homoclinic bifurcations in n dimensional endomorphisms due to expanding periodic points the study of bifurcation mechanisms which determine the transition from regular to chaotic behaviour or route to chaos in endomorphisms maps with a nonunique inverse has reached a good level of theoretical knowledge for one dimensional maps we refer to the book by mira l as a review of the main results in this research area the situation is quite different as regards endomorphisms of dimensions higher than one a few results have been worked out until now 2 81 mainly in the simplest cases of two dimensional endomorphisms however many mathematical models of interest in the applications are ultimately described by maps with a nonunique inverse clearly in the plane ir2 or in higher dimensional spaces we can no longer take advantage of the koenigs lemeray graphical construction and of the qualitative behaviour of the graphs of powers of t x t x t3 x which are powerful tools to identify the local and global bifurcations of one dimensional endomorphisms x t x x e ir however to study bifurcation mechanisms in ir n 2 2 we make use of properties which are the natural generalizations of those used in the one dimensional case indeed a basic tool for studying the dynamics and bifurcation mechanisms of endomorphisms is provided by the analysis of the critical manifolds first introduced by gumowski and mira in 1965 see 2 and references therein the critical manifolds are the natural generalization to ir n 2 2 of the local extrema also called critical points of rank l of one dimensional endomorphisms in this work we deal with homoclinic bifurcations in endomorphisms of ir n 1 l due to expanding periodic points as in one dimensional endomorphisms we will see that critical manifolds play a pre eminent role our purpose is to contribute to the understanding of the bifurcation mechanisms related to homoclinic orbits occurring in n dimensional endomorphisms mechanisms strictly related to the nonuniqueness of the inverse function that is typical of endomorphisms not present in invertible maps although similar to those occurring in these maps in particular this work concerns the global bifurcation homoclinic bifurcations occurring when a repelling fixed point becomes a snap back repeller t sbr henceforth and in general when there exists a critical orbit homoclinic to an expanding

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

fractal dimension of higher dimensional chaotic repellors abstract using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in typical dynamical systems in terms of the lyapunov exponents and the characteristic escape time from the repellor our main example a three dimensional chaotic scattering billiard yields a new structure for its invariant manifolds this system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae

multidimensional unrepetitive configurations note abstract we show that one can associate a letter of a finite alphabet with each lattice point of the n dimensional space so that an r th power free word corresponds to each straight line of the space this result is obtained as a consequence of the existence of bi infinite words on finite alphabets which are r th power free together with a certain class of their subwords an analogous property is obtained for some avoidable sets of patterns

on square free arithmetic progressions in infinite words abstract let w be an infinite word and p 2 a given positive integer denote by w p the infinite arithmetic progression modulo p of w i e it is the substring of w consisting of the concatenation of the letters at positions kp for k 1 2 we show that in the ternary case for each p 3 there exists an infinite square free word w such that also the arithmetic progression w p is square free for the case p 2 there are no solutions

words avoiding repetitions in arithmetic progressions carpi constructed an infinite word over a 4 letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference we show a connection between carpi s construction and the paperfolding words we extend carpi s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference we also construct infinite words avoiding overlaps and infinite words avoiding all sufficiently large squares in arithmetic progressions of odd difference we use these words to construct labelings of the 2 dimensional integer lattice such that any line through the lattice encounters a squarefree resp overlapfree sequence of labels

short term temporal integration evidence for the influence of peripheral compression thresholds for a 6 5 khz sinusoidal signal temporally centered in a 400 ms broadband noise masker were measured as a function of signal duration for normally hearing listeners and listeners with cochlear hearing loss over a range of masker levels for the normally hearing listeners the slope of the function relating signal threshold to signal duration integration function was steeper at medium masker levels than at low or high levels by a factor of nearly 2 for signal durations between 2 and 10 ms while no significant effect of level was found for signal durations of 20 ms and more no effect of stimulus level was found for the hearing impaired listeners at any signal duration for signal durations greater than 10 ms consistent with many previous studies the slope of the integration function was shallower for the hearing impaired listeners than for the normally hearing listeners however for shorter durations there was no significant difference in slope between the results from the hearing impaired listeners and those from the normally hearing listeners in the high and low level masker conditions a model incorporating a compressive nonlinearity representing the effect of basilar membrane bm compression and a short term temporal integrator postulated to be a more central process can account well for changes in the short term integration function with level if it is assumed that the compression is greater at medium levels than at low or high levels by a factor of about 4 this is in reasonable agreement with physiological measurements of bm compression and with previous psychophysical estimates

