stability estimation of high dimensional vibrating systems under state delay feedback control the paper presents a method of assessing the stability of high dimensional vibrating systems under state feedback control with a short time delay it is first proved that if the time delay is sufficiently short ann degree of freedom system with feedback delay maintains 2neigenvalues near those of the corresponding system without feedback delay a perturbation approach is then proposed to determine the first order variation of an arbitrary eigenvalue and corresponding eigenvector of the system with feedback delay by solving a set of linear algebraic equations only the computation in this approach can be simplified to a matrix multiplication provided that the product of the time delay and the modulus of the eigenvalue is much less than 1 furthermore the approach is directly related to the newton raphson iteration in the continuation of eigenvalues for long time delay the efficacy of the approach is demonstrated via a number of case studies on two feedback delay systems of two degrees of freedom and ten degrees of freedom respectively

bifurcation analysis in van der pol s oscillator with delayed feedback in this paper a classical van der pol s equation with generally delayed feedback is considered it is shown that there are bogdanov takens bifurcation triple zero and hopf zero singularities by analyzing the distribution of the roots of the associated characteristic equation in the situation that the zero is as a simple eigenvalue the normal forms of the reduced equations are obtained by the center manifold theory and normal form method for functional differential equation and hence the stability of the fixed point is determined and transcritical and pitchfork bifurcations are found

multiple scales scheme for bifurcation in a delayed extended van der pol oscillator abstract the van der pol vdp oscillator is an essential nonlinear oscillator which has been extensively studied however the results of delayed generalized vdp oscillator are relatively less the issue of bifurcation control for a delayed extended vdp oscillator is fully considered in this paper firstly a nonlinear time delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator and the results of stability of such oscillator are obtained by means of stability switches theory secondly a general explicit formula is derived to establish the properties of hopf bifurcation by utilizing the method of multiple scales moreover the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically finally the effectiveness of theoretical results is verified via a numerical example

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

self excited oscillation under nonlinear feedback with time delay several important applications use nonlinear feedback methods for synthetically inducing self excited oscillations in mechanical systems the van der pol and saturation function type feedback methods are widely used the effects of time delay on the self excited oscillation of single and two degrees of freedom systems under nonlinear feedback have been studied in this paper it is shown that a single degree of freedom oscillator with the van der pol type nonlinear feedback can produce unbounded response in presence of time delay in general an uncontrolled time delay in the feedback changes the state of oscillations in an uncertain manner therefore a bounded saturation type feedback with controllable time delay is proposed for inducing self excited oscillations the feedback signal is essentially an infinite weighted sum of a nonlinear function of the state variables of the system measured at equal intervals in the past more recent is the measurement higher is the weight thus the feedback signal uses a large amount of information about the past history of the dynamics such a control signal can be realized in practice by a recursive means the control law allows three parameters to be varied namely the time delay feedback and recursive gains multiple time scale analysis is used to plot amplitude vs time delay curves time delay can be controlled to vary the amplitude of oscillation as well as to switch the oscillation from one mode to the other in a two degrees of freedom system it is shown that a higher recursive gain can exercise a better and a more robust control on the amplitude of oscillation of the system analytical results are compared with the results of numerical simulations

using delayed state feedback to stabilize periodic motions of an oscillator abstract the paper states how to stabilize the periodic motions of a linear non positively damped system of single degree of freedom through the use of delayed state feedback the study indicates that the delayed state feedback works in certain frequency ranges for a linear undamped system of single degree of freedom the delayed displacement feedback is able to stabilize almost all periodic motions of the system provided that their fundamental frequencies are higher than the natural frequency of the system but it only works in a series of narrower and narrower frequency bands lower than the natural frequency the introduction of delayed velocity feedback can remarkably enlarge the working frequency ranges of delayed displacement feedback however even the delayed state feedback cannot stabilize a periodic motion if the corresponding period is an integral multiple of the natural period of system the criteria of stability switches prove to be a powerful tool to analyze the stabilization problem of a linear system however the stabilization of a linear undamped system of single degree of freedom with delayed velocity feedback only is a degenerate case where the available criteria of stability switches fail to offer any useful information a detailed study in the paper reveals the complexity of the degenerated case where the stabilization conditions can be identified according to the second order derivative of the real part of an arbitrary characteristic root

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

control of friction driven oscillation by time delayed state feedback an experimental set up is designed and fabricated to yield a single degree of freedom friction driven oscillator self excited limit cycles obtained experimentally are used to determine the numerical values of the parameters that appear in the mathematical model under pure slip condition then the control of the limit cycle using a time delayed displacement feedback is investigated the same system under harmonic excitation is also considered a pd control system with identical time delay in the displacement and velocity feedback is analysed only the primary harmonic entrainment region is investigated both the best and worst time delays are determined stability and bifurcation conditions are discussed numerical results are used to verify the analytical results obtained by using the method of multiple time scales and a parametric investigation is included the proper choice of time delay which depends on the model parameters is crucial it is shown that with appropriate choices of time delay and gain factors both the limit cycle and the steady state forced amplitude can be controlled no time delay or inappropriate time delay can enhance the vibration level

delayed feedback control of oscillations in non linear planar systems the use of delayed feedback of position for the control of oscillatory behaviour is analysed it is shown that linear feedback can annihilate limit cycles and stabilize the origin furthermore it is possible to create an asymptotically stable limit cycle with any prescribed amplitude using a simple non linear feedback for almost all values of the delay these feats can be achieved without using derivative information in the feedback the results indicate a possible mechanism by which biological systems accomplish complex control tasks using delayed and partial state information

non periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses abstract we present a non periodic averaging principle for measure functional differential equations and using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales see federson et al 2012 8 we obtain a non periodic averaging result for functional dynamic equations on time scales moreover using the relation between measure functional differential equations and impulsive measure functional differential equations we get a non periodic averaging theorem for these equations also it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales see federson et al 2013 9 therefore applying this correspondence to our averaging principle we obtain a non periodic averaging theorem for impulsive functional dynamic equations on time scales

simplified newton type adaptive estimation algorithms a new adaptive estimation algorithm is presented it is the result of a combination of the lms and the fast newton transversal filters fntf class the main characteristic of the proposed algorithm is its improved convergence rate as compared to lms for cases where it is known that lms behaves poorly this improved characteristic is achieved in expense of a slight increase in the computational complexity while the overall algorithmic structure is very simple lms type the proposed algorithm seems also to compare relatively well against rls and fntf

oscillations in a van der pol equation with delayed argument we consider the van der pol equation d2dt2u t eddtu t eu2 t r ddtu t r u t 0 with the delayed time r where r and e are real parameters with e 0 small and 0 r 2 and prove the existence and stability of a periodic solution with period near 2 and amplitude near 2 cosr with these properties this solution is unique up to translations in time

correlation theory of delayed feedback in stochastic systems below andronov hopf bifurcation here we address the effect of large delay on the statistical characteristics of noise induced oscillations in a nonlinear system below andronov hopf bifurcation in particular we introduce a theory of these oscillations that does not involve the eigenmode expansion and can therefore be used for arbitrary delay time in particular we show that the correlation matrix cm oscillates on two different time scales on the scale of the main period of noise induced oscillations and on the scale close to the delay time at large values of the delay time the cm is shown to decay exponentially only for large values of its argument while for the arguments comparable with the value of the delay the cm remains finite disregarding the delay time the definition of the correlation time of the system with delay is discussed

periodic solutions for van der pol equation with time delay for van der pol equation with distributed delay when the distributed delay kernel is the general gamma distributed delay kernel the existence and the stability of periodic solution are obtained by using the linear chain trick and geometric singular perturbation theory on the other hand for van der pol equation with discrete delay the existence and the stability are proved by employing a technique which is based on inertial manifold finally numerical simulations are also given to illustrate the results

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

an evaluation of the power and conditionality properties of empirical likelihood empirical and dual likelihood are two of a growing array of artificial or approximate likelihoods currently in use in statistics a question of major interest focuses on the performance of these new constructs relative to ordinary parametric likelihoods we consider here two criteria local power and conditional properties looking at tests of a scalar parameter we show that there is no loss of efficiency in using a dual or empirical likelihood model to second order to third order either the artificial likelihood or the true likelihood test could be more efficient this is determined by the fisher information the distance between the null and the alternative hypotheses and the statistical curvature of the models conditionality properties are assessed by comparing empirical likelihood with quasilikelihood if there is overdispersion present or more generally an unknown amount of dispersion then there is little difference in the ancillaries for conditional inference based on empirical likelihood or on other possible likelihoods such as a quasilikelihood if the amount of dispersion is known however it is correct to base inference on the quasilikelihood

controlling friction induced instability by recursive time delayed acceleration feedback a novel time delayed acceleration feedback method of controlling friction induced instability is proposed a single degree of freedom mechanical oscillator on a moving belt represents the basic friction driven system the control force is synthesized based on an infinite weighted sum of the acceleration of the vibrating mass measured at regular intervals in the past such a control force can be effectively produced by the recursive summation of the time delayed acceleration and the time delayed control signal and hence the technique is termed as the recursive time delayed acceleration feedback control the local stability analysis of the equilibrium reveals nontrivial and beneficial influences of the recursive gain on the system performance robustness of the control is shown to improve with the increasing value of the recursive gain a multiple time scale based analysis of the system elucidates the role of the recursive gain in enhancing the amount of dissipation produced by the control action the influences of the time delay and the control gain on the optimized performance of the system are also discussed numerical simulations of the system equations corroborate the analytical results the present method is believed to be applicable to any self excited system having a large degree of instability that is not removable by an ordinary time delayed feedback

vibrational feedback control of time delay systems this paper applies vibrational feedback control to time lag systems averaging theory for differential delay systems is presented and then applied to aid in the design of periodic controllers both stabilization and transient issues are discussed an illustrative example is given which demonstrates that the proposed controller 1 provides superior robustness properties in comparison to time invariant controllers when applied correctly 2 is not robust with respect to unmodeled delays and 3 does not have zero placement capabilities in the sense defined in the paper

nonresonant hopf hopf bifurcation and a chaotic attractor in neutral functional differential equations abstract an algorithm for calculating the third order normal form of a nonresonant hopf hopf singularity in a neutral functional differential equation nfde is established the van der pol equation with extended delay feedback is investigated as an nfde of second order the existence of hopf hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm periodic solutions and quasi periodic solutions are found with the aid of the bifurcation diagram and corresponding numerical illustrations are presented with the breaking down of the 3 torus a chaotic attractor appears in this nfde of second order following the ruelle takens newhouse scenario which usually arises for an ordinary differential equation of order at least 4 this transition is shown via both theoretical and numerical approaches

first passage failure of quasi integrable hamiltonian systems under time delayed feedback control a procedure for studying the first passage failure of quasi integrable hamiltonian systems under time delayed feedback control is proposed the stochastic averaging method for quasi integrable hamiltonian systems under time delayed feedback control is firstly introduced a backward kolmogorov equation governing the conditional reliability function and a set of generalized pontryagin equations governing the conditional moments of first passage time are then established the conditional reliability function the conditional probability density and moments of first passage time are obtained by solving the backward kolmogorov equation and generalized pontryagin equations with suitable initial and boundary conditions an example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure the effects of time delay in feedback control forces on the conditional reliability function conditional probability density and moments of first passage time are analyzed

delay dependent robust stabilization of uncertain systems with multiple state delays this paper deals with the problem of robust stabilization for uncertain systems with multiple state delays the parameter uncertainties are time varying and unknown but are norm bounded and the delays are time varying a new method for achieving robust stabilization is presented for a class of uncertain time delay systems via linear memoryless state feedback control the results depend on the size of the delays and are given in terms of several linear matrix inequalities

robust hurwitz stability test for linear systems with uncertain commensurate time delays the note deals with the robust hurwitz stability of a class of linear systems with uncertain commensurate time delays which is formulated as the robust hurwitz stability of a nonpolytopic family of quasi polynomials a np hard problem from the viewpoint of computational complexity on the basis of the edge theorem and sylvester resultant elimination a necessary and sufficient condition is developed for the robust stability of the whole family and it yields an effective testing procedure an illustrative example is given to show the effectiveness of the method

limit cycle oscillators subject to a delayed feedback the coexistence of two stable limit cycles exhibiting different periods is examined for a nonlinear oscillator subject to a delayed feedback for the case of a weakly nonlinear oscillator we discuss the validity of a previously determined phase equation for the case of a strongly nonlinear oscillator we derive a phase equation and analyze its bifurcation diagram our analysis is motivated by previous experimental studies of chemical oscillators controlled by a delayed feedback

stochastic averaging of quasi integrable hamiltonian systems with delayed feedback control abstract a stochastic averaging method for quasi integrable hamiltonian systems with time delayed feedback control is proposed first a quasi integrable hamiltonian system with delayed feedback control subjected to gaussian white noise excitations is formulated and then transformed into ito stochastic differential equations without time delay then the averaged ito stochastic differential equations for the system are derived and the stationary solution of the averaged fokker planck kolmogorov fpk equation associated with the averaged ito equations is obtained for both non resonant and resonant cases finally three examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback control on the response of the systems

hopf pitchfork bifurcation in van der pol s oscillator with nonlinear delayed feedback abstract first we identify the critical values for hopf pitchfork bifurcation second we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system furthermore we find some interesting phenomena such as the coexistence of two asymptotically stable states two stable periodic orbits and two attractive quasi periodic motions which are verified both theoretically and numerically

zero hopf bifurcation for van der pol s oscillator with delayed feedback in this paper we study the dynamical behaviors of the following van der pol oscillator with delay x e x 2 1 x x eg x t t in the case that its associated characteristic equation has a simple zero root and a pair of purely imaginary roots zero hopf singularity the normal form is obtained by performing a center manifold reduction and by using the normal form theory developed by faria and magalhaes a critical value e 0 of e in 0 2 is obtained to predict the bifurcation diagrams from which saddle node bifurcation pitchfork bifurcation hopf bifurcation the existence and stability of the periodic solutions and heteroclinic bifurcation are determined some examples are given to confirm the theoretical results

the response of a duffing van der pol oscillator under delayed feedback control abstract the nonlinear dynamics of a duffing van der pol oscillator under linear plus nonlinear state feedback control with a time delay are investigated by means of the averaging method and taylor expansion two slow flow equations for the amplitude and phase of the primary resonance response are derived from which the relations between the amplitude and phase of the primary resonance response and all other parameters are obtained respectively the singularity analysis of the equation governing the amplitude of the primary resonance response shows that the bifurcation modes are perturbations of the pitchfork bifurcation conditions preventing multiple solutions corresponding to two different kinds of bifurcation modes are given since cases for which multiple solutions are available should be avoided the stable condition for steady state response is also given by the routh hurwitz criterion it is also shown that coupled nonlinear state feedback control can be replaced by uncoupled nonlinear state feedback control

a fast newton multichannel algorithm for decision feedback equalization fast newton transversal filters fntf are a new class of adaptive algorithms unifying nlms and fast rls this is achieved by allowing the algorithm to exploit a priori information about the associated predictors order this note extends fntf to the multichannel case and demonstrates the applicability of the resulting algorithm to decision feedback equalization

time delayed feedback stabilisation of nonlinear potential systems mechanical systems with nonlinear potential forces and delayed feedback are studied it is assumed that in the absence of control the trivial equilibrium positions of considered systems are stable but they are not attracting ones an approach for the constructing of nonlinear controllers providing the asymptotic stability of the equilibrium positions is proposed by the use of the lyapunov direct method and the razumikhin approach it is proved that for the corresponding closed loop systems the asymptotic stability can be guaranteed even in the cases when delay is unknown and time varying moreover estimates for solutions of closed loop systems are found an example and the results of a computer simulation are presented to demonstrate the effectiveness of the proposed approach

vibration control by recursive time delayed acceleration feedback this paper presents a theoretical basis of time delayed acceleration feedback control of linear and nonlinear vibrations of mechanical oscillators the control signal is synthesized by an infinite weighted sum of the acceleration of the vibrating system measured at equal time intervals in the past the proposed method is shown to have controlled linear resonant vibrations low frequency non resonant vibrations primary and 1 3 subharmonic resonances of a forced duffing oscillator the concept of an equivalent damping and natural frequency of the system is also introduced it is shown that a large amount of damping can be produced by appropriately selecting the control parameters for some combinations of the control parameters the effective damping factor of the system is shown to be inversely related to the time delay in the small delay limit selection of the optimum control parameters for controlling the forced and free vibrations is discussed it is shown that forced vibration is best controlled by unity recursive gain and smaller values of the time delay parameter however the transient response can be optimally controlled by suitably selecting the time delay depending upon the gain the delay values for the optimal forced response may be different from that required for the optimum transient response when both are important a suboptimal choice of the delay parameters with unity recursive gain is recommended

stabilization of vibration systems via delayed state difference feedback this paper presents a study of the stabilization problem via delayed state feedback of pyragas type for unstable vibration systems that have even number of characteristic roots with positive real parts on the basis of stability switches with respect to the time delay a new stabilization criterion is established and an effective procedure for determining the admissible values of the feedback gains and the delay is given two examples are given in detail to demonstrate the efficiency of the theory

dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks abstract we study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single hopf bifurcation oscillator our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips frequency suppression multiple periodic states and chaos such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator we construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms we find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity phase reversals radial trapping phase jumps and spiraling patterns in the amplitude space our results may find useful applications in physical chemical or biological systems

bayesian empirical likelihood summary research has shown that empirical likelihood tests have many of the same asymptotic properties as those derived from parametric likelihoods this leads naturally to the possibility of using empirical likelihood as the basis for bayesian inference different ways in which this goal might be accomplished are considered the validity of the resultant posterior inferences is examined as are frequentist properties of the bayesian empirical likelihood intervals

controlling coherence of noisy and chaotic oscillators by a linear feedback abstract we analyze the dynamics of a noisy limit cycle oscillator coupled to a general passive linear system we analytically demonstrate that the phase diffusion constant which characterizes the coherence of the oscillations can be efficiently controlled theoretical analysis is performed in the framework of linear and gaussian approximations and is supported by numerical simulations we also demonstrate numerically the coherence control of a chaotic system

robust stabilization of uncertain systems with time varying multistate delay the authors deal with the problem of stabilizing a class of uncertain linear systems with time varying multi state delay and subject to norm bounded parameter uncertainty via memoryless linear state feedback some sufficient conditions for the robust stabilizability are derived fur this class of uncertain systems if there exists a positive definite symmetric solution satisfying the algebraic riccati equation or inequality a suitable memoryless state feedback law can be derived also moreover all such parametric algebraic riccati inequalities have been transformed into some linear matrix inequality problems so there is no tuning of the parameters to gain a stabilizing solution

difference feedback can stabilize uncertain steady states the paper is concerned with the stabilization of uncertain steady states by the state difference feedback the feedback method has a peculiar feature that it uses only the difference between the present state x t and the past state x t t considering exact information on the steady state is unavailable hitherto a condition is known under which such stabilization can not be realized the article conversely shows that the state difference feedback can stabilize only if the exclusion condition is not true furthermore a dynamic output difference feedback is shown to be able to stabilize under quite a mild condition that the steady state is not associated with zero eigenvalues the ability of the method is illustrated by using a cart pendulum system which moves along a one dimensional varying slope

vibration amplitude control for a van der pol duffing oscillator with time delay abstract periodic solutions for the fundamental resonance response of a van der pol duffing system under time delayed position and velocity feedbacks are investigated using the asymptotic perturbation method two slow flow equations for the amplitude and phase of the fundamental resonance response are derived their fixed points correspond to limit cycles phase locked periodic solutions for the original system in the uncontrolled system periodic solutions exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude in many cases the amplitudes of periodic solutions do not correspond to the technical requirements it is demonstrated that if the vibration control terms are added stable periodic solutions with arbitrarily chosen amplitude can be accomplished therefore the results obtained show that an effective vibration amplitude control is possible if appropriate time delay and feedback gains are chosen

vibrational control of chemical reactions in a cstr with delayed recycle stream vibrational control is a method of modification of dynamic properties obtained by introducing zero mean oscillations into system parameters this paper shows that by introducing oscillations into the flow rates of a continuous stirred tank reactor with delayed recycle stream it is possible to operate reactions in the vicinity of previously unstable steady states

fundamental theorems of averaging for functional differential equations abstract in this paper we improve on classical averaging theorems for functional differential equations by proposing new averaged models specifically we present averaged models which retain functional dependence on state variables delays and additional parametric dependence as shown in three examples the new models often yield improvements in approximation accuracy

fast newton transversal filters a new class of adaptive estimation algorithms a class of adaptive algorithms for the estimation of fir finite impulse response transversal filters is presented the main characteristic of this class is the fast computation of the gain vector needed for the adaptation of the transversal filters the method for deriving these algorithms is based on the assumption that the input signal is autoregressive of order m where m can be much smaller than the order of the filter to be estimated under this assumption the covariance matrix of the input signal is estimated by extending in a min max way the m order sample covariance matrix this estimate can be regarded as a generalization of the diagonal covariance matrix used in lms and leads to an efficient computation of the gain needed for the adaptation the new class of algorithms contains the lms and the fast versions of ls as special cases the complexity changes linearly with m starting from the complexity of the lms for m 0 and ending at the complexity of the fast versions of ls

control of unstable steady states by time delayed feedback methods we show that time delayed feedback methods which have successfully been used to control unstable periodic orbits provide a tool to stabilize unstable steady states we present an analytical investigation of the feedback scheme using the lambert function and discuss effects of both a low pass filter included in the control loop and nonzero latency times associated with the generation and injection of the feedback signal

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

counting roots of the characteristic equation for linear delay differential systems abstract a formula is given that counts the number of roots in the positive half plane of the characteristic equation for general real constant coefficient linear delay differential systems the formula is used to establish necessary and sufficient conditions for asymptotic stability of the zero solution of linear delay differential systems the formula is potentially useful in verifying stability hypotheses that arise in bifurcation analysis of autonomous delay differential systems application of the formula to hopf bifurcation theory for delay differential systems is discussed and an example application to an equation with two delays is given

coherence of noisy oscillators with delayed feedback a theory ofe ect ofdelayed f on the coherence ofthe noisy self sustained oscillations is developed in the gaussian approximation a closed system ofequations is derived f the phase di usion constant and the mean frequency a comparison with numerics shows that the theory works well for weak feedback and strong noise c

total and partial amplitude death in networks of diffusively coupled oscillators networks of weakly nonlinear oscillators are considered with diffusive and time delayed coupling averaging theory is used to determine parameter ranges for which the network experiences amplitude death whereby oscillations are quenched and the equilibrium solution has a large domain of attraction the amplitude death is shown to be a common phenomenon which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions in addition when the network consists of dissimilar oscillators there exist parameter values for which only parts of the network are suppressed sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities coupling coefficients and connection delays

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

likelihood based confidence intervals for a log normal mean to construct a confidence interval for the mean of a log normal distribution in small samples we propose likelihood based approaches the signed log likelihood ratio and modified signed log likelihood ratio methods extensive monte carlo simulation results show the advantages of the modified signed log likelihood ratio method over the signed log likelihood ratio method and other methods in particular the modified signed log likelihood ratio method produces a confidence interval with a nearly exact coverage probability and highly accurate and symmetric error probabilities even for extremely small sample sizes we then apply the methods to two sets of real life data

on the analysis of the double hopf bifurcation in machining processes via centre manifold reduction the single degree of freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double hopf bifurcation point centre manifold reduction and normal form calculations are performed to investigate the long term dynamics of the cutting process the normal form of the four dimensional centre subsystem is derived analytically and the possible topologies in the infinite dimensional phase space of the system are revealed it is shown that bistable parameter regions exist where unstable periodic and in certain cases unstable quasi periodic motions coexist with the equilibrium taking into account the non smoothness caused by loss of contact between the tool and the workpiece the boundary of the bistable region is also derived analytically the results are verified by numerical continuation the possibility of transient chaotic motions in the global non smooth dynamics is shown

posterior expectation based on empirical likelihoods 5summary posterior expectation is widely used as a bayesian point estimator in this paper we extend it from parametric models to nonparametric models using empirical likelihood and provide a nonparametric analog of james stein estimation we use the laplace method to establish asymptotic approximations to our proposed posterior expectations and show by simulation that they are of 10 ten more efficient than corresponding classical nonparametric procedures especially when the underlying data are skewed

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

adaptive control of unknown unstable steady states of dynamical systems a simple adaptive controller based on a low pass filter to stabilize unstable steady states of dynamical systems is considered the controller is reference free it does not require knowledge of the location of the fixed point in the phase space a topological limitation similar to that of the delayed feedback controller is discussed we show that the saddle type steady states cannot be stabilized by using the conventional low pass filter the limitation can be overcome by using an unstable low pass filter the use of the controller is demonstrated for several physical models including the pendulum driven by a constant torque the lorenz system and an electrochemical oscillator linear and nonlinear analyses of the models are performed and the problem of the basins of attraction of the stabilized steady states is discussed the robustness of the controller is demonstrated in experiments and numerical simulations with an electrochemical oscillator the dissolution of nickel in sulfuric acid a comparison of the effect of using direct and indirect variables in the control is made with the use of the controller all unstable phase space objects are successfully reconstructed experimentally

averaging of linear systems with almost periodic coefficients a time delay approach abstract we study stability of linear systems with fast almost periodic coefficients that are piecewise continuous in time the classical averaging method guarantees the stability of such systems for small enough values of parameter provided the corresponding averaged system is stable however it is difficult to find an upper bound on the small parameter by using classical tools for asymptotic analysis in this paper we introduce an efficient constructive method for finding an upper bound on the value of the small parameter that guarantees a desired exponential decay rate we transform the system into a model with time delays of the length of the small parameter the resulting time delay system is a perturbation of the averaged system the averaged system is supposed to be exponentially stable the stability of the time delay system guarantees the stability of the original one we construct an appropriate lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities lmis the upper bound on the small parameter that preserves the exponential stability is found from the lmis two numerical examples stabilization by vibrational control and by time dependent switching illustrate the efficiency of the method moreover we apply the time delay approach to persistently excited systems that leads to a novel quadratic time independent lyapunov functional for such systems we further extend our method to input to state stability iss analysis finally the results are extended to linear fast varying systems with time varying delays

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

bogdanov takens singularity in van der pol s oscillator with delayed feedback abstract in this paper we study the classical van der pol s equation with delayed feedback we investigate the bogdanov takens singularity and bifurcation occurring in the system with the variation of the original parameters by using the normal form method and predict the suitable choice in the feedback control in order to obtain the asymptotic stable state

delayed feedback control of collective synchrony an approach to suppression of pathological brain rhythms we suggest a method for suppression of synchrony in a globally coupled oscillator network based on the time delayed feedback via the mean field having in mind possible applications for suppression of pathological rhythms in neural ensembles we present numerical results for different models of coupled bursting neurons a theory is developed based on the consideration of the synchronization transition as a hopf bifurcation

continuous control of chaos abstract we present analytical results pertaining to the control of chaos via a continuous feedback as proposed by k pyragas it is shown that the required feedback matrix exists and can be chosen so that other periodic orbits can be stabilized simply by changing the forcing term furthermore it is shown that the method can be generalized to allow greater flexibility in choosing the forcing term

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

realizability of the normal form for the triple zero nilpotency in a class of delayed nonlinear oscillators abstract the effects of delayed feedback terms on nonlinear oscillators have been extensively studied and there are important applications in many areas of science and engineering we study a particular class of second order delay differential equations near a point of triple zero nilpotent bifurcation using center manifold and normal form reduction we show that the three dimensional nonlinear normal form for the triple zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay differential equation

robust stability for time delay systems the edge theorem and graphical tests the robust stability problem is discussed for a class of uncertain delay systems where the characteristic equations involve a polytope p of quasi polynomials i e polynomials in one complex variable and exponential powers of the variable given a set d in the complex plane the goal is to find a constructive technique to verify whether all roots of every quasi polynomial in p belong to d that is to verify the d stability of p first it is demonstrated by counterexample that kharitonov s theorem does not hold for general delay systems next it is shown that under a mild assumption on the set d a polytope of quasi polynomials is d stable if and only if the edges of the polytope are d stable this extends the edge theorem for the d stability of a polytope of polynomials the third result gives a constructive graphical test for checking the d stability of a polytope of quasi polynomials which is especially simple when the set d is the open left half plane an application is given to demonstrate the power of the results

controlling oscillator coherence by delayed feedback we demonstrate that the coherence of a noisy or chaotic self sustained oscillator can be efficiently controlled by the delayed feedback we develop a theory of this effect considering noisy systems in the gaussian approximation we obtain a closed equation system for the phase diffusion constant and the mean frequency of oscillation for weak feedback and strong noise the theory is in good agreement with the numerics we discuss possible applications of the effect for the synchronization control

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

vibrational control of nonlinear time lag systems with bounded delay averaging theory stabilizability and transient behavior this paper develops the theory of vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay averaging theory for fast oscillating differential delay equations is presented and then applied to vibrational control conditions are given which ensure the existence of parametric vibrations that stabilize nonlinear time lag systems transient behavior is also discussed illustrative examples are given which show 1 the feasibility of the theory to important applications and 2 the differences in the theory presented and the existing known theory for vibrational control of ordinary differential equations

stabilizing the fast kalman algorithms the fast kalman algorithms are stabilized by introducing a quantity that measures the accumulation of the roundoff errors this quantity is used to correct the variables of the algorithm at every time step the correction is defined as the solution of a specific minimization problem the resulting algorithm still has the nice complexity properties of the original algorithm linear in the number of parameters to be estimated but it has a much more stable behavior