analytical approximation of cuspidal loops using a nonlinear time transformation method abstract in this work we consider cuspidal loops i e homoclinic orbits to cuspidal singular points we develop an iterative procedure founded on the nonlinear time transformation method to estimate such codimension three global bifurcations up to any wanted order not only in the space of parameters but also in the phase plane as far as we know this is the first time in the literature that this theoretical result is achieved for these global connections the existence and uniqueness of the perturbed solution obtained are proved to illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate takens bogdanov bifurcations excellent agreement is found between our analytical predictions and the corresponding numerical continuations

variable delay feedback control of unstable steady states in retarded time delayed systems we study the stability of unstable steady states in scalar retarded time delayed systems subjected to a variable delay feedback control the important aspect of such a control problem is that time delayed systems are already infinite dimensional before the delayed feedback control is turned on when the frequency of the modulation is large compared to the system s dynamics the analytic approach consists of relating the stability properties of the resulting variable delay system with those of an analogous distributed delay system otherwise the stability domains are obtained by a numerical integration of the linearized variable delay system the analysis shows that the control domains are significantly larger than those in the usual time delayed feedback control and that the complexity of the domain structure depends on the form and the frequency of the delay modulation

deterministic brownian motion generated from differential delay equations this paper addresses the question of how brownian like motion can arise from the solution of a deterministic differential delay equation to study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation our results show that solutions of the deterministic equation with randomly selected initial conditions display a gaussian like density for long time but the densities are supported on an interval of finite measure using these chaotic solutions as velocities we are able to produce brownian like motions which show statistical properties akin to those of a classical brownian motion over both short and long time scales several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation numerical studies suggest that these conjectures could be universal for similar types of chaotic dynamics but we have been unable to prove this

domination by product measures we consider families of 0 1 valued random variables indexed by the vertices of countable graphs with bounded degree first we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site the probability of seeing a 1 at this site is at least a value p which is large enough then this random field dominates a product measure with positive density moreover the density of this dominated product measure can be made arbitrarily close to 1 provided that p is close enough to 1 next we address the issue of obtaining the critical value of p defined as the threshold above which the domination by positive density product measures is assured for the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units this critical value is shown to be 1 k k k 1 k 1 and a discontinuous transition is shown to occur similar critical values of p are found for other classes of probability measures on 0 1 z for the class of k dependent measures the critical value is again 1 k k k 1 k 1 with a discontinuous transition for the class of two block factors the critical value is shown to be 1 2 and a continuous transition is shown to take place in this case thus both the critical value and the nature of the transition are different in the two block factor and 1 dependent cases

the cyclicity of period annuli of a class of quintic hamiltonian systems abstract in this paper we prove that the cyclicity of period annuli for system d x d t y d y d t a x b x 3 c x 5 under perturbations of polynomials with degree n is not more than 54 n 13 taking into account the multiplicity where a b c r with c 0 such that the unperturbed systems have at least one center

hyperchaos in the generalized rossler system introduced as a model for hyperchaos the generalized r ossler system of dimension n is obtained by linearly coupling n ensuremath 3 additional degrees of freedom to the original r ossler equation under variation of a single control parameter it is able to exhibit the chaotic hierarchy ranging from fixed points via limit cycles and tori to chaotic and finally hyperchaotic attractors through the help of a mode transformation we reveal a structural symmetry of the generalized r ossler system the latter will allow us to interpret the number shape and location in phase space of the observed coexisting attractors within a common scheme for arbitrary odd dimension n the appearance of hyperchaos is explained in terms of interacting coexisting attractors in a second part we investigate the lyapunov spectra and related properties of the generalized r ossler system as a function of the dimension n we find scaling properties which are not similar to those found in homogeneous spatially extended systems indicating that the high dimensional chaotic dynamics of the generalized r ossler system fundamentally differs from spatiotemporal chaos if the time scale is chosen properly though a universal scaling function of the lyapunov exponents is found which is related to the real part of the eigenvalues of an unstable fixed point

