strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

the weak metric approximation property we introduce and investigate the weak metric approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property among others we show that if a banach space has the approximation property and is 1 complemented in its bidual then it has the weak metric approximation property we also study the lifting of the weak metric approximation property from banach spaces to their dual spaces this enables us in particular to show that the subspace of c0 constructed by johnson and schechtman does not have the weak metric approximation property

the johnson schechtman space has the 6 bounded approximation property abstract in 2001 g godefroy proved that a subspace x js of c 0 constructed by w b johnson and g schechtman in 1996 has the bounded approximation property with 8 this paper slightly improves godefroy s proof establishing that 6

dichotomies and asymptotic solutions of nonlinear differential systems the problem of asymptotic equivalence between two systems of ordinary differential equations has been studied by many authors l 5 it was first used by levinson in his theorem on asymptotic integration 6 71 in l 51 this problem is studied when system 1 2 is stable a t is constant and g t y b t is integrable in 3 theorems 5 111 it is assumed that the linear system 1 2 has an ordinary dichotomy and the nonlinear term g t y is lipschitz continuous in 2 the previous results are extended to nonlinear perturbations g t y which are not lipschitz continuous such that g t y l i r t y where r t y is monotone in y we emphasize that in all these works only the asymptotic equivalence between bounded solutions is established nothing is said about a possible correspondence between unbounded solutions of 1 1 and 1 2 in this paper we suppose that the nonlinear term g t y satisfies some integrability conditions that we summarize in hypothesis h2 these conditions permit one to consider systems with large nonlinearities for example equations where g t y is an oscillatory function of the variable y no lipschitz conditions or monotonicity properties are required of g t y however we will use dichotomic properties of system 1 2

an examination of initial condition maps for the sinusoidally excited buckled beam modeled by the duffing s equation abstract the initial condition problem is studied for the dynamics of the buckled beam by using the forced holmes duffing s equation in an attempt to understand the route to chaos earlier results generated by moon 1 for different parameter values are reconsidered and new techniques are used in order to understand more of the global behavior of the system in particular fractal basin boundaries are determined for two types of basins of attraction these are correlated with previous results to help provide an explanation for observed chaotic system behavior

new criterion of the approximation property this paper is concerned with the approximation property which is an important property in banach space theory we show that a banach space x has the approximation property if and only if for every banach space y the set of finite rank operators from x to y is dense in the corresponding space of compact operators in the usual topology of uniform convergence on compact sets

asymptotic integration of ordinary different systems an extension and a new proof of hartman and wintner s theorem on asymptotic integration are given

universal transition in a dynamical system forced at two incommensurate frequencies abstract the various routes to chaos are explored for a nonlinear mechanical system mode locked to one of two incommensurate external frequencies two types of transitions are seen in our model system a saddle node transition with its associated intermittency can occur when the mode locking is lost while the attracting two torus remains smooth a less trivial transition can occur in which the attracting torus roughens and the power spectrum of the time series develops singular low frequency components after the breakdown mode locking persists with noisy small scale motions about the former torus at the multicritical point where these two transition lines meet scaling and universal low frequency power spectra are observed a renormalization group treatment is proposed the analysis might also be applicable to transitions in rotationally invariant systems

the approximation property in terms of the approximability of weak weak continuous operators abstract by a well known result of grothendieck a banach space x has the approximation property if and only if for every banach space y every weak weak continuous compact operator t x y can be uniformly approximated by finite rank operators from x y we prove the following metric version of this criterion x has the approximation property if and only if for every banach space y every weak weak continuous weakly compact operator t x y can be approximated in the strong operator topology by operators of norm t from x y as application easier alternative proofs are given for recent criteria of approximation property due to lima nygaard and oja

