forced oscillations of a self oscillating bimolecular surface reaction model the effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal bimolecular surface reaction model in which two vacant sites are required for reaction the forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed and the response of the system is characterized with the aid of the stroboscopic map where a two parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters the various responses include subharmonic quasi periodic and chaotic solutions in addition bistability between one or more of these responses has been observed bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns period doubling hopf bifurcations including hard resonances homoclinic tangles and several different codimension two bifurcations

the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

differential equations on variable time scales abstract we introduce a class of differential equations on variable time scales with a transition condition between two consecutive parts of the scale conditions for existence and uniqueness of solutions are obtained periodicity boundedness and stability of solutions are considered the method of investigation is by means of two successive reductions b equivalence of the system e akalin m u akhmet the principles of b smooth discontinuous flows computers and mathematics with applications 49 2005 981 995 m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear analysis 60 2005 163 178 m u akhmet n a perestyuk the comparison method for differential equations with impulse action differential equations 26 9 1990 1079 1086 on a variable time scale to a system on a time scale a reduction to an impulsive differential equation m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear analysis 60 2005 163 178 m u akhmet m turan the differential equations on time scales through impulsive differential equations nonlinear analysis 65 2006 2043 2060 appropriate examples are constructed to illustrate the theory

growth rates and fractional dimensions in cellular automata abstract this paper concerns some different ways by which a fractional dimension may be associated to a cellular automaton whose transition rule is linear modulus p the growth rate dimensions are shown to equal the kolmogorov dimensions suggestions are made as to the relevance for dendritic crystal growth

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

suppression of period doubling in the dynamics of a bouncing ball abstract we present a theoretical and experimental study of a bouncing ball system where the effect of a near resonant perturbation is to suppress the onset of the first period doubling bifurcation near the bifurcation point the full dynamical equations are reduced to a discrete time map governing the dynamics on a slowly oscillating center manifold the derivation emphasizes the geometry of the phase space dynamics and serves to clarify several points of a recently proposed theory regarding the effects of strong near resonant perturbations our experimental results agree with the simple theory for moderate perturbations for larger perturbations higher order effects must be included

global analysis of self similarity features of cellular automata selected examples abstract the analysis of self similarity properties of the evolution patterns of linear cellular automata is investigated using the theory of hierarchical iterated function systems and substitution systems we present several selected examples

dynamical behavior of vibro impact machinery near a point of codimension two bifurcation a dual component system with vibro impact is considered local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold and normal form method for maps the period n single impact motion and poincare map of the vibro impact system are derived analytically stability and local bifurcation of single impact periodic motion are analyzed by using the poincare map a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behavior of the dual component system with vibro impact near the point of codimension two bifurcation is analyzed it is found that near the point of codimension two bifurcation there exists not only hopf bifurcation of period one single impact motion but also hopf bifurcation of period two double impact motion period doubling bifurcation of period one single impact motion commonly exists near the point of codimension two bifurcation however no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of hopf bifurcation associated with period two fixed points the period two fixed points are symmetrical about the corresponding period one point the results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of hopf bifurcation of period two points two actual examples the impact forming machinery and inertial shaker are chosen to analyze further the phenomena of codimension two bifurcation of maps and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro impact systems near the points of codimension two bifurcations

on the structure of mandelbrot s percolation process and other random cantor sets we consider generalizations of mandelbrot s percolation process for the process which we call the random sierpinski carpet we show that it passes through several different phases as its parameter increases from zero to one the final section treats the percolation phase

periodic solution finder for an impact oscillator with a drift in this paper an efficient semi analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift which explains the progression mechanism in vibro impact systems and can be used to optimize their performance the method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known for example a period may consist of the following sequential phases i contact with progression ii contact without progression iii no contact and iv contact without progression using this information a system of four piecewise linear first order differential equations is transformed to a system of non linear algebraic equations the method allows one to accurately predict a range of control parameters for which the best progression rates are obtained

period doubling cascades of attractors a prerequisite for horseshoes this paper shows that if a horseshoe is created in a natural manner as a parameter is varied then the process of creation involves the appearance of attracting periodic orbits of all periods furthermore each of these orbits will period double repeatedly with those periods going to infinity