different regimes of synchronization in nonidentical time delayed maps abstract we study the synchronization of time delayed nonidentical maps subject to unidirectional master slave coupling the individual dynamics of the maps have a delay n 1 and the coupling acts with a delay n 2 we show analytically that suitably tuning the slave map parameters two distinct synchronization regimes can occur in one regime the lag time between the slave and the master maps is given by the delay of the coupling n 2 while in the other regime is given by the difference between the delays n 1 n 2 we analyze the effect of the coupling strength on the different synchronization regimes in logistic and henon maps

selective monostability in multi stable systems we propose a robust method that allows a periodic or a chaotic multi stable system to be transformed to a monostable system at an orbit with dominant frequency of any of the coexisting attractors our approach implies the selection of a particular attractor by periodic external modulation with frequency close to the dominant frequency in the power spectrum of a desired orbit and simultaneous annihilation of all other coexisting states by positive feedback both applied to one of the system parameters the method does not require any preliminary knowledge of the system dynamics and the phase space structure the efficiency of the method is demonstrated in both a non autonomous multi stable laser with coexisting periodic orbits and an autonomous rossler like oscillator with coexisting chaotic attractors the experiments with an erbium doped fibre laser provide evidence for the robustness of the proposed method in making the system monostable at an orbit with dominant frequency of any preselected attractor

complexity and synchronization in chaotic fiber optic systems abstract in this work we investigate experimentally the complexity of chaotic attractors generated by a semiconductor laser subjected to optical feedback and associate their dimensionality with the synchronization efficiency of the corresponding chaotic transmitter receiver configuration the complexity is characterized by calculating the correlation dimension d 2 of experimental chaotic time series for different values of the optical feedback we present the effect of d 2 on the synchronization efficiency and determine the optimal operating condition that leads to the most complex chaotic carrier and simultaneously to the most successful synchronization lastly we associate and explain our experimental results with theoretical predictions in the research literature

delay differential equation versus 1d map application to chaos control abstract theoretical and experimental results on the control of chaotic motion in an electronic circuit modeled by a delay differential equation are presented the model is derived from a nonlinear optical system with time delay feedback our control method is based on pyragas approach and applied to the infinite dimensional system generalized mean floquet multipliers and mean lyapunov exponents related to delay differential equations are introduced in order to compare a discrete time approximation with the continuous time equation this control method is also used to find the so called isomers related to harmonic periodic solutions predicted by ikeda and matsumoto

perturbation from an elliptic hamiltonian of degree four iii global centre abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a global centre it is proven that the least upper bound of the number of zeros of the related elliptic integral is four and this is a sharp one this result permits to prove the existence of lienard equations of type 3 2 with a quadruple limit cycle with both a triple and a simple limit cycle with two semistable limit cycles with one semistable and two simple limit cycles or with four simple limit cycles

a sharper threshold for bootstrap percolation in two dimensions two dimensional bootstrap percolation is a cellular automaton in which sites become infected by contact with two or more already infected nearest neighbours we consider these dynamics which can be interpreted as a monotone version of the ising model on an n n square with sites initially infected independently with probability p the critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1 2 holroyd determined the sharp first order approximation pc 2 18 log n as n here we sharpen this result proving that the second term in the expansion is log n 3 2 o 1 and moreover determining it up to a poly log log n factor the exponent 3 2 corrects numerical predictions from the physics literature

perturbations from an elliptic hamiltonian of degree four abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree 4 exhibiting a cuspidal loop it is proven that the least upper bound for the number of zeros of the related elliptic integral is four and this upper bound is a sharp one this permits to prove the existence of lienard equations of type 3 2 with at least four limit cycles the paper also contains a complete result on the respective number of small and large limit cycles

anticipated synchronization in coupled chaotic maps with delays abstract we study the synchronization of two chaotic maps with unidirectional master slave coupling both maps have an intrinsic delay n1 and coupling acts with a delay n2 depending on the sign of the difference n1 n2 the slave map can synchronize to a future or a past state of the master system the stability properties of the synchronized state are studied analytically and we find that they are independent of the coupling delay n2 these results are compared with numerical simulations of a delayed map that arises from discretization of the ikeda delay differential equation we show that the critical value of the coupling strength above which synchronization is stable becomes independent of the delay n1 for large delays

