stochastic bifurcation in duffing van der pol oscillators global analysis of bifurcation in the duffing van der pol oscillators under both additive and multiplicative random excitations is explored in detail by the generalized cell mapping method using digraph system parameters are chosen in the range that two co exist attractors and a saddles as an alternative definition stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value it is found that under certain conditions stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation by this global analysis the mechanism of development occurrence and evolution of stochastic bifurcation can be explored clearly and vividly

transition probabilities for degenerate diffusions arising in population genetics we provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong markov process arising in population genetics the markov processes that we study are a generalization of the classical wright fisher process the main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward kolmogorov equation defined on a compact manifold with corners which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities

lyapunov exponents for a duffing oscillator with the help of a parallel computer we perform a systematic computation of lyapunov exponents for a duffing oscillator driven externally by a force proportional to cos t in contrast to the familiar situation in discrete time systems where one finds windows of regularity embedded in intervals of chaos we find the continuous time duffing oscillator to contain a quite regular epetition of relatively self similar islands of chaos i e regions characterized by positive exponents embedded in large seas of regularity negative exponents we also investigate the effect of driving the oscillator with a jacobian elliptic function cn t m for m 0 one has cn t 0 cos t the usual trigonometric pumping for m 1 one has cn t 1 sech t a hyperbolic pumping when 0 m 1 the jacobian function is an intermediary double periodic function with periodicity depending on m and with fourier spectrum consisting of a regular train of narrow lines with varying envelope using m to tune the drive appropriately one may displace the islands of chaos in parameter space thus jacobian pumping provides a possible way of cleaning chaos in regions of the parameter space for periodically driven systems

non linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method abstract the incremental harmonic balance method is improved by i reversing the order of increment and linearization to find the bifurcated solution branches ii using a fast fourier transform to compute the solution increments and the floquet matrix and iii finding the secondary path after a bifurcation by an improved arc length method a system of two duffing oscillators linked with a linear stiffness is taken as a numerical example the qualitative change of solution due to the change of the linking stiffness is studied in detail

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

stochastic averaging of strongly non linear oscillators under combined harmonic and white noise excitations abstract a stochastic averaging procedure of strongly non linear oscillators subject to external and or parametric excitations of both harmonic and white noise forces is developed by using the so called generalized harmonic functions the procedure is applied to a duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises the averaged fokker planck kolmogrov equation is solved by using the path integration method based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration the stochastic jump of the duffing oscillator under combined harmonic and white noise excitations and its bifurcation as the system parameters frequency ratio strength of the non linearity amplitude of harmonic excitation and intensity of white noise change are examined for the first time

chaotic phase similarities and recurrences in a damped driven duffing oscillator we report strong evidence of remarkably close periodic repetitions of the structuring of the parameter space of a damped driven duffing oscillator as the amplitude of the drive increases families of period adding cascades and some intricate networks of periodic oscillations embedded in chaotic phases are also found to recur closely as the driving force grows such surprising regularities suggest that some hitherto unknown renormalization mechanism may be operating in higher codimension controlling the alternation of chaos and order in parameter space of certain flows

poincare maps of duffing type oscillators and their reduction to circle maps i analytic results bifurcation diagrams and plots of lyapunov exponents in the r omega plane for duffing type oscillators x 2rx v x omega t 0 exhibit a regular pattern of repeating self similar tongues with complex internal structure the authors demonstrate here that this behaviour is easily understood qualitatively and quantitatively from the poincare map of the system in action angle variables this map approaches the one dimensional form phi n 1 a c e rt cos phi n t pi omega provided e rt but not necessarily ce rt r and omega are small the authors derive asymptotic for small r omega formulae for a and c for a special class of potentials v they argue that these special cases contain all the information needed to treat the general case of potentials which obey v or 0 at all times the essential tools of the derivation are the use of action angle variables the adiabatic approximation and the introduction of a non oscillating reference solution of duffing s equation with respect to which the action angle variables have to be determined these allow the explicit construction of the poincare map in powers of e rt to first order the authors obtain the phi map which survives asymptotically to second order they obtain the two dimensional i phi map in the i direction it contracts by a factor e rt upon each iteration

