master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

the pim simplex method an extension of the pim triple method to saddles with an arbitrary number of expanding directions abstract the growing interest in non attracting chaotic sets of high dimensional dynamical systems requires the development of numerical techniques for their study the pim triple method h e nusse j a yorke physica d 36 1989 137 is a very good method to obtain trajectories on saddles with one positive lyapunov exponent in this paper we combine the same ideas with an algorithm for finding local extrema of multi variable functions to develop an extension of the method the pim simplex method that is suitable for the study of sets with an arbitrary number of expanding directions

controlling chaos experimentally in systems exhibiting large effective lyapunov exponents we investigate experimentally the performance of the ott grebogi and yorke phys rev lett 64 1196 1989 feedback concept to control chaotic motion the experimental systems are a driven pendulum and a driven bronze ribbon both setups have unstable periodic orbits characterized by large effective lyapunov exponents all control vectors for the feedback control are extracted from the experimental data to do this for the pendulum a global model obtained by the flow field analysis of cremers and h ubler z naturforsch 42a 797 1987 is used and for the bronze ribbon linear approximations in embedding space are exploited we analyze the problems that arise due to the amplification of noise by large effective lyapunov exponents in the determination of the control values as well as in the performance of the experimental control successful control can be achieved in our experiments by applying the local control method which allows a quasicontinuous adjustment of the control parameter in contrast to adjusting the control parameter only once per return time of the poincar e map

control of noisy heteroclinic cycles abstract we develop a control strategy for strongly nonlinear dynamical systems possessing heteroclinic cycles connecting saddle points and subject to small random perturbations in the absence of control attracting cycles lead to solutions which exhibit intermittent behavior consisting of quiescent periods interrupted by rapid bursts we show that suitable feedback control can retain solutions near the saddle points and hence reduce the typical bursting rate the model studied and the form of control are motivated by our interest in reducing turbulence production in the boundary layer

nonlocal symmetries and recursion operators partial differential and differential difference equations abstract a systematic method to derive the nonlocal symmetries for partial differential and differential difference equations with two independent variables is presented and shown that the korteweg de vries kdv and burger s equations volterra and relativistic toda rt lattice equations admit a sequence of nonlocal symmetries an algorithm exploiting the obtained nonlocal symmetries is proposed to derive recursion operators involving nonlocal variables and illustrated it for the kdv and burger s equations volterra and rt lattice equations and shown that the former three equations admit factorisable recursion operators while the rt lattice equation possesses 2 2 matrix factorisable recursion operator the existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential difference equations does not always determine their mathematical structures for example bi hamiltonian representation

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

control of chaotic systems using an on line trained linear neural controller abstract we propose a control system including an on line trained linear neural controller to control chaotic systems the control system stabilizes a chaotic orbit onto an unstable fixed point without using the knowledge of the location of the point and the local linearized dynamics at the point furthermore the control system can track the stabilized orbit to the unstable fixed point whose location and local dynamics vary slowly with a variation of the system parameter this paper extends a previous paper konishi and kokame 1995 for more general situations and improves the neural controller proposed in the previous paper both to simplify the training algorithm and to guarantee the convergence of the neural controller the stability analysis of the control system reveals that some unstable fixed points cannot be stabilized in the control system numerical experiments show that the control system works well for controlling high dimensional chaotic systems

chaotic transitions and low frequency fluctuations in semiconductor lasers with optical feedback abstract this paper examines the dynamical origin of low frequency fluctuations lffs in semiconductor lasers subject to time delayed optical feedback in particular we study chaotic transitions leading to the onset of lffs by numerical integration of lang kobayashi equations for a laser pumped near threshold we construct a bifurcation analysis scheme that enables the classification of the different operation regimes of the laser we use the scheme to study the coexistence of the lffs and stable emission on the maximum gain mode mgm which was the subject of recent experiments t heil i fischer w elsaser phys rev a 60 1999 634 our computations suggest that as the feedback level increases the regime of sustained lffs alternates with regions of transient lffs where the laser can achieve stabilization on the mgm exploration of the parameter space reveals strong dependence of the structure of the lff dynamics and the coexistence regime on the value of the linewidth enhancement factor