bubbling route to strange nonchaotic attractor in a nonlinear series lcr circuit with a nonsinusoidal force we identify an unconventional route to the creation of a strange nonchaotic attractor sna in a quasiperiodically forced electronic circuit with a nonsinusoidal square wave force as one of the quasiperiodic forces through numerical and experimental studies we find that bubbles appear in the strands of the quasiperiodic attractor due to the instability induced by the additional square wave type force the bubbles then enlarge and get increasingly wrinkled as a function of the control parameter finally the bubbles get extremely wrinkled while the remaining parts of the strands of the torus remain largely unaffected resulting in the creation of the sna we term this the bubbling route to the sna we characterize and confirm this creation from both experimental and numerical data using maximal lyapunov exponents and their variance poincar e maps fourier amplitude spectra and spectral distribution functions we also strongly confirm the creation of a sna via the bubbling route by the distribution of the finite time lyapunov exponents

cauchy matrix for linear almost periodic systems and some consequences abstract a novel method to construct the fundamental matrix for a linear almost periodic system is proposed provided that the diagonal terms satisfy an average separation condition and the off diagonal coefficients are l small the idea is to transform the system in a set of riccati type equations and use exponential dichotomy and its consequences it is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients finally our result enables us to obtain i explicit almost periodic matrices q t q 1 t and q t which diagonalize the original system and ii sufficient conditions for the stability two illustrative examples are shown

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

the birth of strange nonchaotic attractors abstract a mechanism is described for the development of strange nonchaotic attractors from two frequency quasiperiodic attractors in quasiperiodically driven maps this mechanism is intimately tied to the phenomenon of torus doubling the transition to strange nonchaotic behavior occurs when a period doubled torus collides with its unstable parent torus as the collision is approached the period doubled torus becomes extremely wrinkled ultimately becoming fractal at the collision the lyapunov exponent remains negative through the collision these collisions are shown to be a new type of attractor merging crisis the new feature is the possibility of nonchaotic attractors taking part in the crisis this mechanism is illustrated via numerical and analytical studies of a quasiperiodically driven logistic map

the convex approximation property of banach spaces abstract we study the convex approximation property of banach spaces to provide a unified approach to various approximation properties including besides the classical ones e g the positive approximation property of banach lattices and the approximation property for pairs of banach spaces our main results concern lifting of metric and weak metric approximation properties from banach spaces to their dual spaces as an easy application it follows that if x or x has the radon nikod m property then the approximation property of x defined by a convex subset of conjugate compact operators containing 0 in particular the positive approximation property of x is metric

the strong approximation property we introduce and investigate the strong approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the weak bounded approximation property among others we show that the weak bounded approximation property is equivalent to a quantitative strengthening of the strong approximation property some recent results on the approximation property of banach spaces and their dual spaces are improved

quasiperiodically forced dynamical systems with strange nonchaotic attractors abstract we discuss the existence and properties of strange nonchaotic attractors of quasiperiodically forced nonlinear dynamical systems we do this by examining a particular model differential equation cos e cos 2 t where is a two frequency quasiperiodic function of t when 0 the analysis of the equation is facilitated since then it can be related to the schrodinger equation with quasiperiodic potential we show that the equation does indeed exhibit strange nonchaotic attractors and we consider the important question of whether these attractors are typical in the sense that they exist on a set of positive lebesgue measure in parameter space the equation also exhibits two and three frequency quasiperiodic behavior we also show that the strange nonchaotic attractors have distinctive frequency spectrum this property might make them experimentally observable

the hartman wintner theorem for functional differential equations abstract in this paper we present a generalization of the hartman wintner theorem about the asymptotic behavior of the solutions of linear ordinary differential equations to functional differential equations