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

infinite ternary square free words concatenated from permutations of a single word abstract we answer a question of harju an infinite square free ternary word with an n stem factorization exists for any n 13 we show that there are square free uniform ternary morphisms of length k for every k 23 this resolves almost completely a problem of the author and rampersad

pressure and fractal indices for the gibbs measures of hyperbolic julia sets the authors give a simple algorithm to compute the topological pressure for the julia sets j of hyperbolic rational endomorphism this allows one to compute the spectra of the generalised dimensions and entropies with respect to the whole class of gibbs measures on j

monaural temporal integration and temporally selective listening in children and adults this study used two paradigms to investigate the development of temporal integration and temporally selective listening experiment 1 measured detection as a function of duration for a pure tone at 1625 or 6500 hz at both frequencies thresholds of children younger than 7 years old were higher than those for older children and adults the pattern of temporal integration was similar across groups for the 6500 hz signal but younger children showed relatively more temporal integration for the 1625 hz signal due to high thresholds for the briefest 1625 hz signal experiment 2 measured detection thresholds for one or for three brief tone pips presented in a noise masker in one set of conditions the noise masker consisted of 100 ms steady bursts interleaved with 10 ms temporal gaps in other conditions the level of the central 50 ms of the 100 ms masking noise bursts was adjusted by either 6 or 6 db children showed higher thresholds but similar temporal integration compared with adults overall these data suggest that children are less efficient than adults in weighting the output of the monaural temporal window at 1625 but not 6500 hz children are efficient in combining energy from brief temporal epochs that are separated by noise

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

annular chaotic areas on etudie l existence et les proprietes d acier chaotiques annulaires pour l application t de r 2 sur lui meme t x y f x y g x y

fractional iterates for n dimensional maps this paper constitutes an extension of results of the paper by narayaninsamy int j bifurcations chaos 6 1 1996 55 67 we study here the solutions of the problem of the fractional iteration for n dimensional maps

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

fractal boundaries in open hydrodynamical flows signatures of chaotic saddles we introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently it is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors the fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set we point out that these boundaries coincide with streak lines passing through appropriately chosen points as an illustrative numerical experiment we consider a model of the von k arm an vortex street a time periodic two dimensional flow of a viscous fluid around a cylinder

a method to construct irregular fractal curves abstract in this paper we present a method of constructing irregular fractal curves this method is based on a sequence n n n of piecewise differentiable maps defined succinctly in narayaninsamy t narayaninsamy on basin boundaries appl math comput 99 1999 261 274 the experimental results concerning the basins of attraction of attractors which are given in gardini mira et al and narayaninsamy l gardini some global bifurcations of two dimensional endomorphisms by use of critical lines nonlinear anal theory meth appl 18 4 1991 361 399 c mira d fournier prunaret l gardini h kawakami j c cathala basin bifurcations of two dimensional noninvertible maps fractalization of basins j bifurcat chaos 2 1994 343 381 t narayaninsamy contribution a l etude de l iteration fractionaire et a celle des endomorphismes bi dimensionnels doctorat mathematiques appliquees de l universite paul sabatier toulouse iii constitute the basic tool of this sequence

chaos induced by regular snap back repellers abstract this paper is concerned with chaos induced by regular snap back repellers one new criterion of chaos induced by strictly coupled expanding maps in compact sets of metric spaces is established by employing this criterion the nondegenerateness assumption in the marotto theorem established in 1978 is weakened in addition it is proved that a regular snap back repeller and a regular homoclinic orbit to a regular expanding fixed point in finite dimensional spaces imply chaos in the sense of li yorke an illustrative example is provided with computer simulations

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

intermingled basins in coupled lorenz systems we consider a system of two identical linearly coupled lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength we verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa we investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

a dynamical meaning of fractal dimension when two atlractors of a dynamical system have a common basin boundary b small changes in initial conditions which lie near b can result in radically different long term behavior of the trajectory a quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary b

the binaural temporal window in adults and children this study investigated the binaural temporal window in adults and children 5 10 5 years of age detection thresholds were estimated for a brief interaurally out of phase s 500hz pure tone signal masked by bandpass 100 2000hz gaussian noise in one set of conditions the masker was consistently either in phase no or out of phase n in another set of conditions the masker changed abruptly in interaural phase non or n no and threshold was estimated at a range of delays with respect to the phase transition masked thresholds were also obtained in further conditions where the masker interaural phase was steady and the signal was of long duration age effects obtained with dynamic maskers could be accounted for by positing that children have a binaural temporal window with a relatively prolonged leading edge or that the children position the binaural temporal window relatively late with respect to the signal modeling of the reduced masking level difference shown by children for a brief s signa

q deformed statistical mechanical property in the dynamics of trajectories en route to the feigenbaum attractor we demonstrate that the dynamics toward and within the feigenbaum attractor combine to form a q deformed statistical mechanical construction the rate at which ensemble trajectories converge to the attractor and to the repellor is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor the values of the q indices involved are given by the unimodal map universal constants while the thermodynamic structure is closely related to that formerly developed for multifractals as an essential component in our demonstration we expose in great detail the features of the dynamics of trajectories that either evolve toward the feigenbaum attractor or are captured by its matching repellor the dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period doubling transition to chaos elements in our analysis are the following i the preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops ii the limiting form of this rank structure results in an infinite number of families of well defined phase space gaps in the positions of the feigenbaum attractor or of its repellor iii the gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions iv the power law with log periodic modulation associated with the rate of approach of trajectories toward the attractor and to the repellor is explained in terms of the progression of gap formation v the relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated vi a mean field evaluation of the atypical partition function a thermodynamic interpretation of the time evolution process and a crossover to ordinary exponential statistics are given we make clear the dynamical origin of the anomalous thermodynamic framework existing at the feigenbaum attractor

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

on basin boundaries in this paper we make a study of a class of basin boundaries we show that an efficient characterization of a given element of this class can be constructed with segments of differentiable curves

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

binaural and monaural auditory filter bandwidths and time constants in probe tone detection experiments auditory filter bandwidths and time constants were obtained with five normal hearing subjects for different masker configurations both in the frequency and time domain for monaural and binaural listening conditions specifically the masking level in the monaural condition and the interaural correlation in the binaural conditions respectively was changed in a sinusoidal stepwise and rectangular way in the frequency domain in the corresponding experiments in the time domain a sinusoidal and stepwise change of the masker was performed from these results a comparison was made across conditions to evaluate the influence of the factors shape of transition monaural versus binaural frequency domain versus time domain and subject also the respective data from the literature were considered using the same model assumptions and fitting strategy as used for the current data the results indicate that the monaural auditory filter bandwidths and time constants fitted to the data are consistent acr

some further results on squarefree arithmetic progressions in infinite words abstract in a recent paper one of us posed three open problems concerning squarefree arithmetic progressions in infinite words in this paper we solve these problems and prove some additional results for instance among other things we show that there exists a squarefree word w over a ternary alphabet such that for every p 3 the subsequence of w indexed by the multiples of p contains a square

turbulence in multistep methods for initial value problems solving a scalar initial value problem with a numerical k step scheme amounts to iterating a k dimensional discrete dynamical system in this paper we investigate regular and chaotic behaviour of some dynamical systems associated with adams bashforth formulas applied to a simple nonlinear test equation

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

some global bifurcations of two dimensional endomorphisms by use of critical lines many studies have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by smale l and gavrilov and shilnikov 2 31 but comparatively few works have been devoted to endomorphisms that is to maps with a nonunique inverse however the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems or maps with a nonunique inverse this is particularly true in the socio economic field 4 61 but examples can also be found in ecology 7 81 biology 9 l l and control systems 121 this work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by gumowski and mira ll 13 141 for two dimensional endomorphisms in particular this is the notion of critical lines or critical curves representing the twodimensional extension of the critical points in one dimensional maps and the notions of absorbing area and chaotic area bounded by elements of critical lines which determine phase plane domains where chaotic solutions are located these notions which will be recalled in the next section together with some bifurcations involving the chaotic areas are the basic concepts and properties that we shall use in the analysis of the subsequent sections the aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms particularly when they are used in combination with more traditional techniques it is known that in diffeomorphisms global bifurcations which qualitatively change the structure of a chaotic attractor called interior crises or catastrophes by some authors 15 181 are characterized by the stable manifold of a saddle approaching and touching the chaotic attractor we shall find that this phenomenology also occurs in the endomorphisms of our model with a particular role being played by the critical lines in section 3 we shall describe our mathematical model a two dimensional piecewise linear map which we call map f together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided endomorphisms of the first class have been the object of a previous work 191 here we shall show in section 4 how several bifurcations involving chaotic areas can be explained by use of critical lines the main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

periodic solutions and chaos in a delay differential equation modelling haematopoesis simple mathematical models have been devised to describe the process of haematopoesis the formation and differentiation of blood cells within the human body l 21 haematopoesis is an example of a biological control system with an inherent delay in it this is due to the time required for primitive blood cells to mature such a delayed response is known to cause oscillatory behaviour in control systems and it has been found that similar oscillatory instabilities in blood cell counts are associated with diseases such as aplastic anaemia 2 this suggest that the study of such equations may be of biological importance mathematically they are of interest because they can exhibit complex aperiodic behaviour a model put forward for consideration by mackey glass 3 is

on the node fractional iterates let m be a bounded convex set of r n let f m r n be a continuous non bijective map we study solutions g of the functional equation g m x g g m 1 x f x m 2 this functional equation represents the problem of the fractional iteration for f

a connection between fractional iteration and graph theory we consider the problem of the fractional iteration for n dimensional maps we have two objectives in this paper the first objective is to establish a connection between fractional iteration and graph theory the second objective is to make a study concerning the problem of fractional iteration for n dimensional maps

basins of attraction in piecewise smooth hamiltonian systems piecewise smooth hamiltonian systems arise in physical and engineering applications for such a system typically an infinite number of quasiperiodic attractors coexist here we use the term attractors to indicate invariant sets to which typically initial conditions approach as a result of the piecewise smoothness of the underlying system these attractors are therefore characteristically different from the attractors in dissipative dynamical systems we find that the basins of attraction of different attractors exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales this practically prevents prediction of attractors from specific initial conditions and parameters the mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems we demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation

unpredictability of the asymptotic attractors in phase coupled oscillators an array of phase coupled oscillators may exhibit multiple coexisting chaotic and nonchaotic attractors the system of coupled circle maps is such an example we demonstrate that it is common for this type of system to exhibit an extreme type of final state sensitivity in both parameter and phase space numerical computations reveal that there exist substantial regions of the parameter space where arbitrarily small perturbations in parameters or initial conditions can alter the asymptotic attractor of the system completely consequently asymptotic attractors of the system cannot be predicted reliably for specific parameter values and initial conditions

on chaos and fractal behavior in a generalized duffing s system abstract in this paper three duffing s equations their attractors and their initial condition maps are examined for elements of universality in the literature all three equations have been examined briefly before but their behavior has only been incompletely described this paper more completely examines these system and answers important questions concerning the reason for chaotic system behavior

uniformly growing k th power free homomorphisms a string is called kth power free if it does not have xk as a nonempty substring for all nonnegative rational numbers k kth power free strings and kth power free homomorphisms are investigated and the shortest uniformly growing square free k 2 and cube free k 3 homomorphisms mapping into least alphabets with three and two letters are introduced it is shown that there exist exponentially many square free and cube free strings of each length over these alphabets sharpening the kth power freeness to the repetitive threshold rt n of n letter alphabets we provide arguments for the nonexistence of various rt n th power free homomorphisms

fractal control boundaries of driven oscillators and their relevance to safe engineering design when a metastable damped oscillator is driven by strong periodic forcing the catchment basin of constrained motions in the space of the starting conditions x 0 0 develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds the four dimensional basin in the phase control space spanned by x 0 0 f where f is the magnitude and the frequency of the excitation will likewise acquire a fractal boundary and we here explore the engineering significance of the control cross section corresponding for example to x 0 0 0 the fractal boundary in this section is a failure locus for a mechanical or electrical system subjected while resting in its ambient equilibrium state to a sudden pulse of excitation we assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well absolute and transient basins are examined giving control space maps analogous to familiar pictures of the mandelbrot set generalizing from this prototype study it is argued that in engineering design against boat capsize or earthquake damage for example a study of safe basins should augment and perhaps entirely replace conventional analysis of the steady state attracting solutions

methods for estimating infant thresholds detection thresholds for 1000 hz 16 ms tone bursts were estimated for 3 and 6 month old infants and for young adults the test procedure used was the observer based psychoacoustic procedure thresholds were estimated using two different adaptive procedures and the method of constant stimuli there was little difference among the average thresholds determined by any of these techniques the least variable thresholds were obtained in the method of constant stimuli in addition 10 infants at each age completed two 30 trial blocks of trials in the method of constant stimuli 22 adults completed 8 blocks of 30 trials for 3 and 6 month olds and for adults there was no significant change in average threshold between blocks individual 6 month olds thresholds rarely changed by more than 5 db between blocks and the correlation between the thresholds obtained in the two blocks was significant individual 3 month olds thresholds however sometimes changed by as much as 10 db between blocks and the correla

dynamic bifurcations in non stationary systems transitions with an unpredictable outcome in nonlinear non stationary systems dynamic bifurcations result in a transition to a qualitatively new state in this paper we examine how the dynamics of transition of such systems may be assessed using the concept of transient basins of attraction we delineate the phenomenon of indeterminate dynamic bifurcations where it is shown that the response after the system passes through critical parameter values may be extremely sensitive to the choice of initial conditions or parameter states this new form of unpredictability in systems whose parameters vary with time is clearly an important concept to be assimilated in the theory of non stationary dynamics