on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

on probabilistic φ contractions on menger spaces abstract recently ciric lj b ciric solving the banach fixed point principle for nonlinear contractions in probabilistic metric spaces nonlinear anal 72 2010 2009 2018 obtained a fixed point theorem with the intention to get a probabilistic version of the boyd wong theorem d w boyd j s w wong on nonlinear contractions proc amer math soc 20 1969 458 464 we give a counterexample to the key lemma of ciric and we establish a corrected version of his main theorem we also discuss the relations between our result and fixed point theorems of browder f e browder on the convergence of successive approximations for nonlinear functional equations nederl akad wetensch proc ser a 71 indag math 30 1968 27 35 and matkowski j matkowski integrable solutions of functional equations dissertationes math rozprawy mat 127 1975 68 pp

tripled coincidence theorems for contractive type mappings in partially ordered metric spaces abstract in this paper we establish tripled coincidence point theorems for a pair of mappings f x x x x and g x x satisfying a nonlinear contractive condition ordered metric spaces presented theorems extend reveral existing results in the literature v lakshmikantham l ciric coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces nonlinear anal 70 2009 4341 4349 and harjani lopez and sadarangani in article j harjani b lopez k sadarangani fixed point theorems for mixed monotone operators and applications to integral equations nonlinear anal 74 2011 1749 1760

the measure of chaos by the numerical analysis of the fundamental frequencies application to the standard mapping abstract the method of analysis of the chaotic behaviour of a dynamical system by the numerical analysis of the fundamental frequencies developed for the study of the stability of the solar system j laskar icarus 88 1990 is presented here with application to the standard mapping this method is well suited for weakly chaotic motion with any number of degrees of freedom and is based on the analysis of the variations with time of the fundamental frequencies of an hamiltonian system it allows to give an analytical representation of the solution when it is regular to detect if an orbit is chaotic over a smaller time span than with the lyapunov exponents and gives also an estimate of the size of the chaotic zones in the frequency domain the frequency analysis also provides a numerical criterion for the destruction of invariant curves its application to the standard mapping shows that the golden curve does not survive for a 0 9718 which is very close and compatible with greene s value ac 0 971635

periodic spatio temporal oscillations governed by a globally controlled ginzburg landau equation abstract a new type of periodic oscillations in a globally controlled subcritical cubic complex ginzburg landau equation formerly observed in numerical simulations is explained and investigated analytically by means of a multiscale perturbation theory using an appropriate class of solutions of the nonlinear schrodinger equation as a starting point we construct a new class of asymptotic solutions of the cubic complex ginzburg landau equation in the limit of large dispersion and nonlinear frequency shift

incidence of nonextensive thermodynamics in temporal scaling at feigenbaum points abstract recently in phys rev lett 95 2005 140601 grassberger addresses the interesting issue of the applicability of q statistics to the renowned feigenbaum attractor he concludes there is no genuine connection between the dynamics at the critical attractor and the generalized statistics and argues against its usefulness and correctness yet several points are not in line with our current knowledge nor are his interpretations we refer here only to the dynamics on the attractor to point out that a correct reading of recent developments invalidates his basic claim

new indicators of chaos an algorithm which allows to recognize whether the evolution of the system is periodic chaotic or hyperchaotic is proposed minimum length orbit is introduced its behavior depends on the kind of solution the calculations are presented for the henon map for the damped driven pendulum for the rossler system and for the hyperchaotic qi system

escape rates in hamiltonian systems particles escape from the vicinity of invariant tori is macroscopically modeled by a nonuniform diffusive process the space dependence of transport coefficients is fixed by using perturbation theory scalings this leads to universal predictions for the escape rates that are then observed in numerical simulations

alignment indices a new simple method for determining the ordered or chaotic nature of orbits we introduce a new simple and efficient method for determining the ordered or chaotic nature of orbits in two dimensional 2d four dimensional 4d and six dimensional 6d symplectic maps the computation of the alignment indices for a given orbit we follow the evolution in time of two different initial deviation vectors computing the norms of the difference d parallel alignment index and the addition d antiparallel alignment index of the two vectors the time evolution of the smaller alignment index reflects the chaotic or ordered nature of the orbit in 2d maps the smaller alignment index tends to zero for both ordered and chaotic orbits but with completely different time rates which allows us to distinguish between the two cases in 4d and 6d maps the smaller alignment index tends to zero in the case of chaotic orbits while it tends to a positive non zero value in the case of ordered orbits the efficiency of the new method is also shown in a case of weak chaos and a comparison with other known methods that separate chaotic from regular orbits is presented