stochastic response of vibro impact duffing oscillators under external and parametric gaussian white noises abstract this study presents a solution procedure for the stationary probability density function pdf of the response of vibro impact duffing oscillators under external and parametric gaussian white noises first the zhuravlev non smooth coordinate transformation is adopted to convert a vibro impact oscillator into an oscillator without barriers the stationary pdf of the converted oscillator is governed by the fokker planck fp equation the fp equation is solved by the exponential polynomial closure epc method illustrative examples are presented with vibro impact duffing oscillators under external and parametric gaussian white noises to show the effectiveness of the solution procedure the parametric excitation is acting in displacement and the constraint is a unilateral zero offset barrier the restitution coefficient of impacts is taken as 0 90 comparison with the simulated results shows that the proposed solution procedure can provide good approximate pdfs for displacement and velocity although a little difference exists in the tail of these pdfs this difference may be due to the weak approximation on the response of the vibro impact oscillators using a continuous markov process when the restitution coefficient is not very close to unity

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

analysis of a random repeated impact process a simple random repeated impact process similar to the classical random walk process is analyzed the process which consists of a ball bouncing on a randomly vibrating surface is analogous to loss of contact situations which can occur in linkages and vibrating tools it also has relevance to rolling contact where the rolling element may separate from the surface and sustain repeated impacts a coefficient of restitution is used to describe impact and some important assumptions limit the analysis to values of this coefficient greater than 0 8 the markov nature of the process is demonstrated by its one step memory it may be regarded as a discrete markov process imbedded in continuous time a difference equation governing the process is developed and analyzed for the case where the vibrating surface has a gaussian distribution of velocities with the one step transition probability density function a linear weighting function is used to account for the weighting effect of the ball s velocity on the distribution of table velocities at impact the resulting integral equation is solved iteratively to yield the probability density function of ball velocities after impact this information may then be used to predict the magnitudes of the impacts and the time between them

fractals their transfer matrices and their eigen dimensional sequences this paper defines and studies a general algorithm for constructing new families of fractals in euclidean space this algorithm involves a sequence of linear interscale transformations that proceed from large to small scales the authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a variable number of addend fractals the addends relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix which they call the transfer matrix of the fractal tmf the authors first demonstrate by a series of examples then prove rigorously that the eigenvalues of our tmfs are real and positive and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of their tmf they say that these dimensions form the overall fractal s eigendimensional sequence the eigenvalues of their tmf are integers in the non random variants of the construction but are non integer in the random variants a geometrical interpretation of the eigenvalues and the eigenvectors is given their tmf have other striking and very special properties that deserve additional attention

on the sensitivity of additive cellular automata in besicovitch topologies we prove that additive cellular automata in the besicovitch topology that have a willson limit set of hausdorff dimension strictly bigger than 1 are sensitive to initial conditions

statistical mechanics of a dissipative dynamical system with small noise abstract a wagon rolls up a slightly inclined plane effective gravity acceleration g and then it rolls down until it collides inelastically coefficient of restitution 1 with a piston oscillating parallel to the plane between 0 and max and with velocity f t where is amplitude is frequency and f is a specified waveform e g sinusoidal in the high frequency limit g there are very many periodic solutions but in the presence of a small noise level e g the standard deviation in none of these are realizable because when this noise is assumed to introduce a random phase tn n mod 2 of the piston each time tn the wagon arrives at max moving down hill with velocity v for mathematical simplicity we confine our attention to the case of a square wave oscillator f 1 and derive an evolution equation for the probability distribution function pn 1 v numerical solutions reveal the approach to a unique statistically steady state n independent of the assumed initial distribution p1 v and independent of the noise level

bouncing ball problem stability of the periodic modes exploring all its ramifications we give an overview of the simple yet fundamental bouncing ball problem which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity the dynamics is modeled on the basis of a discrete map of difference equations which numerically solved fully reveals a rich variety of nonlinear behaviors encompassing irregular nonperiodic orbits subharmonic and chaotic motions chattering mechanisms and also unbounded nonperiodic orbits for periodic motions the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map through numerical examples it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v gamma 1 epsilon where gamma is the nondimensionalized shaking acceleration and epsilon the coefficient of restitution which quantifies the amount of energy lost in the ball table collision

