite set last a polynomial algorithm is given for recognizing when two terms denote the same tail finite ordinal word

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

diffusion approximation for a class of transport processes with physical reflection boundary conditions consider a stochastic process consisting of the pair of a position and a velocity in a piecewise el d dimensional domain in the interior of the domain the dynamics are assigned by a potential and by random changes of the velocity occurring at exponentially distributed times according to a probability distribution which may depend on the current position and velocity on the boundary the process reflects physically the angle of reflection equals the angle of incidence first it is shown that the process is well defined for all times then when the coefficients depend on a diverging parameter n in particular such that the speed and the jump rate of the velocity go to xo with order rn and at least n respectively a diffusion approximation is sought the position process is represented as a solution of a skorohod reflection equation a skewing effect on the boundary results from the interaction between the dynamics and the reflection law so that the direction of reflection is in general oblique the assumption that the mean change of the velocity in the interior is linear in the current velocity up to order at least 1 2 in 1 n ensures that the cone of directions of reflection is independent of n the continuity properties of the skorohod oblique reflection problem enable one to show tightness of the position processes without having to estimate explicitly local times boundary layers and together with a suitable law of large numbers for the velocity allow one to identify the limit stochastic differential equation with oblique reflection the theory is illustrated by several applications in particular one to a mechanical model of brownian motion

finite time stochastic reduction models a new energy based stochastic extension of the schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed an exact closed form solution to the dynamical equation valid for all finite dimensional quantum systems is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs a time change technique is introduced to construct a clock variable that relates the asymptotic and the finite time collapse models by means of a nonlinear transformation

calderia leggett master equation and medium temperatures abstract the markovian approximation of quantum dissipation has been reconsidered in the model of caldeira and leggett physica a 121 1983 587 their high temperature master equation has been generalized to medium temperatures two additional damping terms in the master equation have been derived and shown to assure the lindblad form of the equation a transient term in the density matrix has been found and expressed in a simple form

diffusion approximation of stochastic master equations with jumps in the presence of quantum measurements with direct photon detection the evolution of open quantum systems is usually described by stochastic master equations with jumps heuristically diffusion models can be obtained from these equations as an approximation a condition for a general diffusion approximation for jump master equations is presented this approximation is rigorously proved by using techniques for markov processes which are based on the convergence of markov generators and martingale problems this result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection

numerical simulation of stochastic evolution equations associated to quantum markov semigroups we address the problem of approximating numerically the solutions x t t 0 t of stochastic evolution equations on hilbert spaces h with respect to brownian motions arising in the unraveling of backward quantum master equations in particular we study the computation of mean values of x t ax t where a is a linear operator first we introduce estimates on the behavior of x t then we characterize the error induced by the substitution of x t with the solution x t n of a convenient stochastic ordinary differential equation it allows us to establish the rate of convergence of e to e where x t n denotes the explicit euler method finally we consider an extrapolation method based on the euler scheme an application to the quantum harmonic oscillator system is included

quantum stochastic positive evolutions characterization construction dilation a characterization of the unbounded stochastic generators of quantum completely positive flows is given this suggests the general form of quantum stochastic adapted evolutions with respect to the wiener diffusion poisson jumps or general quantum noise the corresponding irreversible heisenberg evolution in terms of stochastic completely positive cp maps is constructed the general form and the dilation of the stochastic completely dissipative cd equation over the algebra is discovered as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators a unitary quantum stochastic cocycle dilating the subfiltering cp flows over is reconstructed

on the stability properties of a van der pol duffing oscillator that is driven by a real noise abstract in this paper we consider a van der pol duffing oscillator that is excited parametrically by a small intensity real noise which is assumed to be an integrable function of an n dimensional ornstein uhlenbeck vector process that is an output of a linear filter system the stability properties include the moment lyapunov exponent g p x 0 and the maximal lyapunov exponent and the stability in probability are examined to study a model of enhanced generality we remove both the detailed balance condition and the strong mixing condition in the case of an arbitrary finite real number p we employ the perturbation method and a spectrum representation of the fokker planck operator of the linear filter system to construct asymptotic expansions of the pth moment lyapunov exponent and the top lyapunov exponent the same methods are also used for a nonlinear stochastic system to obtain the fpk fokker planck kolmogonov equation for the amplitude process which is identical to the one that is derived from the stochastic averaging method in the case of a broadband noise excitation on the basis of this fpk equation we also examine the almost sure stability condition of the ito stochastic differential equation for the amplitude process which matches the result that is derived from the maximal lyapunov exponent finally the method proposed by lin and cai probabilistic structural dynamics advanced theory and application mcgraw hill new york 1995 is adopted to examine the stability in probability of the amplitude process for the nonlinear ito differential equation

nonresonant hopf bifurcations of a controlled van der pol duffing oscillator abstract the trivial equilibrium of a van der pol duffing oscillator with a nonlinear feedback control may lose its stability via hopf bifurcations when the time delay involved in the feedback control reaches certain values nonresonant hopf hopf interactions may occur in the controlled van der pol duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots with the aid of normal form theory and centre manifold theorem as well as a perturbation method the dynamic behaviour of the nonresonant co dimension two bifurcation is investigated by studying the possible solutions and their stability of the four dimensional ordinary differential equations on the centre manifold in the vicinity of the nonresonant hopf bifurcation the oscillator may exhibit the initial equilibrium solution two periodic solutions as well as a quasi periodic solution on a two dimensional torus depending on the dummy unfolding parameters and nonlinear terms the analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation

normal forms of non resonance and weak resonance double hopf bifurcation in the retarded functional differential equations and applications the norm forms of the double hopf bifurcation for a general dde is established the van der pol duffing oscillator with delayed position and velocity feedback is investigated the dynamical classification near the double hopf bifurcation point is studied the stable periodic doubling solutions and unstable three dimensional torus are found in this paper we firstly present the general framework of calculation of normal forms of non resonance and weak resonance double hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to faria and magalh a es then the dynamical behavior of van der pol duffing oscillator with delayed position and velocity feedback is considered specifically the dynamical classification near the double hopf bifurcation point is investigated by analyzing the obtained normal form finally the numerical simulations support the theoretical results and present some interesting phenomena

numerical integration of stochastic differential equations with variable diffusivity the authors construct two stage second order algorithms for integrating stochastic differential equations with variable diffusivity in one and higher dimensions