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

roughness of h k dichotomies abstract in this paper we prove a roughness theorem for a class of dichotomies of linear systems of differential equations the so called h k dichotomies these include a great variety of dichotomies among them the ordinary and exponential dichotomies exponential ordinary dichotomies levinson dichotomies for diagonal systems etc we prove that under general uniform conditions upon the functions h and k these dichotomies are not destroyed by integrable perturbations this result generalizes the property of roughness of ordinary dichotomies under integrable perturbations

doubling bifurcations of destroyed t 2 tori abstract two duffing oscillators driven with incommensurate frequencies and coupled additively are investigated with this system we observe a host of complex behavior which compete with each other in parameter space in particular the period doubling of an attracting t 2 torus and the destruction of a t 2 torus as the parameters are varied along a curve in parameter space we observe period doubling bifurcations of a destroyed t 2 torus this phenomenon is studied in greater detail via a simple recursive map

absolutely summing operators on c 0 1 as a tree space and the bounded approximation property let x be a banach space for describing the space p c 0 1 x of absolutely summing operators from c 0 1 to x in terms of the space x itself we construct a tree space l1tree x on x it consists of special trees in x which we call two trunk trees we prove that p c 0 1 x is isometrically isomorphic to l1tree x as an application we characterize the bounded approximation property bap and the weak bap in terms of x valued sequence spaces

dichotomies and asymptotic formulas for the solutions of differential equations abstract the new concept of h k dichotomy is studied extensively and is used in asymptotic equivalence problems

noise induced jumps in two coupled duffing oscillators abstract we consider two coupled duffing oscillators with the coexistence of four attractors switching between the coexisting attractors by noises is investigated three types of noises such as gaussian amplitude dependent and dichotomous are used we observed noise induced extrinsic intermittency intermittent dynamics is characterized using power spectrum average and relative residence times on each precoupled attractors and probability distribution of time interval between successive switchings variation of probability distribution of state variables in poincare map is studied as a function of noise strength the effect of various noises is compared the occurrence of crisis induced and type i intermittencies are also presented

the strong approximation property and the weak bounded approximation property abstract we show that the strong approximation property strong ap respectively strong cap and the weak bounded approximation property respectively weak bcap are equivalent for every banach space this gives a negative answer to oja s conjecture as a consequence we show that each of the spaces c 0 and l 1 has a subspace which has the ap but fails to have the strong ap

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

the quasi periodic doubling cascade in the transition to weak turbulence abstract the quasi periodic doubling cascade is shown to occur in the transition from regular to weakly turbulent behaviour in simulations of incompressible navier stokes flow on a three periodic domain special symmetries are imposed on the flow field in order to reduce the computational effort thus we can apply tools from dynamical systems theory such as continuation of periodic orbits and computation of lyapunov exponents we propose a model ode for the quasi period doubling cascade which in a limit of a perturbation parameter to zero avoids resonance related problems the cascade we observe in the simulations is then compared to the perturbed case in which resonances complicate the bifurcation scenario in particular we compare the frequency spectrum and the lyapunov exponents the perturbed model ode is shown to be in good agreement with the simulations of weak turbulence the scaling of the observed cascade is shown to resemble the unperturbed case which is directly related to the well known doubling cascade of periodic orbits

fourier methods for quasi periodic oscillations quasi periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators this paper presents two new numerical methods for the investigation of quasi periodic oscillations both algorithms can be regarded as generalizations of the averaging and the harmonic spectral balance methods the algorithms are easy to implement and require only minimal a priori knowledge of the system most importantly the methods do not depend on an a priori co ordinate transformation the methods are applied to a number of illustrative examples from non linear electrical engineering and the results show that the methods are efficient and reliable in addition these examples show that the presented algorithms can also continue through regions of sub harmonic phase locked resonance even though they are designed only for the quasi periodic case copyright 2006 john wiley sons ltd

duality of the distance to closed operator ideals special operator ideal approximation properties aps of banach spaces are employed to solve the problem of whether the distance functions s dist s i f e and s dist s i e f are uniformly comparable in each space l e f of bounded linear operators here i e f s l e f s i f e stands for the adjoint ideal of the closed operator ideal i for banach spaces e and f counterexamples are obtained for many classical surjective or injective banach operator ideals i by solving two resulting asymmetry problems for these operator ideal aps

global bifurcation destroying the experimental torus t 2 we show experimentally the scenario of a two frequency torus t 2 breakdown in which a global bifurcation occurs due to the collision of a torus with an unstable periodic orbit creating a heteroclinic saddle connection followed by an intermittent behavior