discontinuity mappings for stochastic nonsmooth systems abstract for stability and bifurcation analysis involving recurrent behaviour such as periodic orbits it is important to be able to quantify how nearby trajectories behave by means of a local mapping in smooth systems these mappings can be computed using the system s variational equations for piecewise smooth or hybrid systems the same technique cannot be used without some corrections this is due to the fact that nearby trajectories can be topologically distinct because they can undergo different sequences of events associated with the system s discontinuity boundaries to account for this one can derive zero time discontinuity mappings associated with boundary interactions in this paper we derive zero time discontinuity mappings for piecewise smooth vector fields and hybrid dynamical systems in which the position of the discontinuity boundary has a stochastic component in particular we consider systems with stochastically oscillating boundaries and systems with stochastic imperfections on the discontinuity boundary

chaos in a noisy mechanical system with stress relaxation the influence of periodic and random external excitation on the chaotic behaviour of a self excited system is considered the lyapunov exponents of the noisy system have been defined as random variables the properties of their distribution which allow one to quantify chaos have been determined

diffusion on complex cellular automata patterns we studied random walks on two dimensional patterns formed by the sequence of configurations of complex elementary cellular automata ca with random initial configurations the walkers are allowed to jump between nearest neighbours or next nearest neighbours 1 sites on patterns of rules 22 54 90 122 126 150 and 182 the diffusion is normal 1 2 on patterns 18 and 146 the diffusion is anomalous with 0 415 0 005 and the spectral dimension is ds 1 25 0 1 from the analysis of the diffusion in the horizontal and vertical directions space and time directions of the ca problem respectively of patterns 18 and 146 we obtained x 0 22 and y 0 42 because they have branches of 1 sites which are long in the vertical direction but narrow in the horizontal direction due to the anisotropic diffusion the results do not satisfy the relation ds 2df dw with dw 1 and fractal dimension df 2 considering that the fractal dimension of the region visited by the walker is equal to the dimension of the substrate df we suggest the new scaling relation ds 2 x y this relation is supported by our numerical results and may be generalized to other structures with anisotropic diffusion

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

the equality of fractional dimensions for certain cellular automata abstract a study is made of the development of certain cellular automata with apparently chaotic behavior results are obtained for cellular automata arising from additive rules modulo 2 for the corresponding limit sets it is shown that the growth rate dimension the kolmogorov dimensions and the hausdorff dimensions all coincide this dimension is moreover independent of the choice of finite initial seed and the sampling times used these cellular thus provide test cases for the production of fractals by local processes

optimal imaging parameters for fiber orientation estimation in diffusion mri this study uses monte carlo simulations to investigate the optimal value of the diffusion weighting factor b for estimating white matter fiber orientations using diffusion mri with a standard spherical sampling scheme we devise an algorithm for determining the optimal echo time pulse width and pulse separation in the pulsed gradient spin echo sequence for a specific value of b the monte carlo simulations provide an estimate of the optimal value of b for recovering one and two fiber orientations we show that the optimum is largely independent of the noise level in the measurements and the number of gradient directions and that the optimum depends only weakly on the diffusion anisotropy the maximum gradient strength and the spin spin relaxation time the optimum depends strongly on the mean diffusivity in brain tissue the optima we estimate are in the ranges 0 7 1 0 x 10 9 s m 2 and 2 2 2 8 x 10 9 s m 2 for the one and two fiber cases respectively the best b for estimating the fractional anisotropy is slightly higher than for estimating fiber directions in the one fiber case and slightly lower in the two fiber case to estimate tr d in the one fiber case the optimal setting is higher still simulations suggest that a ratio of high to low b measurements of 5 to 1 is a good compromise for measuring fiber directions and size and shape indices

minimization of imaging gradient effects in diffusion tensor imaging a new sample independent optimization criterion for minimizing the effect of the imaging gradients including the directional awareness they create is defined for diffusion tensor imaging dti experiments the dti linear algebraic framework is expanded to a normed space to design optimal diffusion gradient schemes dgs in an integral fashion without separating the magnitude and direction of the dgs vectors the feasible space of dgs vectors for which the estimation equations are determinate thus a hard constraint for the optimization is parametrized newly generated optimal dgss demonstrate on an isotropic sample and an ex vivo baboon brain that the optimization goals are reached as well as a significant increase in estimation performance

computing fractal dimensions for additive cellular automata abstract efficient methods are given for finding the statistics of symbol blocks occuring in the graphs of additive cellular automata after large numbers of iterations in particular the methods yield fractal dimensions entropies and a recurrence relation for the number of nonzero symbols the methods are based on the construction of a nonnegative integer matrix a closely related to the automation the largest eigenvalue of a tells the fractal dimension while the corresponding eigenvector gives the relative frequencies of blocks

