the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

subharmonic and homoclinic bifurcations in a parametrically forced pendulum abstract depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the melnikov method and of averaging methods it is shown that as a parameter is varied repeated resonances of successively higher periods occur culminating in homoclinic orbits according to the theorem of smale homoclinic bifurcation is the source of the unstable chaotic motions observed for some selected parameter sets the theoretical predictions are tested by numerical calculations very good agreement is found between analytical and numerical results

nondegenerate homoclinic tangency and hyperbolic sets in this paper we show that a nondegenerate homoclinic tangency on a surface is accumulated by a sequence of uniformly hyperbolic sets and in fact it is a boundary point of a nonuniformly hyperbolic set

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

chaotic rotations of a liquid filled solid abstract the disturbed hamiltonian equations of a solid filled with a rotating ellipsoidal mass of a liquid and subjected to small applied moments are revisited using deprit s variables we investigate the chaotic dynamics of the orbiting liquid filled solid and of the liquid filled solid sliding and rolling on a perfectly smooth plane in either energy conservative or energy dissipative conditions when appropriately perturbed criteria for the judgment of potential chaotic rotations of the perturbed system are formulated by means of melnikov holmes marsden mhm integrals strategies for the solution of heteroclinic orbits of the symmetrical liquid filled solid under torque free conditions are outlined theoretically physical parameters that will probably trigger the onset of chaotic motions can be determined accordingly results from mhm algorithms are crosschecked with poincare sections together with lyapunov characteristic exponents

on the capacity achieving distribution of the discrete time noncoherent and partially coherent awgn channels we investigate the communication limits over rapid phase varying channels and consider the capacity of a discrete time noncoherent additive white gaussian noise ncawgn channel under the average power constraint we obtain necessary and sufficient conditions for the capacity achieving input distribution and show that this distribution is discrete and possesses an infinite number of mass points using this characterization of the capacity achieving distribution we compute a tight lower bound on the capacity of the channel based on examining suboptimal input distributions in addition we provide some easily computable lower and upper bounds on the channel capacity finally we extend some of these results to the partially coherent channel where it is assumed that a phase locked loop pll is used to track the carrier phase at the receiver and that an ideal interleaver and de interleaver are employed rendering the tikhonov distributed residual phase errors statistically independent from one symbol interval to another

steadiness of regular semiartinian rings with primitive factors artinian abstract we provide a ring theoretic criterion of steadiness which applies to all regular semiartinian rings with primitive factors artinian

sbil nikov hopf bifurcation abstract we derive a 2 parameter poincare map modeling a 3 dimensional vector flow near the codimension 2 point defined by a sil nikov homoclinic orbit to a fixed point undergoing a hopf bifurcation in one parameter regime the map reproduces glendinning and sparrow s results for the sil nikov bifurcation while in another regime it provides new results for a small hopf cycle with a global reinjection mechanism the map is analyzed finding primary homoclinic tangencies to the small hopf cycle parameter dependence of periodic orbits associated with these tangencies is described global behavior of the tube like unstable manifold of the hopf cycle is studied in depth analytically and numerically locating its simplest subsidiary tangencies

chaotic dynamics for perturbations of infinite dimensional hamiltonian systems where w t z r is the transverse de9ection of the axis of the beam w t 0 w t 1 wzz t 0 wzz t 1 0 is an external load 0 is a ratio indicating the extensional rigidity and is the damping the rst result on the existence of a chaotic dynamics for system h has been given by holmes and marsden in 10 for a speci c periodic forcing perturbation of the type p t w t z f z cos t they use the theory of invariant manifolds and of non linear semigroups in order to extend the classical melnikov approach for planar ordinary di erential equations to system h

on calculating the capacity of an infinite input finite infinite output channel a version of the arimoto blahut algorithm for continuous channels involves evaluating integrals over an entire input space and thus is not tractable two generalized discrete versions of the arimoto blahut algorithm are presented for this purpose instead of calculating integrals both algorithms require only the computation of a sequence of finite sums this significantly reduces numerical computational complexity

calculation of mutual information for partially coherent gaussian channels with applications to fiber optics the mutual information between a complex valued channel input and its complex valued output is decomposed into four parts based on polar coordinates an amplitude term a phase term and two mixed terms numerical results for the additive white gaussian noise awgn channel with various inputs show that at high signal to noise ratio snr the amplitude and phase terms dominate the mixed terms for the awgn channel with a gaussian input analytical expressions are derived for high snr the decomposition method is applied to partially coherent channels and a property of such channels called spectral loss is developed spectral loss occurs in nonlinear fiber optic channels and it may be one effect that needs to be taken into account to explain the behavior of the capacity of nonlinear fiber optic channels

a unified control framework of the non regular dynamics of mechanical oscillators a method for controlling non linear dynamics and chaos based on avoiding homo heteroclinic bifurcations is considered in a unified framework various non linear oscillators both hardening and softening symmetric and asymmetric considered as archetypes of a more general class of single d o f systems are analyzed the melnikov s method is applied to analytically detect the homo heteroclinic bifurcations and the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space the generic character of the optimization problem is highlighted and the problem itself is discussed in detail various control strategies are proposed based on the elimination either of a single bifurcation one side controls or of all bifurcations global control the optimization problems are solved analytically and numerically under various forms by taking into account the physical admissibility of the related optimal excitation and the easiness of implementation in practical applications it is shown that the solutions of one side control problems are always system independent in turn the solutions of global control problems are system independent for the large class of symmetric systems considered in this work although there are other systems whose solutions are system dependent these considerations support the very generic nature of the optimal control

generic diffeomorphisms with superexponential growth of number of periodic orbits let m be a smooth compact manifold of dimension at least 2 and diff r m be the space of c r smooth diffeomorphisms of m associate to each diffeomorphism f 2 diff r m the sequence pn f of the number of isolated periodic points for f of period n in this paper we exhibit an open set n in the space of diffeomorphisms diff r m such for a baire generic diffeomorphism f 2 n the number of periodic points pnf grows with a period n faster than any following sequence of numbers fangn2zc along a subsequence i e pn f a ni for some ni 1 with i 1 in the cases of surface diffeomorphisms i e dim m d 2 an open set n with a supergrowth of the number of periodic points is a newhouse domain a proof of the man result is based on the gontchenko shilnikov turaev theorem gst a complete proof of that theorem is also presented

symbolic extensions and smooth dynamical systems let f x x be a homeomorphism of the compact metric space x a symbolic extension of f x is a subshift on a finite alphabet g y which has f as a topological factor we show that a generic c1 non hyperbolic i e non anosov area preserving diffeomorphism of a compact surface has no symbolic extensions for r 1 we exhibit a residual subset mathcal r of an open set mathcal u of cr diffeomorphisms of a compact surface such that if f in mathcal r then any possible symbolic extension has topological entropy strictly larger than that of f these results complement the known fact that any c diffeomorphism has symbolic extensions with the same entropy we also show that cr generically on surfaces homoclinic closures which contain tangencies of stable and unstable manifolds have hausdorff dimension two

homoclinic bifurcations for the henon map abstract chaotic dynamics can be effectively studied by continuation from an anti integrable limit we use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations we find a bound on the parameter range for which the henon map exhibits a complete binary horseshoe as well as a subshift of finite type we classify homoclinic bifurcations and study those for the area preserving case in detail simple forcing relations between homoclinic orbits are established we show that a symmetry of the map gives rise to constraints on certain sequences of homoclinic bifurcations our numerical studies also identify the bifurcations that bound intervals on which the topological entropy is apparently constant

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

the transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points the complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained it is topologically conjugate to a non bernoulli shift called we also obtain a more or less complete picture referred to as the net weaving bifurcation when the fixed point of such a system is undergoing the generic saddle node bifurcation the idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork transcritical periodic doubling and hopf bifurcations are all homotopically conjugate to our shift dynamics in a neighborhood of a transverse homoclinic orbit

bifurcation of periodics and subharmonics in abstract nonlinear undamped wave equations abstract periodic and subharmonic bifurcation is investigated for certain abstract wave differential equations modelled after a nonlinear beam equation

rotation numbers of periodic orbits in the henon map for invertible area contracting maps of the plane it is common for a basin of attraction to have a fractal basin boundary certain periodic orbits on the basin boundary are distinguished by being accessible by a path from the interior of the basin a numerical study is made of the accessible periodic orbits for the henon family of maps theoretical results on rotary homoclinic tangencies are given which describe the appearance of the accessible saddles and organize them in a natural way according to the continued fractions expansions of their rotation numbers

bifurcations of subharmonics abstract we consider a piecewise linear second order ordinary differential equation which is topologically equivalent to the sine gordon equation the system is subjected to a time harmonic disturbance and the behavior of the periodic solutions is examined it is shown that subharmonic motions with period n times that of the disturbance appear via saddle node bifurcations for all n 1 2 3 these motions then undergo period doubling or pitchfork bifurcations as parameters are varied beyond the saddle node bifurcation values the limit n is considered and is compared with a melnikov calculation for the system due to the piecewise linear nature of the system no approximations are necessary and thus the results are valid for nonsmall parameter values furthermore this provides an answer to the uniformity question in the study of stability of subharmonics

melnikov s criteria and chaos in systems with fractional order deflection in this paper a qualitative analysis of the dynamic systems described with the second order differential equation with fractional order deflection function is considered the existence of fixed points closed orbits and the unions of fixed points and the trajectories connecting them is shown the homoclinic orbit which connects a fixed point with itself and the corresponding stable and unstable manifolds are given in the closed analytical form melnikov s procedure for defining the criteria for transversal intersection of the stable and unstable manifolds is extended for the systems with fractional order deflection function the critical parameter values for chaos are obtained analytically and proved numerically using the lyapunov exponents the bifurcation diagrams are plotted for various values of fractional order and the transition to chaos by period doubling is shown the phase plane diagrams and the poincare maps for certain fractional orders are obtained the control of chaos and the transformation to periodic motion is considered

dynamics of a class of odes more general than almost periodic abstract a temporally global solution if it exists of a nonautonomous ordinary differential equation need not be periodic almost periodic or almost automorphic when the forcing term is periodic almost periodic or almost automorphic respectively an alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here specifically the class of functions consists of uniformly continuous functions defined on the real line and taking values in a banach space which have pre compact ranges besides periodic and almost periodic functions this class also includes many nonrecurrent functions assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts the existence of a special bounded entire solution as well the existence of stable and unstable manifolds of this solution are established moreover it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation in the stable case it is shown that this special solution is the pullback attractor of the system a class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a c 0 semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations

semiartinian v rings and semiartinian von neumann regular rings abstract semiartinian right v rings which we call right sv rings form a special class of von neumann regular rings we characterize these rings by the fact that every factor ring imbeds as a subring in a direct product of right full linear rings containing the socle if r is a semiartinian ring with all primitive factor rings artinian then the condition of being an sv ring is right left symmetrical and is equivalent to being regular on the other hand if r is a right and left sv ring then all primitive factor rings of r are artinian for right sv rings whose proper ideals are prime we show that the condition of being unit regular is equivalent to being directly finite on the other hand we show that there exists a directly finite right sv ring which is not unit regular furthermore we provide two constructions for any given ordinal the first one gives a prime unit regular right sv ring of loewy length 1 which is not a left v ring and is hereditary if is a natural number the second one gives a directly infinite right sv ring not a left v ring whose loewy length is 2 if is a natural number and is 1 otherwise these constructions are general enough to produce a wide supply of sv rings starting from given ones

when every self small module is finitely generated abstract the aim of this paper is to give necessary and sufficient conditions for rings for which every right self small module is finitely generated it is proved that semi simple rings commutative perfect rings and right nonsingular extending rings have this property a right nonsingular semi prime ring has the property if and only if it is semi simple a commutative noetherian ring has the property if and only if it is artinian