multi dimensioned intertwined basin boundaries basin structure of the kicked double rotor abstract using numerical computations on a map which describes the time evolution of a particular mechanical system in a four dimensional phase space the kicked double rotor we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale in particular for the double rotor map if one chooses a restricted region of the phase space and examines the basin boundary in that region then either one observes that the boundary is a smooth three dimensional surface or one observes that the boundary is fractal with dimension d 3 9 and which of these two possibilities applies depends on the particular phase space region chosen for examination furthermore for any region no matter how small for which d 3 9 one can choose subregions within it for which d 3 hence d 3 9 region and d 3 region are intertwined on arbitrarily fine scale other examples will also be presented and analyzed to show how this situation can arise these include one dimensional map cases a map of the plane and the lorenz equations in one of our one dimensional map cases the boundary will be fractal everywhere but the dimension can take on either of two different values both of which lie between 0 and 1 these examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values more specifically let these values be denoted d 1 d 2 d n choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region then our conjecture is that for all typical volume choices the evaluated dimension within the chosen volume will be one of the values d 1 d 2 d n for example in our double rotor map it appears that n 2 and d 1 3 0 and d 2 3 9

controlling chaotic dynamical systems using time delay coordinates abstract the ott grebogi yorke control method is analyzed in the case that the attractor is reconstructed from a time series using time delay coordinates it turns out that the control formula of ogy should be modified in order to apply to experimental systems if time dekay coordinates are used we reveal that the experimental surface of section map does not only depend on the actual parameter but also on the preceding one in order to meet this dependence two modifications are introduced which lead to a better performance of the control to compare their control abilities they are applied to simulations of a duffing oscillator the issue of measurement noise is considered

chaos and crises in more than two dimensions instituto de astronomi ayfi sica del espacio conicet casilla de correo 67 socursal 28 1428 buenos aires argentina received 23 december 1996 noisy chaotic trajectories with nite time lyapunov exponents that uctuate about zero are basicallyunshadowable s dawson c grebogi t sauer and j a yorke phys rev lett 73 1927 1994 this canoccur when periodic orbits with different numbers of unstable directions coexist inside the attractor thepresence of a he non type chaotic saddle i e a nonattracting chaotic set with a structure similar to that of thehe non attractor guarantees such coexistence in a persistent manner s p dawson phys rev lett 76 4348 1996 in this paper we describe how these sets appear naturally in maps of more than two dimensions howthey can be found and what crises they produce s1063 651x 97 13005 7 pacs number s 05 45 1b 05 40 1ji introduction

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

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

on a class of degenerate and singular elliptic systems in bounded domains abstract this paper deals with the nonexistence and multiplicity of nonnegative nontrivial solutions to a class of degenerate and singular elliptic systems of the form div h 1 x u f u x u v in div h 2 x v f v x u v in where is a bounded domain with smooth boundary in r n n 2 and h i 0 h i l loc 1 h i i 1 2 are allowed to have essential zeroes at some points in f u f v f and is a positive parameter our proofs rely essentially on the critical point theory tools combined with a variant of the caffarelli kohn nirenberg inequality in p caldiroli r musina on a variational degenerate elliptic problem nodea nonlinear differential equations appl 7 2000 189 199

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

multiple positive solutions for semilinear dirichlet problems with sign changing weight function in infinite strip domains abstract in this paper existence and multiplicity results to the following dirichlet problem u u f x u q 1 h x u p 1 in u 0 in u 0 on are established where r r n 1 is bounded smooth domain and n 2 here 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 is a positive real parameter the function f among other conditions can possibly change sign in and the function h satisfies suitable conditions the study is based on the comparison of energy levels on nehari manifold

crossing bifurcations and unstable dimension variability a crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied in this letter we describe a global bifurcation in three dimensions which can result in a crisis this bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3 we present evidence of unstable dimension variability as a result of the crisis we then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis we illustrate this new type of bifurcation with a specific example of a three dimensional chaotic attractor undergoing a crisis

master symmetries for volterra equation belov chaltikian and blaszak marciniak lattice equations it is shown how to derive master symmetries for nonlinear lattice equations systematically using the basic principles but without using either their zero curvature equations or the bi hamiltonian structure this has been illustrated for volterra equation two coupled belov chaltikian bc and three coupled blaszak marciniak bm lattice equations the existence of a sequence of master symmetries is one of the characteristics of completely integrable nonlinear partial differential and differential difference equations admitting hamiltonian structure

symmetry and control spatially extended chaotic systems in a recent paper phys rev e 57 1998 1550 it was demonstrated that the symmetries of the evolution equation and the target state have a profound effect on controlling the chaotic behavior in the present paper we extend these results to the cases of time periodic target trajectories and inexact symmetries and apply the developed formalism to the problem of controlling spatiotemporal chaos we use the example of a lattice dynamical system in arbitrary spatial dimension to show that there exists an intimate relationship between the geometry of an extended system and the geometry of feedback control 2000 elsevier science b v all rights reserved

infinitely many conservation laws for the discrete kdv equation rasin and hydon 2007 j phys a math theor 40 12763 73 suggested a way to construct an infinite number of conservation laws for the discrete kdv equation dkdv by repeated application of a certain symmetry to a known conservation law it was not decided however whether the resulting conservation laws were distinct and nontrivial in this paper we obtain the following results 1 we give an alternative method to construct an infinite number of conservation laws using a discrete version of the gardner transformation 2 we give a direct proof that the conservation laws obtained by the method of rasin and hydon are indeed distinct and nontrivial 3 we consider a continuum limit in which the dkdv equation becomes a first order eikonal equation in this limit the two sets of conservation laws become the same and are evidently distinct and nontrivial this proves the nontriviality of the conservation laws constructed by the gardner method and gives an alternative proof of the nontriviality of the conservation laws constructed by the method of rasin and hydon

taming chaotic systems with dithers abstract a simple approach is proposed to tame chaos by injecting high frequency signals commonly called dithers into the chaotic systems based on the relaxed method an appropriate dither is introduced to suppress chaotic motion if the frequency of dither is high enough the trajectory described by the dithered chaotic system and the trajectory of its corresponding mathematical model the relaxed system would be made as close as desired this fact enables us to get a rigorous prediction of the dithered chaotic system s behavior by obtaining the behavior of the relaxed system finally an example with simulations is given to illustrate the concepts discussed throughout this paper

controlling chaos of a dynamical system with discontinuous vector field abstract the paper presents how to control the chaos of dynamical systems with discontinuous vector field through the paradigm of a harmonically forced oscillator having a set up elastic stop it is first shown that the poincare mapping of this oscillator is not smooth near the fixed point corresponding to a periodic motion that grazes the stop thus the current control strategies based on the smooth mapping cannot be directly used to stabilize the chaotic motion near a periodic grazing motion then proposed is a piecewise linear control strategy based on the piecewise linearized poincare mapping reconstructed from sampled data and on the pole assignment in two regions near the fixed point the efficacy of the strategy is finally demonstrated via numerical simulations

adaptive control in nonlinear dynamics abstract we extend an adaptive control algorithm recently suggested by huberman and lumer to multi parameter and higher dimensional nonlinear systems this control mechanism is remarkably effective in returning a system to its original dynamics after a sudden perturbation in the system parameters changes the dynamical behaviour we find that in all cases the recovery time is linearly proportional to the inverse of control stiffness for small stiffness in higher dimensions there is an additional optimization problem since increasing stiffness beyond a certain value can retard recovery the control of fixed point dynamics in systems capable of a wide variety of dynamical behaviour is demonstrated we further suggest methods by which periodic motion such as limit cycles can be adaptively controlled and demonstrate the robustness of the procedure in the presence of additive background noise

optimization of local control of chaos by an evolutionary algorithm an evolutionary algorithm for optimizing local control of chaos is presented based on a lyapunov approach a linear control law and the state space region in which this control law is activated are determined in addition we study a relation between certain adjustable design parameters and a particular measure of the uncontrolled chaotic attractor in the state space region of control ssrc from this relation the objective function to be optimized is derived in that context we assume a linear control law to be given and optimize size and shape of the ssrc using an evolutionary algorithm it is shown by examples how the algorithm can also be applied to higher dimensional systems with possibly more than one positive lyapunov exponent 2000 elsevier science b v all rights reserved

dynamics from multivariate time series abstract multivariate time series data are common in experimental and industrial systems if the generating system has nonlinear dynamics we may be able to construct a model that reproduces the dynamics and can be used for control and other purposes in principle multivariate time series are not necessary for recovering dynamics according to the embedding theorem only one time series should be needed however for real data there may be large gains in using all of the measurements in this paper we examine the issues of how to use multiple data streams most effectively for modeling and prediction for example perhaps the data are redundant in that only a subset of the data streams is useful and how should we embed the data if indeed embedding is required at all we show how these questions can be answered and describe some numerical experiments which show that using multivariate time series can significantly improve predictability we also demonstrate a somewhat surprising synchronization between different reconstructions

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

belov chaltikian and blaszak marciniak lattice equations recursion operators and factorization a systematic investigation on the construction of recursion operators for partial differential difference equations pddes with two independent variables one continuous and one discrete using its generalized symmetries is presented also it is explained how to factorize the obtained recursion operators the applicability of the above procedure have been illustrated for the relativistic toda rt belov chaltikian bc and blaszak marciniak bm lattice equations and shown that the former two lattice equations admit 2 2 matrix recursion operators while the latter one possesses a 3 3 matrix recursion operator furthermore the constructed recursion operators can be written as a factor of 2 distinct invertible matrix operators in each of the lattice equations it is also proved explicitly that the factorized operators are hamiltonian and hence rt bc and bm lattice equations are bi hamiltonian systems

amplification and damping of deterministic noise in coupled cellular arrays abstract large scale spatiotemporal organization and enhanced periodicity are seen when a population of chaotic rossler attractors is coupled through one or more of the variables the resulting periodicity has been analyzed by means of return maps and by the projection of these maps as state variable dispersion histograms self mapping of cells in the array within dispersion criteria yielded evidence of reverse bifurcation sequences f 8 4 2 1 expressed in the behavior of most of the 900 to 25 600 individual cells self mapping is seen in spiral and periodic banded fields but not in turbulent fields or in synchronous chaos coupling of arrays of periodic attractors leads to dispersion in self mapping similar to that seen in the coupled chaotic arrays fixed patterns representing the bifurcation sequences that occur in these arrays were overlaid on the spiral field portraits

similarity reduction generalized symmetries and integrability of belov chaltikian and blaszak marciniak lattice equations the lie point symmetries of belov chaltikian bc and blaszak marciniak bm lattice equations is derived using the symmetries similarity reduction for both bc and bm lattice equations is obtained and show that each of the reduced equation possesses lax representation and satisfies the singularity confinement criteria a discrete version of the painleve property indicating their integrability two particular solutions of both the reduced equations are given explicitly a systematic investigation for nonclassical symmetries of bc and bm lattice equations is carried out further a sequence of conserved densities and generalized symmetries of both bc and bm lattice equation are derived explicitly the existence of a sequence of such symmetries is a predictor for integrability

synchronization threshold in coupled logistic map lattices abstract we consider regular lattices of coupled chaotic maps whose local dynamics is ruled by the logistic map x 4 x 1 x a usual paradigm of chaotic systems through finite time exponents we scrutinize the lattice dynamics in the vicinity of the threshold of complete synchronization we connect dynamical features such as relaxation to the coherent state and intermittency with the statistics of finite time exponents focusing on the implications of the particular statistics related to the logistic map although numerical examples are given for lattice couplings decaying with distance as a power law our results are expected to be valid for a wider class of schemes coupling logistic maps

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability severe obstruction to shadowing of computer generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability that is when the dimension of the unstable eigenspace changes along a trajectory in the invariant set no true trajectory of reasonable length can be found to exist near any numerically generated trajectory an important quantity characterizing the shadowability of numerical trajectories is the shadowing time which measures for how long a trajectory remains valid this time depends sensitively on initial condition here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors an algebraic scaling for short times and an exponential scaling for long times the exponential behavior depends on system details but the small time algebraic behavior appears to be universal we describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

remarks on a brezis nirenberg s result in this note we revisit the equation with sobolev critical exponent which is firstly studied by brezis nirenberg in 4 p u u u p 2u u 2 2u in u h01 where rn is a bounded smooth domain n 3 2 0 r we show that for any m n there exists m 0 such that p has m solutions for any r 0 and every 0 m when n 4 0 we show that there is 0 0 such that problem p has a nontrivial solution for every 0 moreover we get multiple sign changing solutions for problem p

hamiltonian structures and conservation laws of the belov chaltikian lattice hierarchy a hierarchy of nonlinear differential difference equations associated with a 3 3 matrix spectral problem is proposed through discrete zero curvature equations one of the nontrivial equations in this hierarchy is the famous belov chaltikian lattice equation it is shown that the hierarchy possesses infinitely many conservation laws hamiltonian structures for this hierarchy are constructed with the aid of the trace identity

a new approach to controlling chaotic systems abstract given a piecewise monotonic transformation which is strongly transitive and preserves an invariant probability density function f a construction is described which modifies very slightly but which changes f significantly in order to possess desired properties this construction permits control of the global long term behavior of chaotic dynamical systems

non autonomous perturbations for a class of quasilinear elliptic equations on r abstract this paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on r involving the p laplacian with a non autonomous perturbation the first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain pass argument the special features of the problem here is the complex structure of the linear part which in particular oblige to work into the space w 1 p r then one faces problems in the convergence of the sequences of derivatives u n u

application of stochastic control techniques to chaotic nonlinear systems a control algorithm based on stochastic control techniques is devised for chaotic nonlinear systems the algorithm uses a state estimator based on the kalman filter and yields performance improvements in at least some regions of state space with respect to that obtainable by use of a controller utilizing only the conditional mean of the system state vector the method is applied to two typical chaotic nonlinear systems the henon heiles system and the lorenz system and their behavior with control is explored numerically

route to high dimensional chaos we present a route to high dimensional chaos that is chaos with more than one positive lyapunov exponent in this route as a system parameter changes a subsystem becomes chaotic through say a cascade of period doubling bifurcations after which the complementary subsystem becomes chaotic leading to an additional positive lyapunov exponent for the whole system a characteristic feature of this route as suggested by numerical evidence is that the second largest lyapunov exponent passes through zero continuously three examples are presented a discrete time map a continuous time flow and a population model for species dispersal in evolutionary ecology

multiple perturbations of a singular eigenvalue problem abstract we study the perturbation by a critical term and a p 1 superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential by means of variational arguments and a version of the concentration compactness principle in the singular case we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem

strange attractors and chaos control in periodically forced complex duffing s oscillators abstract an interesting and challenging research subject in the field of nonlinear dynamics is the study of chaotic behavior in systems of more than two degrees of freedom in this work we study fixed points strange attractors chaotic behavior and the problem of chaos control for complex duffing s oscillators which represent periodically forced systems of two degrees of freedom we produce plots of poincare map and study the fixed points and strange attractors of our oscillators the presence of chaotic behavior in these models is verified by the existence of positive maximal lyapunov exponent we also calculate the power spectrum and consider its implications regarding the properties of the dynamics the problem of controlling chaos for these oscillators is studied using a method introduced by pyragas phys lett a 170 1992 421 which is based on the construction of a special form of a time continuous perturbation the study of coupled periodically forced oscillators is of interest to several fields of physics mechanics and engineering the connection of our oscillators to the nonlinear schrodinger equation is discussed

multiple positive solutions for a semilinear equation with prescribed singularity variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points some exact local behavior for positive solutions are also given

critical singular problems on infinite cones abstract we prove existence results for non autonomous perturbations of critical singular elliptic boundary value problems the non singular case was treated by tarantello ann inst h poincare analyse non lineaire 9 1992 281 for bounded domains here the singular weight allows for unbounded domains as cones and give rise to a different non compactness picture as was first remarked by caldiroli and musina calc variations pde 8 1999 365

bifurcation control of a chaotic system the viability of controlling chaos by controlling associated bifurcations is demonstrated in the context of a thermal convection loop model washout filter aided feedback controls are employed to delay and to extinguish chaos in the model and to target preferred equilibrium points an important feature of the control laws is that they do not result in any change in the set of equilibria even in the presence of model uncertainty

a family of completely integrable multi hamiltonian systems explicitly related to some celebrated equations by introducing a spectral problem with an arbitrary parameter we derive a kaup newell type hierarchy of nonlinear evolution equations which is explicitly related to many important equations such as the kundu equation the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the burgers equation the modified korteweg devries mkdv equation and the sharma tasso olver equation it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure under the bargann constraint between the potentials and the eigenfunctions the spectral problem is nonlinearized as a finite dimensional completely integrable hamiltonian system the involutive representation of the solutions for the kaup newell type hierarchy is also presented in addition an n fold darboux transformation of the kundu equation is constructed with the help of its lax pairs and a reduction technique according to the darboux transformation the solutions of the ku