cellular automata and multifractals dimension spectra of linear cellular automata abstract the dimension distribution of space patterns of linear cellular automata are studied a dimension spectrum derived from the dimension distribution is shown to correspond to a singularity spectrum of a multifractal using this relation the dimension spectrum is efficiently calculated

probability distribution functions for a single particle vibrating in one dimension experimental study and theoretical analysis abstract we consider the form of the rebound velocity 0 particle velocity and height h probability density functions pdfs for the one dimensional motion of a single particle on a sinusoidally oscillating base the motion is considered in the limit of high excitation vibration frequency collision rate experimentally we find that these pdfs are well approximated by p 0 0 0 exp 0 2 a gaussian p exp 2 and a boltzmann type function p h h exp 2 gh where is a constant and g is the acceleration due to gravity we develop an analytical model which accurately predicts the general form for the rebound velocity pdf the other two pdfs are then analytically shown to follow as a consequence scaling laws for the particle granular temperature with peak base velocity and particle base restitution coefficient determined from previous work can also be predicted from the pdf a fine scale spiky structure in the rebound velocity pdf is found using numerical simulations to be a consequence of resonance phenomena between the particle and vibrating base good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

some dynamical properties of a classical dissipative bouncing ball model with two nonlinearities abstract some dynamical properties for a bouncing ball model are studied we show that when dissipation is introduced the structure of the phase space is changed and attractors appear increasing the amount of dissipation the edges of the basins of attraction of an attracting fixed point touch the chaotic attractor consequently the chaotic attractor and its basin of attraction are destroyed given place to a transient described by a power law with exponent 2 the parameter space is also studied and we show that it presents a rich structure with infinite self similar structures of shrimp shape

graphical limit sets for general cellular automata the existing theory of graphical limit sets for cellular automata relies on algebraic structures and applies only to certain classes of cellular automata that possess this structure we extend this theory to general cellular automata using topological methods the starting point is the observation that the rescaled space time diagrams intersected with an appropriately chosen compact set form sequences in a compact metric space they necessarily possess converging subsequences in the present paper we define graphical limit sets as the collection of the accumulation points the main result is that for a large class of cellular automata the graphical limit set defined in this way carries a group structure which is either the trivial group consisting of one element only or is homeomorphic to s 1 the well known self similar graphical limit sets are representatives of the second class

a mechanical counterexample to kam theory with low regularity abstract we give a mechanical example concerning the fact that some regularity is necessary in kam theory we consider the model given by the vertical bouncing motion of a ball on a periodically moving plate denoting with f the motion of the plate some variants of moser invariant curve theorem apply if f is small in norm c 5 and every motion has bounded velocity this is not possible if the function f is only c 1 indeed we construct a function f c 1 with arbitrary small derivative in norm c 0 for which a motion with unbounded velocity exists

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

periodicity of averaged histories of chaotic oscillators abstract this study is dedicated to demonstrate the periodicities embedded in the averaged responses of chaotic systems with periodic excitations recent studies in the field of non linear oscillations often found random like responses for some deterministic non linear systems with periodic excitations which were then named chaotic systems however in this study by discretizing the initial conditions on a chosen domain and averaging the corresponding responses the averaged response can be calculated for the chaotic motions of duffing van der pol and piecewise linear systems these averaged responses exhibit near periodicities with primary frequency components at excitation frequency odd multiples or half multiples of excitation frequency it is also found that this periodicity becomes more evident as the number of discretized initial conditions over a fixed domain these results were obtained and validated by simulations

dynamical analysis of a two parameter family for a vibro impact system in resonance cases abstract dynamics of a two degree of freedom vibro impact system in resonance is considered the dynamical model and poincare maps are established when a pair of complex conjugate eigenvalues of the jacobian matrix cross the unit circle and satisfy the resonant condition 0 4 1 or 0 3 1 the four dimensional map is reduced to a two dimensional one by the center manifold theorem and the reduced map is put into its normal form by the method of normal form the two parameter unfoldings of local dynamical behavior studied in this paper develop the results of one parameter family analysis the hopf and subharmonic bifurcation conditions of period n 1 motion are given the numerical simulation method confirms the theoretical analysis it is shown that there exists an invariant torus via hopf bifurcation and period 4 4 motions via subharmonic bifurcation as two controlling parameters varying near the critical point for the resonance 0 4 1 and that there exists an invariant circle and unstable fixed points of order 3 bifurcating from the fixed point and the system leads eventually to chaos in the resonance 0 3 1