geometrical properties of local dynamics in hamiltonian systems the generalized alignment index gali method we investigate the detailed dynamics of multidimensional hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits the behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion the number of deviation vectors their linear in dependence and the spectrum of lyapunov exponents the different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on n dimensional tori more specifically we introduce the generalized alignment index of order k galik as the volume of a generalized parallelepiped whose edges are k initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step we show analytically and verify numerically on particular examples of n degree of freedom hamiltonian systems that for chaotic orbits galik tends exponentially to zero with exponents that involve the values of several lyapunov exponents in the case of regular orbits galik fluctuates around non zero values for 2 k n and goes to zero for n k 2n following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus t 2 k n m if 0 m k n and t k n if m k n the galik is a generalization of the smaller alignment index sali gali2 sali however galik provides significantly more detailed information on the local dynamics allows for a faster and clearer distinction between order and chaos than sali and works even in cases where the sali method is inconclusive

stability of oscillating hexagons in rotating convection abstract breaking the chiral symmetry rotation induces a secondary hopf bifurcation in weakly nonlinear hexagon patterns which gives rise to oscillating hexagons we study the stability of the oscillating hexagons using three coupled ginzburg landau equations close to the bifurcation point we derive reduced equations for the amplitude of the oscillation coupled to the phase of the underlying hexagons within these equations we identify two types of long wave instabilities and study the ensuing dynamics using numerical simulations of the three coupled ginzburg landau equations

stabilization of a scroll ring by a cylindrical neumann boundary we study the interaction of phase singularities with homogeneous neumann boundaries in one two and three spatial dimensions for the complex ginzburg landau equation the existence of a boundary induced drift attractor well known for spiral waves in two spatial dimensions is demonstrated for scroll waves in three spatial dimensions we find that a cylindrical neumann boundary can lock a scroll ring thus preventing the collapse of its closed filament

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

multidimensional fixed point theorems in partially ordered complete metric spaces abstract in this paper we propose a notion of coincidence point between mappings in any number of variables and we prove some existence and uniqueness fixed point theorems for nonlinear mappings verifying different kinds of contractive conditions and defined on partially ordered metric spaces these theorems extend and clarify very recent results that can be found in t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 7 2006 1379 1393 v berinde m borcut tripled fixed point theorems for contractive type mappings in partially ordered metric spaces nonlinear anal 74 2011 4889 4897 and m berzig b samet an extension of coupled fixed point s concept in higher dimension and applications comput math appl 63 8 2012 1319 1334

weak chaos and poincare recurrences for area preserving maps the spectrum f t of poincare recurrence times for the standard map exhibits two distinct limits an integrable weak coupling limit with an inverse power law and a chaotic strong coupling limit with exponential decay in the domain where chaotic regions coexist with integrable structures the spectrum f t exhibits a superposition of exponential and power law decay such a law can be proved to occur in a model of area preserving map at the boundary of the mixing and integrable components

random versus deterministic exponents in a rich family of diffeomorphisms we study both numerically and theoretically the relationship between the random lyapunov exponent of a family of area preserving diffeomorphisms of the 2 sphere and the mean of the lyapunov exponents of the individual members the motivation for this study is the hope that a rich enough family of diffeomorphisms will always have members with positive lyapunov exponents that is to say positive entropy at question is what sort of notion of richness would make such a conclusion valid one type of richness of a family invariance under the left action of so n 1 occurs naturally in the context of volume preserving diffeomorphisms of the n sphere based on some positive results for families linear maps obtained by dedieu and shub we investigate the exponents of such a family on the 2 sphere again motivated by the linear case we investigate whether there is in fact a lower bound for the mean of the lyapunov exponents in terms of the random exponents with respect to the push forward of haar measure on so 3 in such a family the family that we study contains a twist map with stretching parameter e in the family we find strong numerical evidence for the existence of such a lower bound on mean lyapunov exponents when the values of the stretching parameter e are not too small even moderate values of e like e 10 are enough to have an average of the metric entropy larger than that of the random map for small e the estimated average entropy seems positive but is definitely much less than the one of the random map the numerical evidence is in favor of the existence of exponentially small lower and upper bounds in the present example with an analytic family finally the effect of a small randomization of fixed size of the individual elements of the family e is considered now the mean of the local random exponents of the family is indeed asymptotic to the random exponent of the entire family as e tends to infinity

dynamics of 4d symplectic maps near a double resonance abstract we study the dynamics of a family of 4 d symplectic mappings near a doubly resonant elliptic fixed point we derive and discuss algebraic properties of the resonances required for the analysis of a takens type normal form in particular we propose a classification of the double resonances adapted to this problem including cases of both strong and weak resonances around a weak double resonance a junction of two resonances of two different orders both being larger than 4 the dynamics can be described in terms of a simple in general non integrable hamiltonian model the non integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder we use a 4 d generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4 d symplectic map we evaluate numerically the volume of a 4 d parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively in good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2 d analog different numerical studies point out that double resonances play a key role to understand arnold diffusion this paper has to be seen also as a first step in this direction

phase space structure of multi dimensional systems by means of the mean exponential growth factor of nearby orbits in this paper we deal with an alternative technique to study global dynamics in hamiltonian systems the mean exponential growth factor of nearby orbits megno that proves to be efficient to investigate both regular and stochastic components of phase space it provides a clear picture of resonance structures location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the lyapunov characteristic number here the megno is applied to a rather simple model the 3d perturbed quartic oscillator in order to visualize the structure of its phase space and obtain a quite clear picture of its resonance structure examples of application to multi dimensional canonical maps are also included

temporal scaling at feigenbaum points and nonextensive thermodynamics we show that recent claims for the nonstationary behavior of the logistic map at the feigenbaum point based on nonextensive thermodynamics are either incorrect or can be easily deduced from well known properties of the feigenbaum attractor in particular there is no generalized pesin identity for this system the existing attempts at proofs being based on misconceptions about basic notions of ergodic theory in deriving several new scaling laws of the feigenbaum attractor thorough use is made of its detailed structure but there is no obvious connection to nonextensive thermodynamics

qualitative analysis of the rossler equations bifurcations of limit cycles and chaotic attractors in this paper we study different aspects of the paradigmatic rossler model we perform a detailed study of the local and global bifurcations of codimension one and two of limit cycles this provides us a global idea of the three parametric evolution of the system we also study the regions of parameters where we may expect a chaotic behavior by the use of different chaos indicators the combination of the different techniques gives an idea of the different routes to chaos and the different kinds of chaotic attractors we may found in this system

hole solutions in the 1d complex ginzburg landau equation abstract the cubic complex ginzburg landau equation cgle has a one parameter family of traveling localized source solutions these so called nozaki bekki holes are dynamically stable in some parameter range but always structurally unstable a perturbation of the equation in general leads to a positive or negative monotonic acceleration or an oscillation of the holes this confirms that the cubic cgle has an inner symmetry as a consequence small perturbations change some of the qualitative dynamics of the cubic cgle and enhance or suppress spatio temporal intermittency in some parameter range an analytic stability analysis of holes in the cubic cgle and a semianalytical treatment of the acceleration instability in the perturbed equation is performed by using matching and perturbation methods furthermore we treat the asymptotic hole shock interaction the results which can be obtained fully analytically in the nonlinear schro dinger limit are also used for the quantitative description of modulated solutions made up of periodic arrangements of traveling holes and shocks

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

correlation between the hurst exponent and the maximal lyapunov exponent examining some low dimensional conservative maps the chirikov standard map and the 2d froeschle map are investigated a few thousand values of the hurst exponent he and the maximal lyapunov exponent mle are plotted in a mixed space of the nonlinear parameter versus the initial condition both characteristic exponents reveal remarkably similar structures in this space a tight correlation between the hes and mles is found with the spearman rank 0 83 and 0 75 for the chirikov and 2d froeschle maps respectively based on this relation a machine learning ml procedure using the nearest neighbor algorithm is performed to reproduce the he distribution based on the mle distribution alone a few thousand he and mle values from the mixed spaces were used for training and then using 2 2 4 105 mles the hes were retrieved the ml procedure allowed to reproduce the structure of the mixed spaces in great detail

on a connection between kam and nekhoroshev s theorems abstract using the iteration of nekhoroshev s theorem as a basic tool we point out the existence of a hierarchic structure of nested domains underlying the phenomenon of diffusion at each level we find that the diffusions speed is exponentially small with respect to the previous level the set of kam tori is the domain characterized by a zero diffusion speed

breaking the hidden symmetry in the ginzburg landau equation in this paper we study localised traveling solutions to a ginzburg landau equation to which we have added a small o e 0 e 1 quintic term we consider this term as a model for the higher order nonlinearities which appear in the derivation of the ginzburg landau equation by a combination of a geometrical approach and an explicit perturbation analysis we are able to relate the family of bekki nozaki solutions of the cubic equation n bekki and b nozaki 1985 phys lett a 110 133 135 to a curve of co dimension 2 homoclinic bifurcations in parameter space thus we are able to interpret the hidden symmetry which has been conjectured to explain the existence of the bekki nozaki solutions from the point of view of bifurcation theory we show that the quintic term breaks this hidden symmetry and that the one parameter family of bekki nozaki solutions is embedded in a two parameter family of homoclinic solutions which exist at a co dimension 1 homoclinic bifurcation furthermore we show mainly by geometrical arguments that the addition of the small quintic term can create large families of traveling localised structures that cannot exist in the cubic case these solutions exist in open subsets of the parameter space and correspond to structurally stable multi or n circuit heteroclinic orbits in an ode reduction and have a monotonically decreasing increasing amplitude except for a number n of relatively fast jumps

on the role of high order resonances in normal forms and in separatrix splitting we discuss the role of high order resonances in the construction of normal forms for e close to integrable hamiltonian systems by heuristic considerations based on standard estimates we show that the remainder of normal forms is dominated by the terms corresponding to the main high order resonances and we provide a general argument to show that the size of such leading terms is exponentially small in 1e we apply this method to the problem of estimating the splitting of separatrices in resonant perturbed systems

a three parametric study of the lorenz model in this paper we analyse the three parametric phase diagram of the classical lorenz model this has been done by the combined use of different techniques the most important one for our objective is the combined use of the recently developed ofli2 chaos indicator and the classical maximum lyapunov exponent which have permitted a fast and complete location of the chaotic parameter region as a result we provide a three parametric model of the chaotic regime for the lorenz system that gives new light on this classical problem

bifurcation structures and dominant modes near relative equilibria in the one dimensional discrete nonlinear schrodinger equation abstract we investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear schrodinger equation dnlse when the period of these orbits and a suitable defect act as bifurcation parameters we find a reduced hamiltonian that gives substantial insight into the dynamics of this system the convexity of this hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions we show that the local loss of convexity in the reduced hamiltonian is determined by the hessian of its integrable part in the family of relative equilibria under study stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions whose origin is suggested by the averaging principle we also show that the return times form an optimal embedding to characterize the system dynamics we show that the power spectral density of a suitable interference signal arising from a ring of bose einstein condensates and described by the dnlse has a single prominent peak at the breather like relative equilibria

a numerical study of arnold diffusion in a priori unstable systems this paper concerns the problem of the numerical detection of arnold diffusion in a priori unstable systems specifically we introduce a new definition of arnold diffusion which is adapted to the numerical investigation of the problem and is based on the numerical computation of the stable and unstable manifolds of the system examples of this arnold diffusion are provided in a model system in this model we also find that arnold diffusion behaves as an approximate markovian process thus it becomes possible to compute diffusion coefficients the values of the diffusion coefficients satisfy the scaling d epsilon simeq epsilon 2 we also find that this law is correlated to the validity of the melnikov approximation in fact the d epsilon simeq epsilon 2 law is valid up to the same critical value of e for which the error terms of melnikov approximations have a sharp increment

common coupled fixed point theorems in cone metric spaces for w compatible mappings in this paper we introduce the concept of a w compatible mappings to obtain coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in cone metric space with a cone having non empty interior coupled common fixed point theorems for such mappings are also proved our results generalize extend and unify several well known comparable results in the literature results are supported by three examples

the cubic complex ginzburg landau equation for a backward bifurcation abstract the one dimensional complex ginzburg landau equation cgle with a destabilizing cubic nonlinearity and no saturating higher order terms has stable bounded solutions we consider a simple pedagogical model exhibiting qualitatively the mechanism which may suppress the divergence of the solutions then we investigate the functional form of the blow up collapse solutions immediately before the divergence from this analysis we find analytic boundaries for the existence of collapse solutions in the parameter space of the cgle a comparison with numerical simulations demonstrates that for parameters without collapse solutions the solutions of the cgle remain bounded for all times finally we discuss the implications of our results for the solutions of the cgle when saturating higher order terms are included

application of the gali method to localization dynamics in nonlinear systems we investigate localization phenomena and stability properties of quasiperiodic oscillations in n degree of freedom hamiltonian systems and n coupled symplectic maps in particular we study an example of a parametrically driven hamiltonian lattice with only quartic coupling terms and a system of n coupled standard maps we explore their dynamics using the generalized alignment index gali which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date

quadruple fixed point theorems for nonlinear contractions the notion of coupled fixed point is introduced by gnana bhaskar and lakshmikantham very recently the concept of tripled fixed point was introduced by berinde and borcut in this manuscript a quadruple fixed point is considered and some new related fixed point theorems are obtained we also give some examples to illustrate our results

multidimensional coincidence point results for compatible mappings in partially ordered fuzzy metric spaces in recent times coupled tripled and quadruple fixed point theorems have been intensively studied by many authors in the context of partially ordered complete metric spaces using different contractivity conditions roldan et al showed a unified version of these results for nonlinear mappings in any number of variables which were not necessarily permuted or ordered introducing the notion of multidimensional coincidence point very recently choudhury et al proved coupled coincidence point results in the context of fuzzy metric spaces in the sense of george and veeramani in this paper using the idea of coincidence point for nonlinear mappings in any number of variables we study a fuzzy contractivity condition to ensure the existence of coincidence points in the framework of fuzzy metric spaces provided with hadzic type t norms then we present an illustrative example in which our methodology leads to the existence of coincidence points but previous theorems cannot be applied

chirikov and nekhoroshev diffusion estimates bridging the two sides of the river abstract we present theoretical and numerical results pointing towards a strong connection between the estimates for the diffusion rate along simple resonances in multidimensional nonlinear hamiltonian systems that can be obtained using the heuristic theory of chirikov and a more formal one due to nekhoroshev we show that despite a wide spread impression the two theories are complementary rather than antagonist indeed although chirikov s 1979 review has thousands of citations almost all of them refer to topics such as the resonance overlap criterion fast diffusion the standard or whisker map and not to the constructive theory providing a formula to measure diffusion along a single resonance however as will be demonstrated explicitly below chirikov s formula provides values of the diffusion coefficient which are quite well comparable to the numerically computed ones provided that it is implemented on the so called optimal normal form derived as in the analytic part of nekhoroshev s theorem on the other hand chirikov s formula yields unrealistic values of the diffusion coefficient in particular for very small values of the perturbation when used in the original hamiltonian instead of the optimal normal form in the present paper we take advantage of this complementarity in order to obtain accurate theoretical predictions for the local value of the diffusion coefficient along a resonance in a specific 3dof nearly integrable hamiltonian system besides we compute numerically the diffusion coefficient and a full comparison of all estimates is made for ten values of the perturbation parameter showing a very satisfactory agreement

fixed point theorems in partially ordered metric spaces and applications abstract we prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order using a weak contractivity type of assumption besides including several recent developments our theorem can be used to investigate a large class of problems as an application we discuss the existence and uniqueness of solution for a periodic boundary value problem

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

the fermi pasta ulam problem and its underlying integrable dynamics an approach through lyapunov exponents fpu models in dimension one are perturbations either of the linear model or of the toda model perturbations of the linear model include the usual beta model perturbations of toda include the usual alpha beta model in this paper we explore and compare two families or hierarchies of fpu models closer and closer to either the linear or the toda model by computing numerically for each model the maximal lyapunov exponent chi more precisely we consider statistically typical trajectories and study the asymptotics of chi for large n the number of particles and small varepsilon the specific energy e n and find for all models asymptotic power laws chi simeq c varepsilon a c and a depending on the model the asymptotics turns out to be in general rather slow and producing accurate results requires a great computational effort we also revisit and extend the analytic computation of chi introduced by casetti livi and pettini originally formulated for the beta model with great evidence the theory extends successfully to all models of the linear hierarchy but not to models close to toda

the production of tsallis entropy in the limit of weak chaos and a new indicator of chaoticity abstract we study the connection between the appearance of a metastable behaviour of weakly chaotic orbits characterized by a constant rate of increase of the tsallis q entropy c tsallis possible generalization of boltzmann gibbs statistics journal of statistical physics 52 1988 479 487 and the solutions of the variational equations of motion for the same orbits we demonstrate that the variational equations yield transient solutions lasting for long time intervals during which the length of deviation vectors of nearby orbits grows in time almost as a power law the associated power exponent can be simply related to the entropic exponent for which the q entropy exhibits a constant rate of increase this analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits that we call average power law exponent aple we compare the aple with other established indicators of the literature in particular we give examples of application of the aple in a a thin separatrix layer of the standard map b the stickiness region around an island of stability in the same map and c the web of resonances of a 4d symplectic map in all these cases we identify weakly chaotic orbits exhibiting the metastable behaviour associated with the tsallis q entropy

the hausdorff fuzzy metric on compact sets we propose a method for constructing a hausdorff fuzzy metric on the set of the nonempty compact subsets of a given fuzzy metric space in the sense of george and veeramani we discuss several important properties as completeness completion and precompactness some illustrative examples are given

invariant spectra of orbits in dynamical systems we show that in deterministic dynamical systems any orbit is associated with an invariant spectrum of stretching numbers i e numbers expressing the logarithmic divergences of neighbouring orbits within one period the first moment of this invariant spectrum is the maximal lyapunov characteristic number lcn in the case of a chaotic domain a single invariant spectrum characterizes the whole domain the invariance of this spectrum allows the estimation of the lcn by calculating for short times many orbits with initial conditions in the same chaotic region instead of calculating one orbit for extremely long times however if part of the initial conditions are in an ordered region the average of the short time calculations may deviate considerably from the lcn invariant spectra appear not only for conservative but also for dissipative systems a few examples are given

some more universal scaling laws for critical mappings we study such nonlinear mappingsxn 1 f xn bcr of an intervali into itself for which the feigenbaum scaling laws hold i e for which bcr is an accumulation point of bifurcation points letx0 be a random variable with some absolutely continuous distribution ini we show in particular that i the geometric average distance ofxn from the nearest point of the attractor decreases liken 1 93387 ii the geometric average of xn x0 increases liken0 60 iii the geometric mean distance xn yn between the iterates of two close by pointsx0 y0 asymptotically tends towards a value x0 y0 0 77 these and other properties are also borne out from a simple probabilistic model which depicts the evolution as a random walklike process

tripled fixed point theorems for contractive type mappings in partially ordered metric spaces abstract in this paper we introduce the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtain existence and existence and uniqueness theorems for contractive type mappings our results generalize and extend recent coupled fixed point theorems established by gnana bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 7 2006 1379 1393 examples to support our new results are given

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

superexponential stability of kam tori we study the dynamics in the neighborhood of an invariant torus of a nearly integrable system we provide an upper bound to the diffusion speed which turns out to be of superexponentially small size exp exp 1 being the distance from the invariant torus we also discuss the connection of this result with the existence of many invariant tori close to the considered one

primary and secondary noninear global instability we study oscillating solutions of the complex ginzburg landau equation in a semi infinite domain asymptotic to a saturated traveling wave at c1 and subject to a homogeneous upstream boundary condition at x d 0 this inlet condition breaks the galilean invariance giving the advection velocity the role of a control parameter we give a criterion for the existence of a nontrivial solution called a nonlinear global mode or a self sustained resonance and we obtain the selected frequency the threshold and the frequency are shown to be determined by the linear absolute instability transition we undertake a singular perturbation analysis which first proves the validity of the criterion and secondly yields scaling laws for the frequency the growth length and the slope at the origin of the nonlinear global modes as functions of the criticality parameter comparisons with direct numerical simulation of the ginzburg landau model validate these predictions furthermore the results are in excellent agreement with numerical simulation of the taylor couette problem with throughflow by buchel et al p buchel m lucke d roth r schmitz pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem taylor vortices in axial flow phys rev e 53 1996 4764 4777 and of rayleigh benard flow with an added poiseuille flow by muller et al h w muller m lucke m kamps convective patterns in horizontal flow europhys lett 10 1989 451 456 h w muller m lucke m kamps transversal convection patterns in horizontal shear flow phys rev a 45 1992 3714 3726 the numerical simulations indicate that the nonlinear global modes are stable if the saturated traveling wave to which the nonlinear global mode is asymptotic i e the asymptotic downstream part of the global modes is not absolutely unstable to perturbations a complete analysis of the dispersion relation of the secondary instability of saturated traveling waves is given 1999 elsevier science b v all rights reserved

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

frequency analysis for multi dimensional systems global dynamics and diffusion abstract frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system it was first devised in order to study the stability of the solar system j laskar icarus 88 1990 266 291 and then applied to the 2d standard mapping laskar et al physica d 56 1992 253 269 it is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps for regular motions it yields an analytical representation of the solutions the analysis of the regularity of the frequency map with respect to the action space and of its variations with respect to time gives rise to two criteria for the regularity of the motion which are valid for multi dimensional systems for a 4d symplectic map plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and reveals that high order resonances are of great importance in understanding the diffusion of non regular orbits through the invariant tori in particular it appears in several examples that diffusion along the resonance lines arnold diffusion is of less importance than diffusion across the resonances lines which can lead to large diffusion due to the phenomenon of overlap of higher order resonances chaotic layers many fine features of the dynamics are also revealed by frequency analysis which would require more theoretical study for a better understanding

a numerical study of the topology of normally hyperbolic invariant manifolds supporting arnold diffusion in quasi integrable systems abstract we investigate numerically the stable and unstable manifolds of the hyperbolic manifolds of the phase space related to the resonances of quasi integrable systems in the regime of validity of the nekhoroshev and kam theorems using a model of weakly interacting resonances we explain the qualitative features of these manifolds characterized by peculiar flower like structures we detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies we find numerically a correlation among these transitions and the speed of arnold diffusion

common fixed point theorems in fuzzy metric spaces using the clrg property the main aim of this paper is to obtain some new common fixed point theorems for weakly compatible mappings in fuzzy metric spaces in the sense of kramosil and michalek satisfying new contractive conditions employing the common limit in the range property our results generalize and extend some recent results contained in jain et al 2012 12 to generalized contractive conditions under some control functions in the process some results about multidimensional common fixed points as coupled tripled quadrupled common fixed point results are derived from our main results

detecting order and chaos in hamiltonian systems by the sali method we use the smaller alignment index sali to distinguish rapidly and with certainty between ordered and chaotic motion in hamiltonian flows this distinction is based on the different behaviour of the sali for the two cases the index fluctuates around non zero values for ordered orbits while it tends rapidly to zero for chaotic orbits we present a detailed study of sali s behaviour for chaotic orbits and show that in this case the sali exponentially converges to zero following a time rate depending on the difference of the two largest lyapunov exponents 1 2 i e sali e

peeling the onion of order and chaos in a high dimensional hamiltonian system abstract coexistence of various ordered chaotic states in a hamiltonian system is studied with the use of a symplectic coupled map lattice besides the clustered states for the attractive interaction a novel chaotic ordered state is found for a system with repulsive interaction characterized by a dispersed state of particles the dispersed and clustered states form an onion like structure in phase space the degree of order increases towards the center of the onion while chaos is enhanced at the edge between ordered and random chaotic states for a longer time scale orbits itinerate over ordered and random states the existence of these ordered states leads to anomalous long time correlation for many quantifiers such as the global diffusion

measuring the mixing efficiency in a simple model of stirring some analytical results and a quantitative study via frequency map analysis we prove the existence of invariant curves for a t periodic hamiltonian system which models a fluid stirring in a cylindrical tank when t is small and the assigned stirring protocol is piecewise constant furthermore using the numerical analysis of the fundamental frequency of laskar we investigate numerically the break down of invariant curves as t increases and we give a quantitative estimate of the efficiency of the mixing

detection of arnold diffusion in hamiltonian systems we detect diffusion along resonances in a quasi integrable system at small values of the perturbing parameter the diffusion coefficient goes to zero as the perturbation parameter goes to zero faster than a power law typical of chirikov diffusion and is compatible with an exponential law expected in the nekhoroshev theorem

standing waves of the complex ginzburg landau equation abstract we prove the existence of nontrivial standing wave solutions of the complex ginzburg landau equation t e i e i k on a bounded domain with dirichlet boundary conditions our result requires that 0 be sufficiently small

spatiotemporal chaos in the one dimensional complex ginzburg landau equation the dynamical behavior of a large one dimensional system obeying the cubic complex ginzburg landau equation is studied numerically as a function of parameters near a supercritical bifurcation two types of chaotic behavior can be distinguished beyond the benjamin feir instability a phase turbulence regime with a conserved phase winding number and no phase dislocations space time defects and a defect regime with a nonzero density of defects the transition between the two can either be continuous or discontinuous hysteretic depending on parameters the spatial decay of the phase correlation function is inferred to be exponential in both regimes with a sharp decrease of the correlation length upon entering the defect phase the temporal decay of correlations is exponential in the defect regime

stability of simple periodic orbits and chaos in a fermi pasta ulam lattice we investigate the connection between local and global dynamics in the fermi pasta ulam fpu beta model from the point of view of stability of its simplest periodic orbits spo s in particular we show that there is a relatively high q mode q 2 n 1 3 of the linear lattice having one particle fixed every two oppositely moving ones called spo2 here which can be exactly continued to the nonlinear case for n 5 3m m 0 1 2 and whose first destabilization e 2u as the energy or beta increases for any fixed n practically coincides with the onset of a weak form of chaos preceding the breakdown of fpu recurrences as predicted recently in a similar study of the continuation of a very low q 3 mode of the corresponding linear chain this energy threshold per particle behaves like e 2u n alpha n 2 we also follow exactly the properties of another spo with q n 1 2 in which fixed and moving particles are interchanged called spo1 here and which destabilizes at higher energies than spo2 since e 1u n alpha n 1 we find that immediately after their first destabilization these spo s have different positive lyapunov spectra in their vicinity however as the energy increases further at fixed n these spectra converge to the same exponentially decreasing function thus providing strong evidence that the chaotic regions around spo1 and spo2 have merged and large scale chaos has spread throughout the lattice since these results hold for n arbitrarily large they suggest a direct approach by which one can use local stability analysis of spo s to estimate the energy threshold at which a transition to ergodicity occurs and thermodynamic properties such as kolmogorov sinai entropies per particle can be computed for similar one dimensional lattices

on the numerical detection of the effective stability of chaotic motions in quasi integrable systems we describe and compare two recent tools for detecting the geometry of resonances of a dynamical system in order to get information about the long term stability of chaotic solutions one of these tools is the so called fast lyapunov indicator fli celest mech dyn astr 67 1997 41 while the other is a recently introduced spectral fourier analysis of chaotic motions discrete contin dyn syst b 1 2001 1 for the first tool we provide new analytical estimates which explain why the fli is a sensitive means of discriminating between resonant and non resonant regular orbits thus providing a method to detect the geometry of resonances of a quasi integrable system the second tool based on a recent theoretical result can test directly whether a chaotic motion is in the nekhoroshev stability regime so that it practically cannot diffuse in the phase space or on the contrary if it is in the chirikov diffusive regime using these two methods we determine the value of the critical parameter at which the transition from the nekhoroshev to the chirikov regime occurs in a quasi integrable model hamiltonian system and standard four dimensional map

some new common fixed point theorems under strict contractive conditions the main purpose of this paper is to give some new common fixed point theorems under strict contractive conditions for mappings satisfying a new property

fixed points in fuzzy metric spaces abstract in this note the well known fixed point theorems of banach and edelstein are extended to fuzzy metric spaces in the settle of kramosil and michalek

examples of fuzzy metrics and applications in this paper we present new examples of fuzzy metrics in the sense of george and veeramani the examples have been classified attending to their construction and most of the well known fuzzy metrics are particular cases of those given here in particular novel fuzzy metrics by means of fuzzy and classical metrics and certain special types of functions are introduced we also give an extension theorem for two fuzzy metrics that agree in its nonempty intersection finally we give an application of this type of fuzzy metrics to color image processing we propose a fuzzy metric that simultaneously takes into account two different distance criteria between color image pixels and we use this fuzzy metric to filter noisy images obtaining promising results this application is also illustrative of how fuzzy metrics can be used in other engineering problems

stickiness and cantori we study the phenomenon of stickiness in the standard map the sticky regions are limited by cantori most important among them are the cantori with noble rotation numbers that are approached by periodic orbits corresponding to the successive truncations of the noble numbers the size of an island of stability depends on the last kam torus as the perturbation increases the size of the kam curves increases but the outer kam curves are gradually destroyed and in general the island decreases higher order noble tori inside the outermost kam torus are also destroyed and when the outermost kam torus becomes a cantorus the size of an island decreases abruptly then we study the crossing of the cantori by asymptotic curves of periodic orbits just inside the cantorus we give an exact numerical example of this crossing non schematic and we find how the asymptotic curves after staying for a long time near the cantorus finally extend to large distances outwards finally we find the relation between the forms of the sticky region and asymptotic curves

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

exact front soliton and hole solutions for a modified complex ginzburg landau equation from the harmonic balance method abstract a novel approach of using harmonic balance hb method is presented to find front soliton and hole solutions of a modified complex ginzburg landau equation three families of exact solutions are obtained one of which contains two parameters while the others one parameter the hb method is an efficient technique in finding limit cycles of dynamical systems in this paper the method is extended to obtain homoclinic heteroclinic orbits and then coherent structures it provides a systematic approach as various methods may be needed to obtain these families of solutions as limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation this approach can be extended further to find analytic solution of complex quintic ginzburg landau equation in terms of fourier series