hopf bifurcation subject to a large delay in a laser system hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system the problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks most of these instabilities are starting from a single hopf bifurcation because of the large delay a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi periodic states we determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double hopf bifurcation points we study numerically different cases of bistability between steady periodic and quasi periodic regimes finally the validity of the hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation

power spectra and dynamical invariants for delay differential and difference equations abstract the mackey glass and ikeda delay differential equations ddes are models of feedback systems in which attractor dimension is proportional to the delay power spectra from these models are investigated analytically and numerically as the delay increases much beyond the system response time an exponentially decaying spectrum with a near periodic superimposed modulation appears it is shown that each peak in this modulation is associated with a mode of oscillation predicted by linear stability analysis around the fixed point the number of such modes within a characteristic autocorrelation time of the solution agrees with the lyapunov dimension of the attractor the disappearance of this modulation at higher delay values is also explained the decay rate of the spectrum in the mid to high frequency range is found to agree with the sum of the positive lyapunov exponents i e with the metric entropy realistic models with a distribution of delays display similar spectra as their memory kernel narrows into a delta function for very large delay the dde approaches an infinite dimensional continuous time difference equation ctde with each point on a delay interval evolving independently and according to a chaotic discrete time map ctde spectra which have many features of finite delay dde spectra can be computed analytically if the spectrum of this map is known for constant initial functions they have a 1 f 2 form at low frequencies our analysis suggests that the spectrum of the chaotic map is at the origin of the peak shapes in dde spectra at low frequency our findings highlight the relevance of the linear prrperties of ddes to the characterization of their nonlinear properties

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

ikeda hopf bifurcation revisited abstract a large variety of passive optical systems subject to a delayed feedback have appeared in the literature and are described mathematically by the same class of scalar delay differential equations ddes these equations include ikeda dde and their solutions are determined in terms of a control parameter distinct from the delay we concentrate on the first hopf bifurcation generated by a fixed delay and determine a general expression for its direction of bifurcation we then examine our result in the two limits of small and large delays for small delays we show that a hopf bifurcation to nearly sinusoidal oscillations is possible provided that the feedback rate is sufficiently high bifurcation from infinity for large delays we complement the early work by chow et al proc roy soc edinburgh a 120 1992 223 229 and hale and huang j diff equ 114 1994 1 23 by comparing analytical and numerical bifurcation diagrams as the oscillations progressively change from sine to square wave

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

effect of very short impulses on nonlinear dissipative kinetics abstract delayed nonautonomous kinetics is considered on the example of the kinetic equation t ax t del b t x 2 t del with b t representing a train of separate impulses dependence on impulse width is discussed for the case of short delay times sufficiently short impulses lead to significant changes in x t appearance of chaotic behaviour absent for longer impulses abrupt drop of the average value of x etc which enables the experimental determination of del it is suggested that there exists a time scale on which the normal macroscopic kinetic description breaks down due to the natural inertia of physical processes related e g to the presence of fast variables averaged out in the macroscopic description hence the effects described in this paper may be used for the phenomenological definition and experimental determination of a new characteristic of the macroscopic dissipative system related to that time scale the inertial relaxation time inertial

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

high dimensional chaos in delayed dynamical systems abstract we introduce a general class of iterative delay maps to model high dimensional chaos in dynamical systems with delayed feedback the class includes as particular cases systems with a linear local dynamics we report analytic and numerical results on the scaling laws of lyapunov spectra with a number of degrees of freedom invariant measure is computed through a self consistent frobenius perron formalism which allows also a recalculation of the maximum lyapunov exponent in good agreement with the one measured directly

analytical approach for the floquet theory of delay differential equations we present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states to this end we start with approximately determining such reference states by extending the poincare lindstedt and the shohat expansions which were originally developed for ordinary differential equations then we systematically elaborate a linear stability analysis around a time periodic reference state this allows us to approximately calculate the floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

heterogeneous k core versus bootstrap percolation on complex networks we introduce the heterogeneous k core which generalizes the k core and contrast it with bootstrap percolation vertices have a threshold r i that may be different at each vertex if a vertex has fewer than r i neighbors it is pruned from the network the heterogeneous k core is the subgraph remaining after no further vertices can be pruned if the thresholds r i are 1 with probability f or k ensuremath geqslant 3 with probability 1 ensuremath f the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation which is an activation process we show that there are two types of transitions in this heterogeneous k core process the giant heterogeneous k core may appear with a continuous transition and there may be a second discontinuous hybrid transition we compare critical phenomena critical clusters and avalanches at the heterogeneous k core and bootstrap percolation transitions we also show that the network structure has a crucial effect on these processes with the giant heterogeneous k core appearing immediately at a finite value for any fg0 when the degree distribution tends to a power law p q q ensuremath ensuremath gamma with ensuremath gamma l3

stability of synchronization in coupled time delay systems using krasovskii lyapunov theory stability of synchronization in unidirectionally coupled time delay systems is studied using the krasovskii lyapunov theory we have shown that the same general stability condition is valid for different cases even for the general situation but with a constraint where all the coefficients of the error equation corresponding to the synchronization manifold are time dependent these analytical results are also confirmed by the numerical simulation of paradigmatic examples

nonlinear dynamics in a laser with a negative delayed feedback abstract asymptotic solutions of nonlinear differential difference equations describing the lasing dynamics with a negative feedback are obtained periodic lasing regimes with a complex relaxational structure are shown to be possible in such lasers the phenomenon of multistability i e the co existence of a great number of structurally different and structurally similar attractors is analytically described asymptotic characteristics of lasing regimes pulse shape and radiation pulsation frequency are found and the domains in the parameter space and the initial conditions for their realization are determined for all types of solution the chaotic dynamics following the sequence of period doubling bifurcations is also numerically determined

delay induced instabilities in nonlinear feedback systems extending general methods developed in synergetics to construct order parameter equations we work out a systematic elimination procedure for the stable modes in nonlinear delay systems it will be shown that in the vicinity of an instability the dynamical behavior of the infinite dimensional delay system is approximately governed by a low dimensional set of order parameter equations which turn out to be of the form of ordinary differential equations i e they no longer contain memory terms the general formalism will then be compared with experimental data obtained from the nonlinear operation of a phase locked loop where the finite propagation time of the signal in the feedback loop is taken into account

multistability in spiking neuron models of delayed recurrent inhibitory loops we consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate and fire neuron model we show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations we observe that the number of fixed points of the so called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns we demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns

metastability for delayed differential equations in systems at phase transitions two phases of the same substance may coexist for a long time before one of them dominates we show that a similar phenomenon occurs in systems with delayed feedback where short term stable oscillatory patterns can also have very long lifetimes before vanishing into constant or periodic steady states

bootstrap percolation in a polluted environment let a low densityp of sites on the lattice z2 be occupied remove a proportionq of them and call the remaining sites empty then update this configuration in discrete time by iteration of the following synchronous rule an empty site becomes occupied by contact with at least two occupied nearest neighbors while occupied and removed sites nerver change their states ifq p2 is large most sites remain unoccupied forever while ifq p2 is small this dynamics eventually makes most sites occupied this demonstrates how sensitive the usual bootstrap percolation rule theq 0 case is to the pollution of space

hopf bifurcation subject to a large delay in a laser system hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system the problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks most of these instabilities start from a single hopf bifurcation because of the large delay a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi periodic states we determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double hopf bifurcation points we study numerically different cases of bistability between steady periodic and quasi periodic regimes finally the validity of the hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation

interface motion in random media we study the pinning phase transition for discrete surface dynamics in random environments a renormalization procedure is devised to prove that the interface moves with positive velocity under a finite size condition this condition is then checked for different examples of microscopic dynamics to illustrate the flexibility of the method we show in our examples the existence of a phase transition for various models including high dimensional interfaces dependent environments and environments with arbitrarily deep obstacles finally our ballisticity criterion is proved to be valid up to the critical threshold for a lipschitz interface model

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

bootstrap percolation on complex networks we consider bootstrap percolation on uncorrelated complex networks we obtain the phase diagram for this process with respect to two parameters f the fraction of vertices initially activated and p the fraction of undamaged vertices in the graph we observe two transitions the giant active component appears continuously at a first threshold there may also be a second discontinuous hybrid transition at a higher threshold avalanches of activations increase in size as this second critical point is approached finally diverging at this threshold we describe the existence of a special critical point at which this second transition first appears in networks with degree distributions whose second moment diverges but whose first moment does not we find a qualitatively different behavior in this case the giant active component appears for any fg0 and pg0 and the discontinuous transition is absent this means that the giant active component is robust to damage and also is very easily activated we also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

existence of chaos in control systems with delayed feedback here f r r denotes some nonlinear function a a positive constant t e r is the independent variable x e r the dependent variable and 1 dx dt the investigation is motivated i by the applications eq 1 1 has found in several areas of applied sciences and ii by the observation of intensely structured periodic and aperiodic oscillations in numerical simulations for a certain class of nonlinearities f we show that these structures are not numerical artefacts but properties of the solution manifold of eq 1 1 in the applications eq 1 1 has been used to explain normal and pathological behavior of control systems in the physiology of blood cell production and respiration 7 8 10 111 and periodic 2 or irregular s activity in neural networks computer simulations to eq 1 1 may be found in 3 the mathematical analysis of this equation so far has been restricted to prove the existence of slowly oscillating simple periodic solutions 1 4 61 a periodic solution is called slowly oscillating if successive

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

front dynamics in a delayed feedback system with external forcing a scalar nonlinear delay differential equation of the kind arising in optically bistable systems is reduced in some physically relevant limit to a partial differential equation first and then to a system of ordinary differential equations these simplified equations are studied analytically we describe the continuous deformation of periodic solutions from sine waves to square waves away from the oscillation threshold and determine their stability we also study the effect of a weak periodic modulation it is found that a resonant forcing can stabilize a very large number of oscillation modes taking the form of square waves with missing front pairs

high dimensional chaotic behavior in systems with time delayed feedback abstract the nature of high dimensional chaos exhibited by a class of delay differential equation is investigated by various methods this delay differential equation models systems with time delayed feedback such as nonlinear optical resonators we first describe briefly the bifurcation phenomena exhibited by the system with an increase in a control parameter representing the energy flow rate periodic states bi multi furcate themselves successively forming a hierarchy of multistable periodic states with increasing complexities finally each branch of multistable periodic states makes transition to chaos the possibility of applying such a multistability as a memory device for complicated information is discussed in the chaotic regime each of the bifurcated states in turn merge successively into fewer sets of states with larger attractor dimensions and finally a single developed chaos with a very large attractor dimension is formed lyapunov analysis is introduced to study the high dimensional chaotic states the lyapunov vectors as well as lyapunov spectrum are shown to be very useful to understand the underlying mechanism of the successive merging process mentioned above characteristics of developed chaos are investigated by high pass filtered time series hfts the intermittency characteristics of the hfts changes markedly at a certain frequency the lyapunov analysis reveals that this frequency corresponds to a characteristic lyapunov mode number this characteristic number can be looked upon as the dimension of subspace in which active chaotic information is generated and is different from the attractor dimension in the customary sense

synchronization of noisy delayed feedback systems with delayed coupling synchronization of delayed coupled and stochastically perturbed systems with delayed nonlinear feedback is studied using as an example circular chains of three and four delayed coupled ikeda oscillators it is proved that in the case of multiplicative noise the exact synchronization in the mean occurs for sufficiently large coupling and an analytic estimate of the sufficient coupling is given the sufficiency condition is compared with numerical computations and typical effects of noise on the exact and some generalized types of synchronization are illustrated

some basic remarks on eigenmode expansions of time delay dynamics abstract we describe in elementary terms how eigenmode expansions can be used to deal with differential difference equations as particular applications we present the full analytical solution of linear stochastic time delay systems and the weakly nonlinear analysis of nonlinear differential difference equations in the limit of large time delay our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the lambert w function and on the explicit formula for the eigenfunctions of the adjoint equation

bifurcations of limit cycles for a quintic hamiltonian system with a double cuspidal loop in this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic hamiltonian vector field x h y x x 3 x 2 1 y under small perturbations of the form e a bx 2 cx 4 y y where 0 e 1 and a b c are real constants using picard fuchs equations for related abelian integrals asymptotic expansion of these integrals about critical level curves of h and some geometric properties of the curves defined by ratios of two especial integrals we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x is 2 taking into account their multiplicities for h 0 1 6 and this upper bound is a sharp one this implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for e 0 inside an eye figure loop is less than or equal to 2

reconstruction of time delayed feedback systems from time series for various classes of time delay systems we propose the methods of their model delay differential equation reconstruction from time series the methods are based on the characteristic location of extrema in the time series of time delay systems and the projection of infinite dimensional phase space of these systems to suitably chosen low dimensional subspaces we verify our methods by using them for the recovery of time delay differential equations from their chaotic solutions and for modelling experimental systems with delay induced dynamics from their chaotic time series

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

hopf bifurcations in time delay systems with band limited feedback abstract we investigate the steady state solution and its bifurcations in time delay systems with band limited feedback this is a first step in a rigorous study concerning the effects of ac coupled components in nonlinear devices with time delayed feedback we show that the steady state is globally stable for small feedback gain and that local stability is lost generically through a hopf bifurcation for larger feedback gain we provide simple criteria that determine whether the hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the taylor expansion of the nonlinearity furthermore the presence of double hopf bifurcations of the steady state is shown which indicates possible quasiperiodic and chaotic dynamics in these systems as a result of this investigation we find that ac coupling introduces fundamental differences to systems of ikeda type k ikeda k matsumoto high dimensional chaotic behavior in systems with time delayed feedback physica d 29 1987 223 235 already at the level of steady state bifurcations e g bifurcations exist in which limit cycles are created with periods other than the fundamental period 2 mode found in ikeda type systems

lattice embeddings in percolation does there exist a lipschitz injection of d into the open set of a site percolation process on d if the percolation parameter p is sufficiently close to 1 we prove a negative answer when d d and also when d 2 if the lipschitz constant m is required to be 1 earlier work of dirr dondl grimmett holroyd and scheutzow yields a positive answer for d d and m 2 as a result the above question is answered for all d d and m our proof in the case d d uses tucker s lemma from topological combinatorics together with the aforementioned result for d d one application is an affirmative answer to a question of peled concerning embeddings of random patterns in two and more dimensions

exact bound on the number of zeros of abelian integrals for two hyper elliptic hamiltonian systems of degree 4 abstract this paper concerns the exact bound on the number of zeros of abelian integrals associated with two hyper elliptic hamiltonian systems of degree 4 the upper and lower bounds for the two systems have been obtained in several previous works however the sharp bounds are still unknown in this paper we provide a proof to show that the exact bound is 3 for both systems the basic idea of our method is to bound the parameter space in r 3 to obtain two parameter sets which might yield maximal 4 zeros of abelian integrals corresponding to the two sets further we show that the existence of 4 zeros on the two sets is not possible and thus the sharp bound is 3

spatiotemporal structures in a model with delay and diffusion pattern formation described by differential difference equations with diffusion is investigated it is shown that an arbitrarily small diffusion induces space time turbulence just at the instability threshold of the homogeneous stationary solution we prove this property by deriving a complex ginzburg landau equation on the basis of normal form analysis well above threshold such turbulent structures give way to synchronized states ordered by spirals and targets this secondary instability can be understood with an asymptotic method representing the system as a cellular automaton network

delay and periodicity systems with time delay play an important role in modeling of many physical and biological processes in this paper we describe generic properties of systems with time delay which are related to the appearance and stability of periodic solutions in particular we show that delay systems generically have families of periodic solutions which are reappearing for infinitely many delay times as delay increases the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions we also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers we show that the spectrum of multipliers can be split into two parts pseudocontinuous and strongly unstable the pseudocontinuous part of the spectrum mediates destabilization of periodic solutions

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

multiple scale analysis of delayed dynamical systems abstract the model equations of a semiconductor laser with delayed feeback are studied as a prototype of general delayed systems in the case of long delay thermodynamic limit it is shown that its behaviour in the vicinity of the hopf bifurcation can be described by suitable amplitude equations the derivation of the final model follows naturally from the space time representation of the evolution and by the corresponding re formulation of the linear stability problem regions characterized by both direct and inverse hopf bifurcations are found and in particular parameter values are identified for which a chaotic evolution is expected

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

tool to recover scalar time delay systems from experimental time series we propose a method that is able to analyze chaotic time series gained from exp erimental data the method allows to identify scalar time delay systems if the dynamics of the system under investigation is governed by a scalar time delay differential equation of the form dy t dt h y t y t tau 0 the delay time tau 0 and the functi on h can be recovered there are no restrictions to the dimensionality of the chaotic attractor the method turns out to be insensitive to noise we successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators

proof of straley s argument for bootstrap percolation we prove thatpc 1 for bootstrap percolation with large void instabilities in particular ifm 3 on the square lattice

polluted bootstrap percolation with threshold two in all dimensions in the polluted bootstrap percolation model the vertices of a graph are independently declared initially occupied with probability p or closed with probability q at subsequent steps a vertex becomes occupied if it is not closed and it has at least r occupied neighbors on the cubic lattice mathbb z d of dimension d ge 3 with threshold r 2 we prove that the final density of occupied sites converges to 1 as p and q both approach 0 regardless of their relative scaling our result partially resolves a conjecture of morris and contrasts with the d 2 case where gravner and mcdonald proved that the critical parameter is q p 2

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

order parameters for class b lasers with a long time delayed feedback abstract a local dynamics for class b laser systems with optoelectronic feedback is considered in the case of a very long time delay we determine the conditions for the bifurcation of infinite dimension it is shown that the ordinary delay system can be described by a systematically derived complex ginzburg landau equation without delay terms the complicated quasi periodic dynamics near equilibrium can be demonstrated analytically on the basis of space time representation

the perturbations of a class of hyper elliptic hamilton systems with a double homoclinic loop through a nilpotent saddle abstract in this paper we consider the upper bounds of a number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 under the perturbations of 0 1 x 2 x 2 3 x 3 4 x 4 y y where 0 1 and i r i 0 4 the unperturbed system has a double homoclinic loop with a nilpotent saddle of order 1 the sharp upper bounds are obtained for each of the cases of 1 4 0 1 3 0 2 3 0 and 3 4 0 when abelian integrals are defined in the bounded period annuli

embedding the dynamics of a single delay system into a feed forward ring we investigate the relation between the dynamics of a single oscillator with delayed self feedback and a feed forward ring of such oscillators where each unit is coupled to its next neighbor in the same way as in the self feedback case we show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring in particular if for the single oscillator the periodic solution is resonant to the delay it can be embedded into a ring with instantaneous couplings we discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring as a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling finally we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of fitzhugh nagumo type

on the eigenvalue spectrum for time delayed floquet problems abstract a linear homogeneous scalar differential difference equation with harmonic time dependence is investigated the associated eigenvalue problem is solved in terms of a continued fraction expansion for the characteristic equation the dependence of the largest eigenvalue on the system parameters being relevant for stability of periodic states in delay systems is discussed in detail the competition between the two timescales the delay and the external period cause intricate structures the result suggests features to improve control of chaos by time delayed feedback schemes with time dependent control amplitudes

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

study of a high dimensional chaotic attractor the nature of a very high dimensional chaotic attractor in an infinite dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high dimensional attractor in the phase space we introduce two complementary bases from which the attractor is observed one the lyapunov basis composed of the lyapunov vectors and the another the fourier basis composed of the fourier modes we introduce the exterior subspaces on the basis of the lyapunov vectors and observe the chaotic motion projected onto these exteriors it is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace goes out of the attractor the origin of such a phenomenon is attributed to more fundamental features of our attractor which become manifest when the attractor is observed from the lyapunov basis a counterpart of the phenomenon can be observed also on the fourier basis because there is a statistical one to one correspondence between the lyapunov vectors and the fourier modes in particular a statistical property of the high pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor

coupled map lattices as models of deterministic and stochastic differential delay equations we discuss the probabilistic properties of a class of differential delay equations dde s by first reducing the equations to coupled map lattices and then considering the spectral properties of the associated transfer operators the analysis is carried out for the deterministic case and a stochastic case perturbed by additive or multiplicative white noise this scheme provides an explicit description of the evolution of phase space densities in dde s and yields an evolution equation that approximates the analog for delay equations of the generalized liouville and fokker planck equations it is shown that in many cases of interest for both stochastic and deterministic delay equations the phase space densities reach a limit cycle in the asymptotic regime this statistical cycling is observed numerically in continuous time systems with delay and discussed in light of our analytical description of the transfer operators 1995 the american physical society

sharp metastability threshold for an anisotropic bootstrap percolation model bootstrap percolation models have been extensively studied during the two past decades in this article we study the following anisotropic boot strap percolation model the neighborhood of a point m n is the set m 2 n m 1 n m n 1 m 1 n m 2 n m n 1 at time 0 sites are occupied with probability p at each time step sites that are occupied remain occupied while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied we prove that it exhibits a sharp metastability threshold this is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model

periodic solutions of a singularly perturbed delay differential equation abstract a singularly perturbed differential delay equation of the form 1 x t x t f x t 1 exhibits slowly oscillating periodic solutions sops near the first period doubling bifurcation point of the underlying map obtained by setting 0 for extremely small values of these periodic solutions resemble square waves which consist of sharp o transition layers connecting intervals of approximately unit length in this article we obtain analytic expressions for these square wave periodic solutions by solving the corresponding transition layer equations and show that they are in excellent agreement with numerical solutions for a range of values of and we also derive analytic expressions for other periodic solutions which are odd harmonics of the sops and numerically exhibit their instability near the first period doubling bifurcation point of the map the numerical computations were performed using a high accuracy chebyshev spectral scheme we give a brief description together with a study of its accuracy and efficiency

routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro optic oscillators the response of a nonlinear optical oscillator subject to a delayed broadband bandpass filtering feedback is studied experimentally numerically and analytically the oscillator loop is characterized by a high cutoff frequency with a response time tau approximately 10 ps and by a low cutoff frequency with a response time theta approximately 1 micros moreover the optoelectronic feedback also consists of a significant delay taud of the order of 100 ns depending on two key physical parameters the loop gain beta and the nonlinearity operating point phi a large variety of multiple time scale regimes are reported including slow or fast periodic oscillations with different waveforms regular or chaotic breathers slow time envelope dynamics complex and irregular self pulsing and fully developed chaos many of these regimes are exhibiting new features that are absent in the classical first order scalar nonlinear delay differential equations ddes which differ in the modeling by the low cutoff only nearly all kinds of solutions are recovered numerically by a new class of integro dde idde that take into account both the high and low cutoff frequencies of the feedback loop for moderate feedback gain asymptotic solutions are determined analytically by taking advantage of the relative values of the time constants tau theta and taud we confirm the experimental observation of two distinct routes to oscillatory instabilities depending on the value of phi one route is reminiscent of the square wave oscillations of the classical first order dde but the other route is quite different and allows richer wave forms for higher feedback gain these two distinct regimes merge leading to complex nonperiodic regimes that still need to be explored analytically and numerically finally we investigate the theoretical limits of our idde model by experimentally exploring phenomena at extreme physical parameter setting namely high frequency locking at strong feedback gain or pulse packages for very large delays the large variety of oscillatory regimes of our broadband bandpass delay electro optic oscillator is attractive for applications requiring rich optical pulse sources with different frequencies and or wave forms chaos based communications random number generation chaos computing and generation of stable multiple ghz frequency oscillations

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop ii abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x for h 1 6 is three and for h 0 is four counted with multiplicity for all parameters and this implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for e 0 outside an eye figure loop is less than or equal to three

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude