existence and uniqueness of fixed points for some mixed monotone operators abstract in this paper we introduce the e concave convex operator without any compact or continuous assumptions we prove the existence and uniqueness of fixed points giving monotone iterative sequences for the unique fixed point for the operator finally we apply the results to an integral equation of polynomial type which possesses items of measurable functions

outlier detection and time series modeling some statistics used in regression analysis are considered for detection of outliers in time series approximations and asymptotic distributions of these statistics are considered a method is proposed for distinguishing an observational outlier from an innovational one a four step procedure for modeling time series in the presence of outliers is also proposed and an example is presented to illustrate the methodology

approximate case influence for the proportional hazards regression model with censored data a method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the cox proportional hazards model observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time an example from a cancer clinical trial is given

stock market dynamics abstract we elucidate on several empirical statistical observations of stock market returns moreover we find that these properties are recurrent and are also present in invariant measures of low dimensional dynamical systems thus we propose that the returns are modeled by the first poincare return time of a low dimensional chaotic trajectory this modeling which captures the recurrent properties of the return fluctuations is able to predict well the evolution of the observed statistical quantities in addition it explains the reason for which stocks present simultaneously dynamical properties and high uncertainties in our analysis we use data from the s p 500 index and the brazilian stock telebras

basin bifurcations in quasiperiodically forced coupled systems we study the effect of quasiperiodic forcing on a system of coupled identical logistic maps upon a variation of system parameters a variety of different dynamical regimes can be observed including phenomena such as bistability and multistability at the bifurcation to bistability in a manner reminiscent of attractor expansion at interior crises there is an abrupt change in the size of attractor basins in the bistable region attractor basins undergo additional bifurcations wherein holes and islands are created within the basins when system parameters change these can be understood by examining critical surfaces for the coupled system

fluctuations of finite time stability exponents in the standard map and the detection of small islands some statistical properties of finite time stability exponents in the standard map can be estimated analytically the mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants whereas the mean carries information about the strength of the interaction and only indirect information about dynamical correlations the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength in particular the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time scalings the average of the stability matrix inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter the usual accelerator modes and the small islands created through double saddle node bifurcations which come halfway between the positions in interaction strength of the usual accelerator modes are clearly visible in the variance whose time scaling is capable of detecting the presence of islands as small as 0 01 of the phase space we study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions we also discuss the nature of this local approximation in some detail

statistics of finite time lyapunov exponents in a random time dependent potential the sensitivity of trajectories over finite time intervals t to perturbations of the initial conditions can be associated with a finite time lyapunov exponent lambda obtained from the elements m ij of the stability matrix m for globally chaotic dynamics lambda tends to a unique value the usual lyapunov exponent lambda infinity for almost all trajectories as t is sent to infinity but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity we compute for a particle moving in a randomly time dependent one dimensional potential how the distribution function p lambda t approaches the limiting distribution p lambda infinity delta lambda lambda infinity our method also applies to the tail of the distribution which determines the growth rates of moments of m ij the results are also applicable to the problem of wave function localization in a disordered one dimensional potential

a partial robust ridge based regression model with fuzzy predictors responses abstract this paper extends the conventional semi parametric partial linear regression model with fuzzy predictors and fuzzy response in cases where 1 outliers occur in data set 2 multicollinearity exists in predictors for this purpose the classical kernel fitting method is first extended for estimating unknown fuzzy smooth function then inspired by the conventional local linear smoothing method a kernel based weighted criterion is suggested for determining the exact coefficients on the basis of ridge methodology some common goodness of fit criteria are also used to examine the performance of the proposed method the effectiveness of the proposed method is then illustrated through two numerical examples including a simulation study the proposed method is also compared with some common fuzzy multiple regression models with fuzzy predictors and fuzzy responses the numerical results clearly indicate that 1 the proposed method is not sensitive to the outliers 2 the proposed method provides sufficiently accurate results in cases where multicollinearity happens among predictors and 3 comparing to existing fuzzy multiple regressions the proposed model provides more efficient predictions

statistics of stretching fields in experimental fluid flows exhibiting chaotic advection stretching fields and their statistical properties are studied experimentally for four distinct two dimensional time periodic confined fluid flows exhibiting chaotic advection a random vortex array for two different reynolds numbers a set of parallel shear layers and a vortex lattice the flows are driven electromagnetically and they are studied by means of precise particle velocimetry we find that for a given flow the probability distributions of log s where s is the local stretching in n cycles can be nearly superimposed for different n when log s is rescaled using the geometrical mean of the stretching distribution the rescaled stretching fields for a given flow at various n are highly correlated spatially when n is large finally the scaled distributions for different flows are similar though there are some differences connected to the degree of spatial symmetry and time reversibility of the flows

