responses of strongly non linear oscillator parametrically excited by random narrow band noise the principal response of a strongly van der pol duffing oscillator subjected to parametric random narrow band excitation is investigated the technique of modified lindstedt poincare mlp method is used to transform the strongly non linear system to a weak one by introducing a new expansion parameter and then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response of the system the effect of damping detuning and bandwidth on the dynamic behaviors such as stability bifurcation are examined by computing the maximum lyapunov exponent analytically also the numerical simulation is carried out to verify the analytical results and random jump phenomenon may be observed in the region of the parameters of the system the excellent agreement between theoretical results and numerical ones can be found immediately and so the present method in this paper is applicable to solve strongly non linear problems

asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects global minimization via monte carlo the asymptotic behavior of the systems x n 1 x n a n b x n xi n a n sigma x n psi n and dy bar b y dt sqrt a t sigma y dw is studied where psi n is i i d gaussian xi n is a correlated bounded sequence of random variables and a n approx a 0 log a 1 n without xi n such algorithms are versions of the simulated annealing method for global optimization when the objective function values can only be sampled via monte carlo the discrete algorithm is a combination of stochastic approximation and simulated annealing our forms are appropriate the psi n are the annealing variables and xi n is the sampling noise for large a 0 a full asymptotic analysis is presented via the theory of large deviations mean escape time after arbitrary time n from neighborhoods of stable sets of the algorithm mean transition times after arbitrary time n from a neighborhood of one stable set to another a

the response envelope probability density function of a duffing oscillator with random narrow band excitation abstract a duffing oscillator excited by narrow band random noise can exhibit multi valued behaviour with random jumps between two pseudo stable states in this paper an approximate form of the probability density function or the envelope of the response is obtained state equations for the response envelope with the random excitation envelope as input are obtained by stochastic averaging the joint probability density function of response envelope and input envelope can be found approximately for certain ranges of the system parameters the response envelope probability density function shows two maxima in the cases where the response is multi valued numerical simulation confirms the results

multiple time scaling of the response of a duffing oscillator to narrow band random excitation the response of a duffing oscillator to narrow band random excitation is considered two approaches multiple time scaling and stochastic averaging are used both lead to the same set of three state equations for the slowly varying local mean square response or envelope and two other variables the stationary solution of these equations is similar to results previously obtained by using statistical linearization the stationary and nonstationary from the three state equations results compare favourably with digital simulations random jumps are shown to occur if the excitation bandwidth is sufficiently small

nonlinear knowledge in kernel approximation prior knowledge over arbitrary general sets is incorporated into nonlinear kernel approximation problems in the form of linear constraints in a linear program the key tool in this incorporation is a theorem of the alternative for convex functions that converts nonlinear prior knowledge implications into linear inequalities without the need to kernelize these implications effectiveness of the proposed formulation is demonstrated on two synthetic examples and an important lymph node metastasis prediction problem all these problems exhibit marked improvements upon the introduction of prior knowledge over nonlinear kernel approximation approaches that do not utilize such knowledge

beam beam interaction models under narrow band random excitation abstract the aim of this paper is to continue our previous investigations on beam beam interaction models in particle accelerators by considering these models under narrow band random excitation in this investigation the multiple scale and the moment methods are used the equations of modulation of the amplitude and the phase are derived and perturbation techniques are used to seek approximate steady state solutions two special cases are considered to illustrate this study and excellent agreement is found between analytical and numerical results other cases can be similarly studied the effects of noise and detuning parameters are examined numerically and theoretically and we find that the system demonstrates a diffused limit cycle as the intensity of noise increases the numerical results show that the multiple scale method is effective for these stochastic systems

dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation abstract we investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation the method of multiple scales is used to derive the analytical expression of first order uniform expansion of the solution based on the largest lyapunov exponent the almost sure stability of the trivial steady state solution is examined meanwhile we obtain the first order and the second order steady state moments for the non trivial steady state solutions specially we discuss the first mode theoretically and numerically results show that under the same conditions of the parameters as the intensity of the random excitation increases non trivial steady state solution fluctuation will become strenuous which will result in the non trivial steady state solution lose stability and the trivial steady state solution can be a possible in the case of parametric principal resonance the stochastic jump is observed for the first mode which indicates that the stationary joint probability density concentrates at the non trivial solution branch when the random excitation is small but with the increase of intensity of the random excitation the probability of the trivial steady state solution will become larger this phenomenon of stochastic jump can be defined as a stochastic bifurcation

subharmonic response of a single degree of freedom nonlinear vibroimpact system to a randomly disordered periodic excitation the subharmonic resonant response of a single degree of freedom nonlinear vibroimpact oscillator with a one sided barrier to narrow band random excitation is investigated the analysis is based on a special zhuravlev transformation which reduces the system to one without impacts or velocity jumps thereby permitting the applications of asymptotic averaging over the period for slowly varying random processes the averaged equations are solved exactly and algebraic equation of the amplitude of the response is obtained in the case without random disorder a perturbation based moment closure scheme is proposed and an iterative calculation equation for the mean square response amplitude is derived in the case with random disorder the effects of damping nonlinear intensity detuning and magnitudes of random excitations are analyzed the theoretical analyses are verified by numerical results theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detuning or large nonlinear stiffness and when intensity of the random disorder increase the steady state solution may change from a limit cycle to a diffused limit cycle and even change to a chaos one

largest lyapunov exponent for second order linear systems under combined harmonic and random parametric excitations the principal resonance of a second order linear stochastic oscillator to combined harmonic and random parametric excitations is investigated the method of multiple scales is used to determine the equations of modulation of amplitude and phase the effects of damping detuning bandwidth and magnitudes of random excitation are analyzed the method of path integration is used to obtain the steady state probability density function of the system and then the largest lyapunov exponent is calculated the almost sure stability or instability of the stochastic system depends on the sign of the largest lyapunov exponent the theoretical analyses are verified by numerical results

adaptive learning with nonlinear dynamics driven by dependent processes we provide a convergence theory for adaptive learning algorithms useful for the study of learning by economic agents our results extend the framework of ljung 1977 previously utilized by marcet sargent 1989a b and woodford 1990 by permitting nonlinear laws of motion driven by stochastic processes that may exhibit moderate dependence such as mixing and mixingale processes we draw on previous work by kushner and clark 1978 to provide readily verifiable and or interpretable conditions ensuring algorithm convergence chosen for their suitability in the context of adaptive learning

knowledge based kernel approximation prior knowledge in the form of linear inequalities that need to be satisfied over multiple polyhedral sets is incorporated into a function approximation generated by a linear combination of linear or nonlinear kernels in addition the approximation needs to satisfy conventional conditions such as having given exact or inexact function values at certain points determining such an approximation leads to a linear programming formulation by using nonlinear kernels and mapping the prior polyhedral knowledge in the input space to one defined by the kernels the prior knowledge translates into nonlinear inequalities in the original input space through a number of computational examples including a real world breast cancer prognosis dataset it is shown that prior knowledge can significantly improve function approximation

random superharmonic and subharmonic response multiple time scaling of a duffing oscillator narrowband random third order superharmonic and one third order subharmonic response of a duffing oscillator is discussed a multiple time scale approach is used the work is a more rigorous approach than that already published on random superharmonic response by the authors and the results obtained corroborate the previous ones in particular random jumps in amplitude can occur but are less likely to occur as the bandwidth of the excitation is increased the results are further corroborated by comparison with digital simulation in addition new results on random subharmonic response are also presented obtained by using the same approach

response of a duffing oscillator to combined deterministic harmonic and random excitation the response of duffing oscillator to combined deterministic harmonic and random excitation is investigated the method of harmonic balance and the method of stochastic averaging are used to determine the response of the system theoretical analyses and numerical simulations show that when the intensity of the random excitation increases the non trivial steady state solution may change from a limit cycle to a diffused limit cycle under some conditions the system may have two steady state solutions and jumps may exist

phase plane for narrow band random excitation of a duffing oscillator abstract the response of a duffing oscillator to narrow band random excitation is consedered it is shown that multiple values of the response can occur at some frequencies for very narrow excitation bandwidths but these reduce to a single value as the bandwidth is increased time histories of quantities such as mean square displacement are obtained from the time dependent fokker planck equation smoothed time histories of mean square velocity and displacement are plotted on a phase plane the phase plane can show either one sink or two sinks and a saddle point as in the sinusoidal case but again excitation bandwith determines whether or not multiple sinks occur

nonlinear knowledge based classification in this brief prior knowledge over general nonlinear sets is incorporated into nonlinear kernel classification problems as linear constraints in a linear program these linear constraints are imposed at arbitrary points not necessarily where the prior knowledge is given the key tool in this incorporation is a theorem of the alternative for convex functions that converts nonlinear prior knowledge implications into linear inequalities without the need to kernelize these implications effectiveness of the proposed formulation is demonstrated on publicly available classification data sets including a cancer prognosis data set nonlinear kernel classifiers for these data sets exhibit marked improvements upon the introduction of nonlinear prior knowledge compared to nonlinear kernel classifiers that do not utilize such knowledge

a group of knowledge incorporated multiple criteria linear programming classifiers classification is a main data mining task which aims at predicting the class label of new input data on the basis of a set of pre classified samples multiple criteria linear programming mclp is used as a classification method in the data mining area which can separate two or more classes by finding a discriminate hyperplane although mclp shows good performance in dealing with linear separable data it is no longer applicable when facing with nonlinear separable problems a kernel based multiple criteria linear programming kmclp model is developed to solve nonlinear separable problems in this method a kernel function is introduced to project the data into a higher dimensional space in which the data will have more chance to be linear separable kmclp performs well in some real applications however just as other prevalent data mining classifiers mclp and kmclp learn only from training examples in the traditional machine learning area there are also classification tasks in which data sets are classified only by prior knowledge i e expert systems some works combine the above two classification principles to overcome the faults of each approach in this paper we provide our recent works which combine the prior knowledge and the mclp or kmclp model to solve the problem when the input consists of not only training examples but also prior knowledge specifically how to deal with linear and nonlinear knowledge in mclp and kmclp models is the main concern of this paper numerical tests on the above models indicate that these models are effective in classifying data with prior knowledge

response of a strongly non linear oscillator to narrowband random excitations abstract the principal resonance of a van der pol duffing oscillator subject to narrowband random excitations has been studied by introducing a new expansion parameter e e e u 0 the method of multiple scales is adapted for the strongly non linear system the behavior of steady state responses together with their stability and the effects of system damping and the detuning and magnitude of the random excitation on steady state responses are analyzed in detail theoretical analyses are verified by some numerical results it is found that when the random noise intensity increases the steady state solution may change form a limit cycle to a diffused limit cycle and the system may have two different stable steady state solutions for the same excitation under certain conditions the results obtained for the strongly non linear oscillator complement previous results in the literature for weakly non linear systems

saturation and resonance of nonlinear system under bounded noise excitation the principal resonance of a 2dof nonlinear oscillator due to bounded random excitations is investigated equations of modulation of response amplitude and phase are derived by the method of multiple scales steady state moments for the response amplitude of the system are determined through the linearized ito differential equation the results of theoretical analyses are verified by numerical simulations saturation phenomena are found in the random counterpart some recommendations for potential applications of this random saturation phenomenon to vibration control problems are given at the end of the paper

largest lyapunov exponent and almost certain stability analysis of slender beams under a large linear motion of basement subject to narrowband parametric excitation the first order approximate solutions of a set of non liner differential equations which is established by using kane s method and governs the planar motion of beams under a large linear motion of basement are systematically derived via the method of multiple scales the non linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3 1 internal resonance between the first two modes is taken into consideration are analyzed in detail the largest lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system

explanation on the importance of narrow band random excitation characters in the response of a cantilever beam a detailed theoretical investigation into the first and second mode response of a parametrically excited slender cantilever beam where the narrow band random excitation characters are taken into consideration is presented the method of multiple scales is used to determine a uniform first order expansion of the solution of the nonlinear integro differential equations the first order moment frequency and force response data curves of a specimen beam tested by other investigators are obtained further comparisons have been made and results show that whether the first order moment frequency response data curves or the first order moment force response data curves of the first two modes are all in agreement with other investigators experimental results furthermore the stochastic jump and bifurcation have been investigated for the first modal parametric principal resonance by using the stationary joint probability of amplitude and phase results show that stochastic jump occurs mainly in the region of triple valued solution for the frequency response domain if the bandwidth is a variable and others keep constant the basic phenomena indicate that the most probable motion is around the higher branch when the bandwidth is smaller whereas the most probable motion gradually approaches the lower one when the bandwidth becomes higher if the excitation central frequency f is a variable and others keep constant the basic phenomena imply that the higher is f the more probable is the jump from the higher branch to the lower one once f exceeds an certain value for the force response domain there is a region of excitation acceleration a within which the joint probability density has two peaks an upper peak and a lower peak results show that the upper peak decreases while the lower peak increases as the value of a decreases

knowledge based linear programming we introduce a class of linear programs with constraints in the form of implications such linear programs arise in support vector machine classification where in addition to explicit datasets to be classified prior knowledge such as expert experience in the form of logical implications are imposed on the classifier the overall problem can be viewed either as a semi infinite linear program or as a linear program with equilibrium constraints which in either case can be solved by an equivalent simple linear program under mild assumptions

non linear resonance in strings under narrow band random excitation part i planar response and stability non linear planar response of a string to planar narrow band random excitation is investigated in this paper a response equation for the mean square deflection 2 is obtained under a single mode approximation by using the equivalent linearization technique it is shown that the response is triple valued as in the case of harmonic excitation if the centre frequency of excitation lies in a certain specified range the triple valued response occurs only if the excitation bandwidth is smaller than a critical value crit which is a monotonically increasing function of the intensity of excitation an approximate method of investigating the almost sure asymptotic stability of the solution is presented and regions of instability in the 2 plane have been charted it is shown that planar response can become unstable either due to an unbounded growth of the in plane component of motion or due to a spontaneous appearance of an out of plane component