randomly disordered periodic induced chaos in a piezoelectric vibration energy harvester system with fractional order physical properties abstract the chaotic behavior of the piezoelectric vibration energy harvester veh system with fractional order physical properties under randomly disordered periodic excitations is investigated by using random melnikov method a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system the results indicate that the increase of the noise intensity will result in the occurrence of chaos and the changes of the possible chaotic region in phase space which first enlarging and then shrinking with a change in trend the threshold of amplitude of randomly disordered periodic for the onset of chaos is determined by the numerical calculation via the largest lyapunov exponent the effects of noise intensity on chaos are also investigated through the largest lyapunov exponent phase portraits poincare maps at the same time the effects of intensity of random frequency on the mean square voltage are further discussed which show that the square voltage is in positive proportion to the size of the chaotic region it is demonstrated that the essential changes of the dynamical behavior of the piezoelectric energy harvester system with fractional order physical properties will occur through changing the noise intensity which can not only induce or suppress the onset of the chaos but also raise or reduce the mean square voltage finally the 0 1 test of responses is used to quantify the responses of veh system which further supports the effects of noise intensity on chaotic behavior of the system

on response of nonlinear oscillators with random frequency of excitation abstract some types of nonlinear oscillators for which the frequency of excitation is stochastic are investigated the paper consists of two parts in the first part equations of motion are linearized with the aid of stochastic averaging differential equations for the mean and variance of the process are obtained and integrated numerically this approach is applicable for weakly nonlinear oscillators the case of strong nonlinearity is considered in the second part making use of computer simulation a number of stochastic realizations of the process are computed the stochastic process is characterized by the mean and standard deviation of these realizations calculations have been carried out for the duffing ueda and van der pol equations and for forced vibrations of a pendulum these calculations show that if attractors exist then the deterministic vibrations which may be chaotic turn regular by adding noise and the motion terminates in a stable fixed point or on a limit cycle

stability transitions of certain exact periodic responses in undamped helmholtz and duffing oscillators abstract a linear stability analysis in which phase integral theory is used is applied to certain exact subharmonic responses of undamped helmholtz and duffing oscillators which are valid beyond the limits of perturbation theory infinite sequences of stability transitions are identified in terms of the mathieu parameters and their limiting behaviours are investigated

a continuation algorithm for discovering new chaotic motions in forced duffing systems we study a dynamical system described by the duffing equation under static plus large periodic excitation the bifurcation diagram in terms of the amplitude of the periodic excitation reveals that there are at least six cascades of period doubling bifurcations with the other parameters fixed at specific values these six sequences are further investigated by a continuation algorithm which is based on the principles of the shooting method combined with the newton method for solving nonlinear equations a runge kutta huta method has been used for solving the system of differential equations the final conclusion is that each of the six sequences is governed by feigenbaum s number 4 6692 from universality theory beyond the limit values derived for the amplitude of the periodic excitation new strange attractors are found

the action of outer automorphisms on bundles of chiral blocks on the bundles of wzw chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine lie algebra these bundle isomorphisms respect the knizhnik zamolodchikov connection and have finite order when all primary fields are fixed points the isomorphisms are endomorphisms in this case the bundle of chiral blocks is typically a reducible vector bundle a conjecture for the trace of such endomorphisms is presented the proposed relation generalizes the verlinde formula our results have applications to conformal field theories based on non simply connected groups and to the classification of boundary conditions in such theories

wavelet analysis of time series for the duffing oscillator the detection of order within chaos we consider a wavelet analysis of various time series for the duffing oscillator for which there is a potential maximum and a harmonic forcing term and we focus on time series that return to the region of the potential maximum when the dynamics is chaotic and the time series is highly nonstationary there are many significant higher harmonics in a fourier expansion and the usual fourier analysis is problematic especially for short total times we show that the wavelet analysis is a robust tool that may be used to obtain qualitative information for highly nonstationary time series specifically that it may be used to detect a small amplitude harmonic forcing term even when the dynamics is chaotic and even for short total times

stochastic averaging of strongly non linear oscillators under bounded noise excitation abstract a stochastic averaging method for strongly non linear oscillators under external and or parametric excitation of bounded noise is proposed by using the so called generalized harmonics functions the method is then applied to study the primary resonance of duffing oscillator with hardening spring under external excitation of bounded noise the stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase subsequently the method is applied to study the dynamical instability and parametric resonance of duffing oscillator with hardening spring under parametric excitation of bounded noise the primary unstable region is delineated by evaluating the lyapunov exponent of linearized system and the response and jump of non linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase

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

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

quasiperiodic resonance and bifurcations of tori in the weakly nonlinear duffing oscillator abstract we study analytically and numerically the weakly nonlinear quasiconservative duffing oscillator under 2 periodic forcing utilizing the secondary averaging approach we derive approximate analytical expressions for the quasiperiodic response of the oscillator and for conditions of torus bifurcations the theoretical results are in good agreement with the numerical ones the structure of the bifurcation sets in the oscillator parameter space is similar to that of a one dimensional map with two extrema

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

principal response of duffing oscillator to combined deterministic and narrow band random parametric excitation abstract the principal resonance of a duffing oscillator to combined deterministic and narrow band random parametric excitations is investigated in particular the case in which the parametric terms share close frequencies is examined the method of multiple scales is used to determine the equations of modulation of amplitude and phase the behavior stability and bifurcation of steady state response are studied by means of qualitative analyses jumps are shown to occur if the random excitation is small the effects of damping detuning and magnitudes of deterministic and narrow band parametric excitations are analyzed the theoretical analyses are verified by numerical results

experimental mimicry of duffing s equation abstract extensive analytical and numerical investigations have focused on duffing s equation however experimental work in a mechanics context has been limited to studying systems the stiffness characteristics of which can be approximated by a non linear cubic restoring force e g a buckled beam excited transversely or a rigid pendulum undergoing moderately large amplitude motion this work describes a novel experimental approach whereby a particle rigid body is contrived to mimic the behavior of duffing s equation this is a direct extension of the concept of a ball rolling on a double well potential energy surface both free and forced oscillations are considered illustrating familiar non linear dynamics features including competing steady state attractors hysteresis sensitivity to initial conditions subharmonic oscillations and chaos

structure in the bifurcation diagram of the duffing oscillator we identify four levels of structure in the bifurcation diagram of the two well periodically driven duffing oscillator plotted as a function of increasing control parameter t the period of the driving term the superstructure or bifurcation peninsula repeats periodically as t increases by ensuremath sim 2 ensuremath pi beginning and ending with symmetric period one orbits whose local torsions differ by 2 within each bifurcation peninsula there is a systematic window structure the primary window structure is due to newhouse and newhouse like orbits fine structure is due to a farey sequence of well ordered orbits between the primary windows hyperfine structure consists of very narrow windows associated with non well ordered orbits we construct a template for the duffing oscillator a two dimensional return map and a one dimensional return map which describes the systematics of orbit creation and annihilation all structures are identified by topological indices our predictions are based on and compatible with numerical computations

steady state response of axially moving viscoelastic beams with pulsating speed comparison of two nonlinear models abstract principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed a nonlinear partial differential equation governing the transverse vibration is derived from the dynamical constitutive and geometrical relations under certain assumption the partial differential reduces to an integro partial differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams the method of multiple scales is applied to two equations to calculate the steady state response closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms the stability of straight equilibrium and nontrivial steady state response are analyzed by use of the lyapunov linearized stability theory numerical examples are presented to highlight the effects of speed pulsation viscoelascity and nonlinearity and to compare results obtained from two equations

duffing equation with two periodic forcings the phase effect a weak additional sinusoidal perturbation is applied to the periodically forced nonlinear oscillator to suppress chaos numerical simulations show that the phase difference between the two sinusoidal forces plays a very important role in controlling chaos when the frequencies of these forces deviate from the resonance condition slightly a different type of intermittency alternation from regular motion to chaotic motion called breather here is observed if the phase difference follows a wiener process conventional intermittency is observed