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

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

optimization of electromechanical control of beam dynamics analytical method and finite differences simulation this paper deals with the optimization of the electromechanical control of beams dynamics using the mode expansion formalism an analytical investigation is carried out to find adequate electrical parameters to reduce the vibration amplitude to control the snap through instability and horseshoe chaos the analytical investigation is validated and complemented by the numerical simulation of the modal equations as well as by the direct numerical simulation of the beam nonlinear partial differential equation under control

structural stability of linear dynamically varying ldv controllers abstract ldv systems are linear systems with parameters varying according to a nonlinear dynamical system this paper examines the robust stability of such systems in the face of perturbations of the nonlinear system three classes of perturbations are examined differentiable functions lipschitz continuous functions and continuous functions it is found that in the first two cases the system remains stable whereas if the perturbations are among continuous functions the closed loop may not be asymptotically stable but instead is asymptotically bounded with the diameter of the residual set bounded by a function that is continuous in the size of the perturbation it is also shown that in the case of differential perturbations the resulting optimal ldv controller is continuous in the size of the perturbation an example is presented that illustrates the continuity of the variation of the controller in the case of a nonstructurally stable dynamical system

reduction of the semisimple 1 1 resonance the method of averaging is often used in hamiltonian systems of two degrees of freedom to find periodic orbits such periodic orbits can be reconstructed from the critical points of an associated reduced hamiltonian on a reduced space this paper details the construction of the reduced space and the reduced hamiltonian for the semisimple 1 1 resonance case the reduced space will be a 2 sphere in r3 and the reduced differential equations will be euler s equations restricted to this sphere the orbit projection from the energy surface in phase space to this sphere will be the hopf map the results of the paper are related to problems in physics on degeneracies due to symmetries of classical two dimensional harmonic oscillators and their quantum analogues for the hydrogen atom

persistent homoclinic tangencies in the he non family abstract homoclinic tangencies in the henon family f x y x 2 by x for the parameter values b 0 3 and 1 270 1 420 are investigated our main observation is that there exist three intervals comprising 93 percent of the values of the parameter such that for a dense set of parameter values in these intervals the henon family possesses a homoclinic tangency therefore one should expect long parameter intervals where the henon family is not structurally stable strong numerical support for this observation is provided

conditions for a monotonic channel capacity motivated by results in optical communications where the performance can degrade dramatically if the transmit power is sufficiently increased the channel capacity is characterized for various kinds of memoryless vector channels it is proved that for all static point to point channels the channel capacity under an equal power constraint is a nondecreasing function of power as a consequence maximizing the mutual information over all input distributions with a certain power is for such channels equivalent to maximizing it over the larger set of input distributions with upperbounded power the channel coding theorem is formally proved for an equal power constraint for interference channels such as optical wavelength division multiplexing systems the primary channel capacity is always nondecreasing with power if all interferers transmit with identical distributions as the primary user also if all input distributions in an interference channel are optimized jointly then the achievable sum rate capacity is again nondecreasing the results generalize to the channel capacity as a function of a wide class of costs not only power

almost sure and moment lyapunov exponents for a stochastic beam equation the transverse vibrations of an euler bernoulli beam with axial tension p and axial white noise forcing are given by mytt m yt eiyxxxx pyxx 0yxxw t 0 x l with hinged endpoints y 0 t yxx 0 t y l t yxx l t 0 the solution may be written y x t n 1 sin n xl cn t where the amplitude cn t of the nth mode of vibration is given by a two dimensional linear stochastic ordinary differential equation the main results give formulas for the almost sure and moment lyapunov exponents for the spde in terms of the almost sure and moment lyapunov exponents for each of the sdes corresponding to modes which are present in the initial condition it is shown that the almost sure and moment lyapunov exponents for the spde depend sensitively on the initial distribution of energy amongst the infinitely many modes of vibration and that they cannot be well approximated using only finitely many modes of vibration

period doubling and quadrupling for some second order difference schemes may exhibit stable n cycles that bifurcate into stable 2n cycles as a increases creating period doubling bifurcation sequences each beginning with a cycle of odd period these dynamics were first described by may i 21 and have since been investigated by many others see 3 61 for example a number of universal features of this bifurcation process have been observed primarily for quadratic maps 7 lo by comparison much less is known concerning the dynamics of higher order nonlinear schemes although there have been many numerical investigations of specific second order equations see i l 181 for example the full range of dynamics possible for such problems has not yet been determined in this work we investigate some general second order nonlinear difference schemes sufficient conditions are derived for the existence stability and bifurcations of periodic cycles for certain classes of such problems firstly the correspondence between periodic cycles of 1 and those of the second order problem x k l f xk l 3

cycles and measure of bifurcation sets for two dimensional diffeomorphisms we describe a large class of one parameter families of two dimensional diffeomorphisms which arestable for 0 exhibit acycle for 0 and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals 0 a main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

analytic solution of the system of two coupled differential equations with the fifth order non linearity abstract in this paper an approximate method for solving a system of two coupled second order non linear differential equation with strong and weak non linearity is developed the strong non linearities are of the third and of the fifth order the suggested new method is partially based on the elliptic krylov bogolubov method and on the power series method developed for solving the weak non linear differential equations the one frequency solution is considered for the case of certain relations between the equation parameters p and m and initial conditions as a special case the oscillatory motion of the physical system described with a fourth order strong non linear differential equation is considered a wide analysis of the corresponding equation is done the results of the suggested analytic procedure are compared with numerical ones they show a good agreement

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

the newhouse set has a positive hausdorff dimension the newhouse phenomenon of infinitely many coexisting periodic attractors is studied in its simplest form one shows that the corresponding parameter set the newhouse set jn has a strictly positive hausdorff dimension this result is stronger than that of tedeschini lalli and yorke commun math phys 106 635 1986 concerning the lebesgue measure of the newhouse set and is complementary to our knowledge on the topological properties ofjn namely it is a residual set hence uncountable and everywhere dense in a parameter interval

normally elliptic singular perturbations and persistence of homoclinic orbits abstract we consider a dynamical system possibly infinite dimensional or non autonomous with fast and slow time scales which is oscillatory with high frequencies in the fast directions we first derive and justify the limit system of the slow variables assuming a steady state persists we construct the stable unstable center stable center unstable and center manifolds of the steady state of a size of order o 1 and give their leading order approximations finally using these tools we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems

explosive solutions of stochastic nonlinear beam equations with damping abstract this study considers a class of damped stochastic nonlinear beam equations driven by multiplicative noise by an appropriate energy inequality we provide sufficient conditions such that the local solutions of the stochastic equations blow up with a positive probability or are explosive in an l 2 sense we also derive estimates of the upper bound of the blow up time

multi pulse jumping orbits and homoclinic trees in a modal truncation of the damped forced nonlinear schro dinger equation abstract in this paper we prove the existence of multi pulse orbits homoclinic to a slow manifold in a two mode truncation of the damped forced nonlinear schrodinger equation first suggested by bishop et al these orbits are jumping i e the corresponding solutions keep switching in time between neighborhoods of the two charateristic breathers of the integrable limit in the case of no damping we find multi pulse smale horseshoes in the two mode model while in the dissipative case we establish the existence of structurally stable multi pulse heterolinic connections between two unstable equilibria the orbits we construct are not amenable to melnikov type perturbation methods in both the hamiltonian and the dissipative case we find homoclinic trees which describe the repeated bifurcations of multi pulse solutions to illustrate the theoretical predictions we also present visualizations of these complicated structures

metamorphoses sudden jumps in basin boundaries in some invertible maps of the plane that depend on a parameter boundaries of basins of attraction are extremely sensitive to small changes in the parameter a basin boundary can jump suddenly and as it does change from being smooth to fractal such changes are calledbasin boundary metamorphoses we prove under certain non degeneracy assumptions that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency

when cantor sets intersect thickly the thickness of a cantor set on the real line is a measurement of its size thickness conditions have been used to guarantee that the intersection of two cantor sets is nonempty we present sharp conditions on the thicknesses of two cantor sets which imply that their intersection contains a cantor set of positive thickness

simple models for bifurcations creating horseshoes we present a class of simple models for global bifurcations creating horseshoes some properties known for henon mappings are easily obtained for these models such as e g the existence of nontrivial hyperbolic sets kneading sequences techniques allow us to exhibit explicit differences with the global bifurcation diagram for maps of the interval explicit examples displaying wild hyperbolic sets and infinitely many sinks are also given as an illustration of the simplicity of these models

stochastic nonlinear beam equations an extensible beam equation with a stochastic force of a white noise type is studied lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution

chaotic perturbations of kdv i rational solutions abstract rational pole solutions of a perturbed kdv equation describing nonlinear ion acoustic plasma waves become chaotic in time when a small perturbation is periodically driven a mcgehee transformation blows up a degenerate stationary point at infinity and the smale birkhoff homoclinic theorem adapted to submanifolds in phase permits the use of the melnikov method with pole solutions being homoclinic orbits

existence theory and invariant manifolds of bubble clouds this work is motivated by a desire to understand the dynamics of a cloud of bubbles in the wake of a ship in particular we are interested in understanding how the sound waves from the propellors excite the cloud and how the cloud in turn modifies these waves many applications of bubble cloud dynamics exist ranging from cavitation damage to shock wave propagation in bubbly liquids

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

how often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency abstract here we study an amazing phenomenon discovered by newhouse s newhouse non density of axiom a a on s 2 in proc sympos pure math vol 14 amer math soc 1970 pp 191 202 s newhouse diffeomorphisms with infinitely many sinks topology 13 1974 9 18 s newhouse the abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms publ math inst hautes etudes sci 50 1979 101 151 it turns out that in the space of c r smooth diffeomorphisms diff r m of a compact surface m there is an open set u such that a baire generic diffeomorphism f u has infinitely many coexisting sinks in this paper we make a step towards understanding how often does a surface diffeomorphism have infinitely many sinks our main result roughly says that with probability one for any positive d a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by d or those whose period is relatively large compared to its cyclicity it verifies a particular case of palis conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are baire generic they have probability zero one of the key points of the proof is an application of newton interpolation polynomials to study the dynamics initiated in v kaloshin b hunt a stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms i ann of math in press 92 pp v kaloshin a stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms ii preprint 85 pp

optimal alphabets and binary labelings for bicm at low snr optimal binary labelings input distributions and input alphabets are analyzed for the so called bit interleaved coded modulation bicm capacity paying special attention to the low signal to noise ratio snr regime for 8 ary pulse amplitude modulation pam and for 0 75 bit symbol the folded binary code results in a higher capacity than the binary reflected gray code brgc and the natural binary code nbc the 1 db gap between the additive white gaussian noise awgn capacity and the bicm capacity with the brgc can be almost completely removed if the input symbol distribution is properly selected first order asymptotics of the bicm capacity for arbitrary input alphabets and distributions dimensions mean variance and binary labeling are developed these asymptotics are used to define first order optimal foo constellations for bicm i e constellations that make bicm achieve the shannon limit 1 59 db it is shown that the eb n0 required for reliable transmission at asymptotically low rates in bicm can be as high as infinity that for uniform input distributions and 8 pam there are only 72 classes of binary labelings with a different first order asymptotic behavior and that this number is reduced to only 26 for 8 ary phase shift keying psk a general answer to the question of foo constellations for bicm is also given using the hadamard transform it is found that for uniform input distributions a constellation for bicm is foo if and only if it is a linear projection of a hypercube a constellation based on pam or quadrature amplitude modulation input alphabets is foo if and only if they are labeled by the nbc if the constellation is based on psk input alphabets instead it can never be foo if the input alphabet has more than four points regardless of the labeling

genesis of bursting oscillations in the hindmarsh rose model and homoclinicity to a chaotic saddle abstract we present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems 1 that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking and 2 that the bursting corresponds to a homoclinicity to this unstable chaotic state these propositions are supported by a numerical test on the hindmarsh rose model a prototype of its kind we conclude by a unified view for three types of complex multi modal oscillations homoclinic systems bursting and the pomeau manneville intermittency

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

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

homoclinic tangencies in the dynamics of articulated pipes conveying fluid abstract numerical evidence is presented showing homoclinic tangencies to periodic orbits occuring in a four dimensional system of ordinary differential equations that describes the dynamics of a pair of articulated pipes conveying fluid bifurcation sequences of mixed mode oscillations are discovered in two separate regions of parameter space in one region the equations have z 2 symmetry and a symmetric pair of heteroclinic tangencies is implied in both cases satisfactory agreement is found between the observed bifurcation sequences and the local theory of homoclinic tangencies global branches of mixed mode orbits are computed it is discovered that some orbits are generated by either sil nikov or figure of eight homoclinic orbits to stationary points whereas others form isolas that feature characteristic towers of saddle node and period doubling bifurcations

periodic nonlinear waves on a half line nontrivial solutions of the equationutt uxx g u which are 2 periodic int and which decay asx are shown to exist ifg a 0 andg 0 1 breather like solutions which also decay asx can be interpreted as homoclinic solutions in thex dynamics their existence is still in question for generalg

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

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

wild hyperbolic sets yet no chance for the coexistence of infinitely many klus simple newhouse attracting sets the phenomenon of the coexistence of infinitely many sinks for two dimensional dissipative diffeomorphisms is a result due to newhouse ne1 ne2 in fact for each parameter value at which a homoclinic tangency is formed nondegenerately there exist intervals in the parameter space containing dense sets of parameter values for which there are infinitely many coexisting sinks robinson r the structure of the sinks constructed by newhouse is limited simple newhouse parameter values are values at which there are infinitely many sinks having some special well defined property concerning the structure a result due to tedeschini lalli and yorke ty says that the lebesgue measure of the set of simple newhouse parameter values is zero when the tangencies are due to the standard affine horseshoe map it is argued in ty and pr that a more general derivation of this measure zero result would be desirable the main result of this paper is that the lebesgue measure of the set of klus simple parameter values including the simple newhouse parameter values is zero for saddle hyperbolic basic sets forming tangencies

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

post buckling longterm dynamics of a forced nonlinear beam a perturbation approach abstract the aim of this paper is to determine by a singular perturbation approach the dynamic response of a harmonically forced system experiencing a pitchfork bifurcation the model of an extensible beam forced by a harmonic excitation and subject to an axial static buckling is space discretized by a galerkin approach and studied by the normal form method for different values of equation parameters influencing the nonlinear dynamic behavior like damping coefficient load amplitude and frequency a relevant issue in the perturbation methods is the concept of small and zero divisors which are related to the possibility to build a transformation that simplifies the original studied problem i e to obtain the normal form by eliminating as much as possible nonlinearities in the equations for nonconservative systems like structural damped systems there are no conditions in the prior literature that define what small means relatively to a divisor in the present paper some conditions about the order of magnitude of the divisors with respect to the perturbation entity are given and related to some physical parameters in the governing equations in order to estimate the relevance of some nonlinear effects

on properties of the support of capacity achieving distributions for additive noise channel models with input cost constraints we study the classical problem of characterizing the channel capacity and its achieving distribution in a generic fashion we derive a simple relation between three parameters the input output function the input cost function and the noise probability density function one which dictates the type of the optimal input in layman terms we prove that the support of the optimal input is bounded whenever the cost grows faster than a cutoff growth rate equal to the logarithm of the inverse of the noise probability density function evaluated at the input output function furthermore we prove a converse statement that says whenever the cost grows slower than the cutoff rate the optimal input has necessarily an unbounded support in addition we show how the discreteness of the optimal input is guaranteed whenever the triplet satisfy some analyticity properties we argue that a suitable cost function to be imposed on the channel input is one that grows similarly to the cutoff rate our results are valid for any cost function that is super logarithmic they summarize a large number of previous channel capacity results and give new ones for a wide range of communication channel models such as gaussian mixtures generalized gaussians and heavy tailed noise models that we state along with numerical computations

chaotic vibrations of the one dimensional wave equation due to a self excitation boundary condition part i controlled hysteresis the study of nonlinear vibrations oscillations in mechanical and electronic systems has always been an important research area while important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear odes arising from nonlinear springs and electronic circuits the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete the 1 dimensional vibrating string satisfying wtt wxx 0 on the unit interval x 0 1 is an infinite dimensional harmonic oscillator consider the boundary conditions at the left end x 0 the string is fixed while at the right end x 1 a nonlinear boundary condition wx wt w3 t 0 takes effect this nonlinear boundary condition behaves like a van der pol oscillator causing the total energy to rise and fall within certain bounds regularly or irregularly we formulate the problem into an equivalent first order hyperbolic system and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates the problem is reduced to a discrete iteration problem of the type un 1 f un where f is the nonlinear reflection relation we say that the pde system is chaotic if the mapping f is chaotic as an interval map algebraic asymptotic and numerical techniques are developed to tackle the cubic nonlinearities we then define a rotation number following j p keener 11 and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of m i malkin 17 to determine the chaotic regime of for the nonlinear reflection relation f thereby rigorously proving chaos nonchaotic cases for other values of are also classified such cases correspond to limit cycles in nonlinear second order odes numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics received by the editors july 20 1995 and in revised form october 16 1996 1991 mathematics subject classification primary 35l05 35l70 58f39 70l05 the first and third authors work was supported in part by nsf grant dms 9404380 texas arp grant 010366 046 and texas a m university interdisciplinary research initiative iri 96 39 work completed while the first author was on sabbatical leave at the institute of applied mathematics national tsing hua university hsinchu 30043 taiwan r o c the second author s work was supported in part by grant nsc 83 0208 m 007 003 from the national council of science of the republic of china c 1998 american mathematical society

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

on models with non rough poincare homoclinic curves abstract the possibility of an a priori complete description of finite parameter models including systems with structurally unstable poincare homoclinic curves is studied the main result reported here is that systems having a countable set of moduli of equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the newhouse regions of non stability we discuss the question of correctly setting a problem for the analysis of models of such type

oscillations and instability of a shallow arch under two frequency excitation a shallow sinusoidal elastic arch with pinned ends is considered it is subjected to a distributed load with sinusoidal spatial distribution and an amplitude comprised of two harmonic components the equation for the response amplitude contains quadratic and cubic non linearities as well as damping and it is solved by numerical integration the effects of the arch rise and the excitation frequencies on the critical load are investigated including the influence of primary subharmonic superharmonic and combination resonances to study the response of the arch time histories phase plane portraits and power spectra are plotted for a number of loading conditions various types of limit cycles occur and in some cases the motion is chaotic

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3