outlier induced collinearities when an observation in a regression analysis has very large values on two or more predictor variables artificial collinearities can be induced the effects of such collinearities on a regression analysis are not well documented although they can be shown to be similar in many respects to those resulting from approximate linear dependencies among the columns of predictor variable values the purpose of this article is to explore the effects of outlier induced collinearities on the estimation of regression coefficients

fixed point theorems in partially ordered metric spaces for operators with ppf dependence abstract in this paper fixed point theorems of a nonlinear operator are developed where the domain space c a b e is different from the range e which is a metric space with partial order an application of the fixed point theorem to a periodic boundary value problem with delay is also given

characterization of the domain chaos convection state by the largest lyapunov exponent using numerical integrations of the boussinesq equations in rotating cylindrical domains with realistic boundary conditions we have computed the value of the largest lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths we study in particular the domain chaos state which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects we compare our results with those from egolf et al nature 404 733 2000 who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short lived spatially localized dislocation nucleation events we also show that the quantity lambda1 is not intensive for aspect ratios gamma over the range 20 gamma 40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

characteristic distributions of finite time lyapunov exponents we study the probability densities of finite time or local lyapunov exponents in low dimensional chaotic systems while the multifractal formalism describes how these densities behave in the asymptotic or long time limit there are significant finite size corrections which are coordinate dependent depending on the nature of the dynamical state the distribution of local lyapunov exponents has a characteristic shape for intermittent dynamics and at crises dynamical correlations lead to distributions with stretched exponential tails while for fully developed chaos the probability density has a cusp exact results are presented for the logistic map stackrel ensuremath rightarrow x 4x 1 ensuremath x at intermittency the density is markedly asymmetric while for typical chaos it is known that the central limit theorem obtains and a gaussian density results local analysis provides information on the variation of predictability on dynamical attractors these densities which are used to characterize the nonuniform spatial organization on chaotic attractors are robust to noise and can therefore be measured from experimental data

implications of influence function analysis for sliced inverse regression and sliced average variance estimation sliced inverse regression sliced inverse regression ii and sliced average variance estimation are three related dimension reduction methods that require relatively mild model assumptions as an approximation for the relative influence of single observations from large samples the influence function is used to compare the sensitivity of the three methods to particular observational types the analysis carried out here helps to explain why there is a lack of agreement concerning the preferability of these dimension reduction procedures in general an efficient sample version of the influence function is also developed and evaluated

fixed point theorems in partially ordered metric spaces and applications abstract we prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order using a weak contractivity type of assumption besides including several recent developments our theorem can be used to investigate a large class of problems as an application we discuss the existence and uniqueness of solution for a periodic boundary value problem

detecting influential measurements in a growth curves model this paper comprises a study of the influence of deleting measurements on growth curve estimates the problem of influential measurements is considered both at the inference level and at the design level a measure of detecting influential vs uninfluential measurements is proposed the influence analysis envisaged also serves as a means for comparing the robustness of various models to missing measurements and to different study designs the paper concludes with an illustrative application

bayes factors for outlier models using the device of imaginary observations abstract suppose we think that most observations in a sample have been generated from a distribution with density f x but we fear that a few outliers from a distribution with density g x may have contaminated our sample in many situations we might assume that f x is a density depending on a parameter and that g x is of the same form as f but with parameter or a number of bayesian models for this problem when f is normal have been discussed by freeman he points out that with a vague improper prior for contaminating parameters most posterior weight is put on the model allowing for the largest number of outliers he therefore confines attention to proper priors when trying to answer the question of how many outliers however in many situations we do not have very certain information on the contaminating parameters and would like to make inferences about outliers when using improper priors for the parameters of the model in this article we apply the ideas of spiegelhalter and smith to

fixed point theorems for generalized contractions in partially ordered metric spaces with semi monotone metric abstract in this paper we study the existence and uniqueness of coupled fixed points for mixed monotone mappings in partially ordered metric spaces with semi monotone metric as an application we prove the existence and uniqueness of the solution for a first order differential equation with periodic boundary conditions

statistics of finite time lyapunov exponents in the ulam map the statistical properties of finite time lyapunov exponents at the ulam point of the logistic map are investigated the exact analytical expression for the autocorrelation function of one step lyapunov exponents is obtained allowing the calculation of the variance of exponents computed over time intervals of length n the variance anomalously decays as 1 n 2 the probability density of finite time exponents noticeably deviates from the gaussian shape decaying with exponential tails and presenting 2 n 1 spikes that narrow and accumulate close to the mean value with increasing n the asymptotic expression for this probability distribution function is derived it provides an adequate smooth approximation to describe numerical histograms built for not too small n where the finiteness of bin size trims the sharp peaks

fixed point theorems for mixed monotone operators with ppf dependence abstract in this paper fixed point theorems for mixed monotone operators are investigated by weakening the requirements in the contractive assumption and strengthening the metric space utilized with a partial order taking the domain space c a b e different from the range e and using the method of reflection operators

analytical signal analysis of strange nonchaotic dynamics we apply an analytical signal analysis to strange nonchaotic dynamics through this technique it is possible to obtain the spectrum of instantaneous intrinsic mode frequencies that are present in a given signal we find that the second mode frequency and its variance are good order parameters for dynamical transitions from quasiperiodic tori to strange nonchaotic attractors snas and from snas to chaotic attractors phase fluctuation analysis shows that snas and chaotic attractors behave identically within short time windows as a consequence of local instabilities in the dynamics in longer time windows however the globally stable character of snas becomes apparent this methodology can be of great utility in the analysis of experimental time series and representative applications are made to signals obtained from rossler and duffing oscillators

sensitivity of long periodic orbits of chaotic systems the properties of long numerically determined periodic orbits of two low dimensional chaotic systems the lorenz equations and the kuramoto sivashinsky system in a minimal domain configuration are examined the primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite amplitude parameter perturbations to address this question an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed the expectation of period averages floquet exponents and sensitivities over such set is then obtained it is shown that all these quantities converge to a limiting value as the orbit period is increased however while period averages and floquet exponents appear to converge to analogous quantities computed from chaotic trajectories the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state similar to observations made with other shadowing algorithms

a simple derivation of deletion diagnostic results for the general linear model with correlated errors a general simple and intuitive derivation is provided for diagnostics associated with the deletion of arbitrary subsets for the linear model with general covariance structure these are seen to be most simply expressed even for the well studied case of independent and identically distributed data in terms of a residual known variously as the conditional residual the deletion prediction residual and the cross validation residual particularly simple specializations arise when the subsets are of size 1 and of size 2 but the method is easy to apply for all subsets and to conditional deletions

influence diagnostics for linear measurement error models summary influence functions are used to show that extreme observations affect linear measurement error model estimators in directions orthogonal to and along the fitted plane rather than vertically and horizontally as with least squares estimators influence diagnostics patterned after those for least squares and derived for generalized linear models by pregibon 1981 are adapted to assess the influence of extreme observations on linear measurement error model estimators perturbation arguments and asymptotic theory are used to support the use of these diagnostics it is well known that least squares estimators are influenced by extreme observations in the response and predictor variables geometrically these influences are in directions defined by the axis of the response variable and the plane defined by the predictors as shown in 3 measurement error model estimators are affected in directions that are perpendicular to and along the fitted plane thus least squares influence diagnostics that measure the effects of observations in the vertical and the horizontal directions may fail to accurately indicate the true effects of extreme observations on measurement error model estimators kelly 1984 derives an influence function for a structural measurement error model in which the error distribution has finite fourth moments and the error covariance matrix is known up to a multiple with the aid of this influence function diagnostics such as cook s d are defined by analogy to the corresponding least squares equivalents fuller 1987 defines a measurement error model estimator of the hat matrix using estimated predictor variable values he also proposes using diagnostics and plotting techniques analogous to those of least squares with estimated model errors and estimated leverage values from the hat matrix miller 1989 provides partial theoretical justification for the use of these least squares based techniques by showing that the estimated model errors asymptotically are a gaussian process apart from these efforts little research has been conducted on influence diagnostics for measurement error models the derivation of influence diagnostics for measurement error models in this paper proceeds as follows measurement error model estimators are briefly described in 2 a more comprehensive discussion of these estimators is given by fuller 1987 ch 2 in 3 influence function arguments are used to establish the need for measurement error model diagnostics measurement error model estimators are expressed as iteratively reweighted least squares estimators in 4 leading to the adaptation of diagnostics for

assessing influence in multiple linear regression with incomplete data the problem of assessing influence and detecting influential cases in multiple linear regression with incomplete data is considered a case is said to be influential if appreciable changes in fitted regression coefficients occur when it is removed from the data a one step influence measure is derived based on the em algorithm for detecting cases that are influential in the maximum likelihood estimation of the regression coefficients results are compared with the complete data cook s distance measure techniques are demonstrated by examples

non gaussian fluctuations of local lyapunov exponents at intermittency in intermittent dynamical systems the distributions of local lyapunov exponents are markedly non gaussian and tend to be asymmetric and fat tailed a comparative analysis of the different time scales in intermittency provides a heuristic explanation for the origin of the exponential tails for which we also obtain an analytic expression deriving from a more quantitative theory application is made to several examples of discrete dynamical systems displaying intermittent dynamics

synchronization threshold in coupled logistic map lattices abstract we consider regular lattices of coupled chaotic maps whose local dynamics is ruled by the logistic map x 4 x 1 x a usual paradigm of chaotic systems through finite time exponents we scrutinize the lattice dynamics in the vicinity of the threshold of complete synchronization we connect dynamical features such as relaxation to the coherent state and intermittency with the statistics of finite time exponents focusing on the implications of the particular statistics related to the logistic map although numerical examples are given for lattice couplings decaying with distance as a power law our results are expected to be valid for a wider class of schemes coupling logistic maps

bifurcations and transitions in the quasiperiodically driven logistic map abstract we discuss several bifurcation phenomena that occur in the quasiperiodically driven logistic map this system can have strange nonchaotic attractors snas in addition to chaotic and regular attractors on snas the dynamics is aperiodic but the largest lyapunov exponent is nonpositive there are a number of different transitions that occur here from periodic attractors to snas from snas to chaotic attractors etc we describe some of these transitions by examining the behavior of the largest lyapunov exponent distributions of finite time lyapunov exponents and the invariant densities in the phase space

influential observations and outliers in regression statistics offered by cook 1977 and andrews and pregibon 1978 purport to reveal influential observations in a regression analysis detailed examination of these statistics shows that two different types of influence are being measured and this is illustrated with examples derived from a set of data given by mickey dunn and clark 1967 recommendations are given for obtaining the best use of the statistics available

k clustering as a detection tool for influential subsets in regression this article describes a new methodology for the detection of influential subsets in regression the method is based on an adaptation of computational and graphical techniques used in cluster analysis and makes use of some general properties of influential subsets but it is independent of any specific measure of influence for small to moderate data sets the proposed method is computationally efficient compared to existing search methods and it identifies subset candidates that merit attention according to some or all measures of joint influence that have appeared in the literature to date examples are given illustrating the method applied to two data sets previously analyzed in published studies

generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces abstract in this paper we extend the coupled fixed point theorems for mixed monotone operators f x x x obtained in t g bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 7 2006 1379 1393 by significantly weakening the contractive condition involved our technique of proof is essentially different and more natural an example as well as an application to periodic bvp is also given in order to illustrate the effectiveness of our generalizations

optimum influence of initial observations in regression models with ar 2 errors abstract this article considers an optimal procedure of identifying the influential observations in a regression model with ar 2 errors it develops some diagnostic techniques using a hat matrix the likelihood ratio and f test statistic have been discussed and an example of a constant mean model with ar 2 error shows its practical application in time series

local predictability and nonhyperbolicity through finite lyapunov exponent distributions in two degrees of freedom hamiltonian systems received 13 september 2007 revised manuscript received 5 november 2008 published 4 december 2008by using nite lyapunov exponent distributions we get insight into both the local and global properties ofa dynamical ow including its nonhyperbolic behavior several distributions of nite lyapunov exponentshave been computed in two prototypical four dimensional phase space hamiltonian systems they have beencomputed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals weanalyze how such distributions serve or not for tracing the orbit nature and local ow properties such as theunstable dimension variability as the axes are allowed or not to tend to the largest stretching direction therelationship between the largest and closest to zero exponent distribution is analyzed it shows a linear depen dency at short intervals related to the number of degrees of freedom of the system finally the hyperbolicityindexes associated to the shadowing times are calculated they provide interesting information at very localscales even when there are no gaussian distributions and the values cannot be regarded as random variables doi 10 1103 physreve 78 066204 pacs number s 05 45 a

transition to weak generalized synchrony in chaotically driven flows we study regimes of strong and weak generalized synchronization in chaotically forced nonlinear flows the transition between these dynamical states can occur via a number of different routes and here we examine the onset of weak generalized synchrony through intermittency and blowout bifurcations the quantitative characterization of this dynamical transition is facilitated by measures that have been developed for the study of strange nonchaotic motion weak and strong generalized synchronous motion show contrasting sensitivity to parametric variation and have distinct distributions of finite time lyapunov exponents

the conditional predictive ordinate for the normal distribution summary the conditional predictive ordinate cpo is a bayesian diagnostic which detects surprising observations it has been used in a variety of situations such as univariate samples the multivariate normal distribution and regression models results are presented about the most surprising observation which has minimum cpo for the multivariate normal distribution it is shown that the most surprising observation must lie at one of the vertices of the convex hull it is also shown that the observation with maximum mahalanobis distance from the sample mean must lie on the convex hull results are given for the expected number of vertices on the convex hull when the sample is contaminated an alternative closely related diagnostic the ratio ordinate measure is presented a numerical comparison of the two measures is given

diagnostic regression analysis and shifted power transformations diagnostic displays for outlying and influential observations are reviewed in some examples apparent outliers vanish after a power transformation of the data interpretation of the score statistic for transformations as regression on a constructed variable makes diagnostic methods available for detection of the influence of individual observations on the transformation methods for the power transformation are exemplified and extended to the power transformation after a shift in location for which there are two constructed variables the emphasis is on plots as diagnostic tools

influence and a quadratic form in the andrews pregibon statistic a quadratic form found in the andrews pregibon ap statistic may be decomposed into two terms one of the terms has a simple geometric interpretation that relates naturally to the expression of influence found in cook s statistic the measure of influence derived from the quadratic form is easy to compute once the two factors of the ap statistic have been found this measure suggests a computing strategy for diagnostics and leads to an exact relation between the ap statistic and cook s statistic

zero lyapunov exponent in the vicinity of the saddle node bifurcation point in the presence of noise we consider a behavior of the zero lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics we analytically deduce the laws for the dependence of this lyapunov exponent on the control parameter both above and below the bifurcation point the developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators we find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations we also discuss how the revealed regularities are expected to take place in other relevant physical circumstances

approximate inferences in multiresponse regression analysis summary we present an approximate inference procedure for the multiresponse regression model this can be used to determine approximate confidence regions or intervals and to formulate diagnostic tools such as standardized residuals and influence measures the overall approximation is based on approximations at two stages at the first stage the multiresponse regression model is replaced by a generalized least squares model using the estimated variance covariance matrix of the error terms at the second stage the nonlinear model functions are replaced by linear approximations

influential variables in linear regression based on a statistic proposed by cook 1977 for the detection of influential observations we propose a measure for the influence of variables in linear regression models and we introduce the leverage of a variable the new diagnostic can be used to identify important collinearities and in the checking of a fitted model to assess the effect of additional variables on the parameter estimates

high dimensional influence measure influence diagnosis is important since presence of influential observations could lead to distorted analysis and misleading interpretations for high dimensional data it is particularly so as the increased dimensionality and complexity may amplify both the chance of an observation being influential and its potential impact on the analysis in this article we propose a novel high dimensional influence measure for regressions with the number of predictors far exceeding the sample size our proposal can be viewed as a high dimensional counterpart to the classical cook s distance however whereas the cook s distance quantifies the individual observation s influence on the least squares regression coefficient estimate our new diagnosis measure captures the influence on the marginal correlations which in turn exerts serious influence on downstream analysis including coefficient estimation variable selection and screening moreover we establish the asymptotic distribution of the proposed influence measure by letting the predictor dimension go to infinity availability of this asymptotic distribution leads to a principled rule to determine the critical value for influential observation detection both simulations and real data analysis demonstrate usefulness of the new influence diagnosis measure

influence diagnostics for linear longitudinal models abstract influence diagnostics are important for analyzing cross sectional regression studies because they allow the analyst to understand the impact of individual observations on the estimated regression model in this article we consider the role of influence diagnostics in subject specific longitudinal models diagnostics are proposed under both fixed and random subject effects our approach is based on subject deletion which in this setting involves deleting a group of correlated observations we develop partial influence statistics to understand the combined impact of observations from a subject on population parameters simple computational formulas make the procedures feasible finally we illustrate the use of our new influence statistics by examining a dataset to model a taxpayer s charitable givings

analysis of high dimensional non hyperbolic coupled systems through finite time lyapunov exponents abstract in this work we investigate the relations among the unstable dimension variability udv and phase space dimensions in a coupled map lattice with diffusive coupling studying a simple system with udv at low dimensions we give theoretical support and numerical evidence to the statement that from some fixed dimensional value onwards there is no udv more precisely we construct a high dimensional non hyperbolic system without udv

using lyapunov exponents to predict the onset of chaos in nonlinear oscillators an analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed the method is based on the lyapunov type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the lyapunov characteristic exponents the concept of temporally local lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable and therefore the existence of chaotic attractors is impossible the procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point by considering the inverse conditions of asymptotic stability this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems including many cases of practical interest the examples of externally excited one and two well duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method as well as a good agreement of the theoretical predictions with the results of numerical experiments the comparison of the proposed method with melnikov s criterion shows a potential advantage of using the former one at high values of dissipation parameter and or multifrequency type of excitation in dynamical systems