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

stochastic jump and bifurcation of a duffing oscillator under narrow band excitation abstract the variance of the displacement the probability densities of the displacement velocity and amplitude and the joint probability density of the displacement and velocity of the stationary response of a duffing oscillator to narrow band gaussian random excitation for a number of combinations of the parameters of the oscillator and excitation obtained from digital simulation are presented based on these results the stochastic jump and bifurcation of the system are examined it is shown that for each combination of the parameters all the statistics of the stationary response are unique and independent of initial conditions in a certain domain of the parameter space there are two more probable motions in the stationary response and jumps may occur the jump is a transition from one more probable motion to another or vice versa outside of this domain the stationary response is either nearly gaussian or like a diffused limit cycle as the parameters cross the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation it is pointed out that some previous results and explanations on this problem are incorrect

relaxation in a duffing potential abstract the solution of the equation of motion for a particle in a duffing potential v x 1 x 2 2 2 x 4 4 1 2 0 for arbitrary anharmonicity strength is characterized by the presence of odd frequencies which implies that velocity and position autocorrelation functions of such an oscillator in a microcanonical ensemble are also characterized by odd frequencies it is however non trivial to determine whether such discrete frequencies also characterize the autocorrelation functions in a canonical ensemble as discussed recently by fronzoni et al j stat phys 41 1985 553 we recover and extend upon the results of fronzoni et al to show analytically via mori lee theory that essentially discrete i e well defined peaks with finite but small width temperature dependent frequencies characterize the autocorrelation functions in a canonical ensemble

c0 estimates and smoothness of solutions to the parabolic equation defined by kimura operators abstract kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics their infinitesimal generator is an elliptic differential operator whose second order coefficients matrix degenerates on the boundary of the domain in this article we consider the inhomogeneous initial value problem defined by generators of kimura diffusions and we establish c 0 estimates which allows us to prove that solutions to the inhomogeneous initial value problem are smooth up to the boundary of the domain where the operator degenerates even when the initial data is only assumed to be continuous

using the extended melnikov method to study the multi pulse global bifurcations and chaos of a cantilever beam the aim of this paper is to investigate the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended melnikov method in the resonant case first the extended melnikov method for studying the shilnikov type multi pulse homoclinic orbits and chaos in high dimensional nonlinear systems is briefly introduced in the theoretical frame then this method is utilized to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the cantilever beam how to employ this method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications is demonstrated through this example finally the results of numerical simulation are given and also show that the shilnikov type multi pulse chaotic motions can occur for the nonlinear non planar oscillations of the cantilever beam which verifies the analytical prediction

galerkin methods for natural frequencies of high speed axially moving beams in this paper natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges in the supercritical transport speed regime the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions the governing equations of coupled planar is reduced to two nonlinear models of transverse vibration for motion about each bifurcated solution those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform the natural frequencies are investigated for the beams via the galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary differential equations under the simple support boundary numerical results indicate that the nonlinear coefficient has little effects on the natural frequency and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro partial differential equation yields results quantitatively closer to those of the coupled equations

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

influences of harmonic coupling on bifurcations in duffing oscillator with bounded potential well abstract the dynamics of bifurcations in a dufhng oscillator with a bounded potential well is investigated special attention is paid to the prominent phenomena of repeated sets of symmetry and reverse symmetry breaking the harmonic balance method and floquet theory are employed to account for this peculiar phenomena

abelian and non abelian branes in wzw models and gerbes we discuss how gerbes may be used to set up a consistent lagrangian approach to the wzw models with boundary the approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted chan paton gauge field using the geometric approach we present a complete classification of the branes that conserve the diagonal current algebra symmetry in the wzw models with simple compact but not necessarily simply connected target groups such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry abelian or non abelian twisted gauge fields the latter situation occurs for the conjugacy classes with fundamental group 2 2 in so 4n 2 the branes supported by such conjugacy classes have to be equipped with a projectively flat twisted u 2 gauge field in one of the two possible wzw models differing by discrete torsion we show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the wzw models with non simply connected target groups

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

optimal escape from potential wells patterns of regular and chaotic bifurcation abstract the patterns of bifurcation governing the escape of periodically forced oscillations from a potential well over a smooth potential barrier are studied by numerical simulation both the generic asymmetric single well cubic potential and the symmetric twin well potential duffing oscillator are surveyed by varying three parameters forcing frequency forcing amplitude and damping coefficient the close relationship between optimal escape and nonlinear resonance within the well is confirmed over a wide range of damping subtle but significant differences are observed at higher damping ratios the possibility of indeterminate outcomes of jumps to and from resonance near optimal escape is cmppletely suppressed above a critical level of the damping ratio about 0 12 for the asymmetric single well oscillator coincidentally at almost the same level of damping the optimal escape condition becomes distinct from the apex in the f plane of the bistable regime this corresponds to the appearance of chaotic attractors which subsume both resonant and non resonant motions within one well at higher damping levels further changes occur involving conversions from chaotic saddle to regular saddle bifurcations these changes in optimal escape phenomena correspond to codimension three bifurcations at exceptional points in the space of three parameters these bifurcations are described in terms of homoclinic and heteroclinic structures of invariant manifolds and changes in accessible boundary orbits the same sequence of codimension three bifurcations is observed in both the twin well duffing oscillator and the asymmetric single well escape equation within the codimension three bifurcation patterns governing escape one particular codimension two global bifurcation involves a chaotic attractor explosion or interior crisis compounded with a blue sky catastrophe or boundary crisis of the exploded attractor this codimension two bifurcation has structure containing a form of predictive power knowledge of attractor bifurcations in part of the codimension two pattern permits inference of the attractor and basin bifurcations in the remainder this predictive power is applicable beyond the context of escape from potential wells quantitative correlation of bifurcation patterns between the two equations according to simple scaling laws is tested the unstable periodic orbits which figure most prominently in the major attractor basin bifurcations are of periods one and three their linking is conveniently interpreted by a three layer spiral horseshoe structure for the folding action in phase space within a well the structure of this 3 shoe implies a partial ordering among order three subharmonic saddle node bifurcations this helps explain the sequence of codimension three bifurcations near optimal escape some bifurcational precedence relations are known to follow from the linking of periodic orbits in a braid on a 3 shoe additional bifurcational precedence relations follow from a quantitative property of generic potential wells the dynamic hilltop saddle has a very large expanding multiplier over one cycle of forcing near fundamental resonance this quantitative property explains the close coincidence of codimension three bifurcations near the suppression of indeterminate outcomes an experimentalist s approach to identifying the three layer template structure from time series data is discussed including a consistency check involving poincare indices the bifurcation patterns emerging at higher damling values create favorable conditions for realizing experimental strategies to recognize optimal escape and locate it in parameter space strategies based solely on observations of quasi steady behavior while remaining always within one well are discussed

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

oscillatory behavior of solutions of forced second order differential equations with alternating coefficients abstract in this paper we study the oscillatory behavior of solutions of the forced second order nonlinear ordinary functional differential equation x t g x t p t d dt sufficient conditions are established for all solutions of the equation to be oscillatory as well as bounded

multidimensional lindstedt poincare method for nonlinear vibration of axially moving beams abstract the multidimensional lindstedt poincare mdlp method is extended to the nonlinear vibration analysis of axially moving systems galerkin method is used to discretize the governing equations the forced response of an axially moving beam with internal resonance between the first two transverse modes is studied the fundamental harmonic resonance is studied the response curves exhibit the same internal resonance characteristics as that of non transferring thin plates and beams because all these systems possess cubic nonlinearity and similar frequency distribution the examples show that the results of the mdlp method agree reasonably well with that obtained by the incremental harmonic balance ihb method however the former is more straightforward and efficient for obtaining the solution

the feynman kac formula and harnack inequality for degenerate diffusions we study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics the so called generalized kimura diffusion operators epstein and mazzeo siam j math anal 42 2010 568 608 degenerate diffusion operators arising in population biology 2013 princeton university press applied mathematics research express 2016 our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower order coefficients and the proof of the scale invariant harnack inequality for nonnegative solutions to the kimura parabolic equation the stochastic representation of solutions that we establish is a considerable generalization of the classical results on feynman kac formulas concerning the assumptions on the degeneracy of the diffusion matrix the boundedness of the drift coefficients and the a priori regularity of the weak solutions

conformal blocks verlinde formula and diagram automorphisms abstract the verlinde formula computes the dimension of the space of conformal blocks associated to simple lie algebras and stable pointed curves if a simply laced simple lie algebra admits a nontrivial diagram automorphism then this automorphism acts on the space of conformal blocks naturally we prove an analogue of the verlinde formula for the trace of the diagram automorphism on the space of conformal blocks along the way we get an analogue of the kac walton formula for the trace of the diagram automorphism we also get a twining type formula between the conformal blocks for the pair s l 2 n 1 s p 2 n

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

from dynkin diagram symmetries to fixed point structures any automorphism of the dynkin diagram of a symmetrizable kac moody algebra g induces an automorphism of g and a mapping between highest weight modules of g for a large class of such dynkin diagram automorphisms we can describe various aspects of these maps in terms of another kac moody algebra the orbit lie algebra g in particular the generating function for the trace of over weight spaces which we call the twining character of g with respect to the automorphism is equal to a character of g the orbit lie algebras of untwisted affine lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory orbit lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory

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

bifurcation analysis of two coupled periodically driven duffing oscillators bifurcation diagrams and phase diagrams of two coupled periodically driven identical duffing oscillators are presented it is shown that the global pattern of bifurcation curves in parameter space consists of repeated subpatterns similar to the superstructure observed for single periodically driven strictly dissipative oscillators the subpattern itself however is different from that of a single duffing oscillator due in particular to hopf bifurcations that are newly added to the bifurcation scenario

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

continuous extension of the geometric control method in this paper we give a continuous version valid for flows of the geometric control method introduced by toroczkai et al 1994 and constructed for mappings on 2d poincare sections this method does not require explicit knowledge of the dynamics just a rough location of the periodic orbit and a single parameter easily computed from four data points applicability in numerical and real experiments is discussed

stability analysis of an axially accelerating string the dynamic response of an axially accelerating string is investigated the time dependent velocity is assumed to vary harmonically about a constant mean velocity approximate analytical solutions are sought using two different approaches in the first approach the equations are discretized first and then the method of multiple scales is applied to the resulting equations in the second approach the method of multiple scales is applied directly to the partial differential system principal parametric resonances and combination resonances are investigated in detail stability boundaries are determined analytically it is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system or when the frequency is close to the sum of any two natural frequencies when the velocity variation frequency is close to zero or to the difference of two natural frequencies however no instabilities are detected up to the first order of perturbation numerical results are presented for a band saw and a threadline problem

the fundamental solution to 1d degenerate diffusion equation with one sided boundary abstract in this work we adopt a combination of probabilistic approach and analytic method to study the fundamental solution to a certain type of one dimensional degenerate diffusion equation to be specific we consider a diffusion equation on 0 whose diffusion coefficient vanishes at the boundary 0 equipped with the cauchy initial data and the dirichlet boundary condition one such diffusion equation that has been extensively studied is the one whose diffusion coefficient vanishes linearly at 0 our main goal is to extend the study to cases when the diffusion coefficient has a general order of degeneracy with a primary focus on the fundamental solution to such a degenerate diffusion equation in particular we study the regularity properties of the fundamental solution near 0 and investigate how the order of degeneracy of the diffusion operator and the dirichlet boundary condition jointly affect these properties we also provide estimates for the fundamental solution and its derivatives near 0

complex dynamics of high speed axially moving systems in this paper the dynamic response of a simply supported travelling beam subjected to a transverse load is investigated in the super critical speed range the well known axially moving beam theory is considered and a simple viscous damping mechanism has been introduced the displacement field is expanded in a series of the buckling modes a sine series and different techniques have been used in analyzing the response of the dynamical system periodic oscillations are studied by means of continuation techniques while non stationary dynamics are investigated through direct simulations a comparison with the literature and a convergence test on the series expansion are performed a sample case of a physical beam is developed and numerical results are presented concerning bifurcation analysis and stability and direct simulations of global postcritical dynamics a complex scenario of alternate regular and chaotic motions is found in a large range of the main parameters