multifractal decompositions of digraph recursive fractals we prove that the multifractal decomposition behaves as expected for a family of sets k known as digraph recursive fractals using measures of markov type for each value of a parameter between a minimum min and maximum max we define multifractal components k of k and show that they are fractals in the sense of taylor the dimension f of k is computed from the data of the problem the typical concave multifractal f dimension spectrum curve results

replication in one dimensional cellular automata abstract in a cellular automaton ca replication is the ability to indefinitely generate copies of a finite collection of patterns starting from finite seeds a transparent feature of additive ca replication mechanisms are less clear in the absence of additivity this paper investigates such dynamics through several examples for the 1 or 2 rule and its generalizations replication is inevitable and we investigate self organization properties in the perturbed exactly 1 rule we study frequency of replicators and the new phenomenon is called quasireplication the last ca is the extended 1 or 3 rule which allows for replication on different backgrounds we employ a mixture of rigorous and empirical techniques

bifurcations and chaos in a system with impacts abstract a vibro impact system is considered a body bounces on a flat horizontal surface of the vibro bench hopf bifurcations of the vibro impact system in two kinds of strong resonance cases 03 1 and 04 1 are investigated a poincare map of the system is established the period 1 single impact motion of the system and its stability are studied by analytical methods and hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined a center manifold theorem technique is applied to reduce the poincare map to a two dimensional one which is put into normal form by theory of normal forms by the theory of hopf and subharmonic bifurcations of fixed points in r 2 strong resonance dynamic behavior of the vibro impact system near points of resonance is analyzed in the resonance case of 03 1 the system generally exhibits unstable period 3 three impact motion in the resonance case of 04 1 the system can exhibit stable period 4 four impact motion and quasi periodic motion the theoretical analyses are verified by numerical solutions routes to chaos in two kinds of strong resonance cases considered are obtained by numerical simulations quasi periodic impacts in non resonance and weak resonance cases doubling periodic bifurcations and chaotic motions are also stated briefly by numerical analyses

topology of vibro impact systems in the neighborhood of grazing abstract the grazing bifurcation is considered for the newtonian model of vibro impact systems a brief review on the conditions sufficient for the existence of a grazing family of periodic solutions is given the properties of these periodic solutions are discussed a plenty of results on the topological structure of attractors of vibro impact systems is known however since the considered system is strongly nonlinear these attractors must be very sensitive to changes of parameters of the system on the other hand they are observed in experiments and numerical simulations we offer theorem 2 an approach which allows to explain this contradiction and give a new robust mathematical model of the non hyperbolic dynamics in a neighborhood of grazing

effects of diffusion weighting schemes on the reproducibility of dti derived fractional anisotropy mean diffusivity and principal eigenvector measurements at 1 5t diffusion tensor imaging dti is used to study tissue composition and architecture in vivo to increase the signal to noise ratio snr of dti contrasts studies typically use more than the minimum of 6 diffusion weighting dw directions or acquire repeated observations of the same set of dw directions simulation based studies have sought to optimize dti acquisitions and suggest that increasing the directional resolution of a dti dataset i e the number of distinct directions is preferable to repeating observations in an equal scan time comparison however it is not always clear how to translate these recommendations into practice when considering physiological noise and scanner stability furthermore the effect of different dw schemes on in vivo dti findings is not fully understood this study characterizes how the makeup of a dw scheme in terms of the number of directions impacts the precision and accuracy of in vivo fractional anisotropy fa mean diffusivity md and principal eigenvector pev findings orientation dependence of dti reliability is demonstrated in vivo and a principled theoretical framework is provided to support and interpret findings with simulation results as long as sampling orientations are well balanced differences in dti contrasts due to different dw schemes are shown to be small relative to intra session variability these differences are accentuated at low snr while minimized at high snr this result suggests that typical clinical studies which use similar protocols but different well balanced dw schemes are readily comparable within the experimental precision

repeated impacts on a sinusoidally vibrating table reappraised the traditional simplification for one commonly used example of repeated collisions is shown to be frequently erroneous for a wide range of coefficients of restitution bounded errors arise from an artificial speed constraint whose circumvention is largely but not exclusively unpredictable therefore exact equations whose periodic solutions are detailed should be generally used

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems