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

time series investigations on an experimental system driven by phase transitions in the present paper the analysis of an intermittent time series obtained from a nonlinear system is presented the setup is driven by phase transitions and has no external periodic driving the observed states range from stable ones over periodicity and intermittence to chaotic behavior the data in the intermittent regime are analyzed with respect to the optical shape the frequency distribution of the laminar lengths and the ensuremath gamma parameter the results of these analyses are compatible with the interpretation of the presence of intermittence type ii in the system under investigation additionaly the reconstruction of the radius dynamics of the experimental data is attempted the corresponding results also indicate type ii intermittence some weight is put on figuring out the principal problems arising in the reconstruction of the ne stackrel mathrm ifmmode breve else u fi mathrm ifmmode imath else i fi mark sacker dynamics from a scalar time series

spiralling self avoiding walks an exact solution an exact solution is presented to a problem of spiralling self avoiding walks on the square lattice recently proposed by privman 1983 for n to infinity the number of n step spiral walks increases as cn approximately 2 23 5 4 pi n 7 4 exp 2 pi n 3 1 2 and their root mean square end to end distance behaves as rn approximately 1 2 square root pi 1n1 2 log n

upper bounds in phase synchronous weak coherent chaotic attractors an approach is presented for coupled chaotic systems with weak coherent motion from which we estimate the upper bound value for the absolute phase difference in phase synchronous states this approach shows that synchronicity in phase implies synchronicity in the time of events a characteristic explored to derive an equation to detect phase synchronization based on the absolute difference between the time of these events we demonstrate the potential use of this approach for the phase coherent and the funnel attractor of the rossler system as well as for the spiking bursting rulkov map

oscillation for a class of odd order delay parabolic differential equations some sufficient conditions and some sufficient and necessary conditions are established for the oscillation of a class of odd order delay parabolic differential equations of the form nu x t tn a t u i 1npi x t u x t i j 1mqi x t u x t j h t f u x t r1 u x t rl 0 x t t0 g t0 r where n is an odd integer is a bounded domain in rm with a smooth boundary and is the laplacian operation with three different boundary conditions to some extent our results extended and improved the oscillatory results of some references meanwhile we corrected some mistakes in a main conclusion of reference j comput appl math 147 2002 263

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset nonlinear dynamics in coupled fuzzy control systems ii abstract a new type of onset of chaos in a multidimensional dissipative system is reported that has been observed under a circumstance in which a delicate balance of order formation and chaos is materialized two limit cycles are connected directly to a sequence of strange attractors the widths in the bifurcation feigenbaum diagram of which converge to zero in a range close to the onset the measures of the extent of chaos are also zero there and the trajectories behave regularly in a macroscopic scale the chaos zones appear alternately with the windows of resonance frequency locking in the bifurcation diagram whose number of periodicity decreases as a system parameter is increased from an infinity in the manner of arithmetical progression 28 n 2 with n being integers down to unity concomitantly the extent of chaos develops and thereby grows measurable also the ranges of both the chaos and resonance zones in the bifurcation diagram become wider two lorenz plots of quantum numbers n and n 1 are always necessary to characterize the n th chaos zone for a chaotic state they appear in such a manner as to sandwich the identity map y x this chaos is readily interrupted by a limit cycle since one of the lorenz maps is crossed with y x by a small change of the control parameter moreover the tangential parts of the individual lorenz plots to y x have as many folds as their quantum numbers they form clusters each of which consists of many stable points being located densely at the limit of onset n both the intervals between these stable points and the folds lying between two neighboring stable points converge to zero this is the origin of the unmeasurable degree of chaos

homoclinic tangles for noninvertible maps not all naturally occurring iterated systems have the property that time is reversible in which case the dynamics must be modelled by a noninvertible map rather than by a di eomorphism while one dimensional theory describes dynamics of noninvertible maps higher dimensional theory has focussed on di eomorphisms though there are many higher dimensional examples of natural systems modelled by noninvertible maps for example noninvertible maps occur in the study of population dynamics 2 time one maps of delay equations 25 control theory algorithms 1 5 6 neural networks 16 and iterated di erence methods 14 in order to use dynamical systems theory in applications for which maps are noninvertible it is important to know how the theory di ers from the di eomorphism case one of the principal indications of chaotic dynamics in deterministic systems is the existence of transverse homoclinic orbits this paper examines the existence of homoclinic tangles and the associated dynamic behavior for noninvertible maps in particular we give examples of maps which have transverse intersection of stable and unstable manifolds but no horseshoe furthermore we show that this is a codimension two phenomenon papers by steinlein and walther 24 25 and lerman and shil nikov 13 both establish conditions for a transverse homoclinic point of a noninvertible map to be contained in a horseshoe this is discussed further in section 4 the work of laura gardini 10 addressed transverse homoclinic orbits for planar maps however she missed the subtle distinction for noninvertible maps between a transverse homoclinic point and

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

critical properties of directed self avoiding walks the generating functions and mean displacements of various two dimensional directed self avoiding walk models are calculated exactly by a simple transfer matrix approach asymptotically the authors find r sub n approximately rn2 1 2 approximately n and rperpendicular to n2 1 approximately n1 2 where n is the number of steps in the walk and and perpendicular to refer to projections of the displacement parallel and perpendicular to the preferred axis of the walk respectively some general properties of directed self avoiding walks for arbitrary dimensions are discussed as well

resonance regions for mathieu type dynamical systems on a torus abstract mathieu type dynamical systems on a torus are torus maps or differential equations whose right hand side components are trigonometric polynomials the asymptotic behaviour of resonance regions for mathieu type dynamical systems close to the integrable ones is drastically different from the behaviour of generic systems because the scaling law of resonance regions depends on the degree of the trigonometric polynomial or its newton polyhedron we prove this scaling law and compare our results with numerical experiments

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

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

spatio temporal dimensionality in the overall complex dynamics of an experimental cable mass system an experimental model of an elastic cable mass hanging at in phase or out of phase vertically moving supports is considered system parameters are adjusted to produce two different conditions of multiple internal resonance nonregular dynamics are analyzed in various frequency ranges including meaningful external resonance conditions attention is devoted to characterization of system dimensionality in terms of both time and spatial complexity the aims of this paper are i to give a general overview of the richness and robustness of different quasiperiodic and homoclinic bifurcation scenarios to chaos in various regions of the control parameter space ii to characterize steady nonregular response through delay embedding technique for attractor reconstruction and proper orthogonal decomposition of spatio temporal flow and iii to identify spatial configuration variables experimental eigenfunctions contributing mostly to nonregular dynamics thus obtaining hints about possible reduced models for reproducing complex regimes system dimensionality will be evaluated both by relating the dimension of attractors to the dimension of the linear phase space and from the dominating proper orthogonal modes

bifurcation structure of the c type period doubling transition abstract the period doubling transition to chaos along the edge of an arnold tongue is known to display unusual organization and scaling behavior kuznetsov et al 2005 7 it is also known that forced period doubling systems may be associated with the appearance of so called period doubled tori arneodo et al 1983 15 using the rossler system as an example we present a detailed analysis of the bifurcation structure associated with the forcing of a three dimensional period doubling system we explain how this structure is related to the recently discovered phenomenon of multi layered tori and discuss different bifurcation scenarios that transform a resonance torus into a period doubled ergodic torus similar bifurcation phenomena have recently been observed in a biologically relevant model of kidney blood flow regulation in response to fluctuations in arterial pressure

forced oscillations of solutions of systems of hyperbolic equations of neutral type sufficient conditions are obtained for forced oscillations of solutions of a class of systems of neutral hyperbolic equations with deviating arguments

behavior of a three torus in truncated navier stokes equations abstract the presence of a three torus in a seven mode truncation of the three dimensional navier stokes equations is investigated numerically by means of cross section and power spectra furthermore by taking advantage of particular features of the model rotation vectors circle maps and torus maps can be computed with high accuracy and used to study the dynamics in particular some interesting phenomena of partial phase locking are described in deep detail the three torus which arises via a hopf bifurcation and persists in a wide parameter range is found to break and originate a strange attractor the onset of chaos and the associated bifurcation point can be defined quite precisely

general framework for phase synchronization through localized sets we present an approach which enables one to identify phase synchronization in coupled chaotic oscillators without having to explicitly measure the phase we show that if one defines a typical event in one oscillator and then observes another one whenever this event occurs these observations give rise to a localized set our result provides a general and easy way to identify ps which can also be used to oscillators that possess multiple time scales we illustrate our approach in networks of chemically coupled neurons we show that clusters of phase synchronous neurons may emerge before the onset of phase synchronization in the whole network producing a suitable environment for information exchanging furthermore we show the relation between the localized sets and the amount of information that coupled chaotic oscillator can exchange

routes to complete synchronization via phase synchronization in coupled nonidentical chaotic oscillators we study the transition route to complete synchronization through phase synchonization in generic coupled nonidentical chaotic oscillators through numerical studies two routes are found i e one via lag synchronization the other via the intermittent chaotic burst state without lag synchronization we claim that these two routes are universal as evidence we analyze several examples on the basis of the conventional theory of intermittency in the presence of noise

characterization in bi parameter space of a non ideal oscillator abstract we investigate the dynamical behavior of a non ideal duffing oscillator a system composed of a mass spring pendulum driven by a dc motor with limited power supply to identify new features on duffing oscillator parameter space due to the limited power supply we provide an extensive numerical characterization in the bi parameter space by using lyapunov exponents following this procedure we identify remarkable new organized distribution of periodic windows the ones known as arnold tongues and also shrimp shaped structures in addition we also identify intertwined basins of attraction for coexisting multiple attractors connected with tongues

from quasiperiodicity to toroidal chaos analogy between the curry yorke map and the van der pol system the van der pol attractor exhibits a wide variety of behavior depending on the control parameter values limit cycles quasiperiodic motion on a torus mode locking period doubling banded chaos boundary crises torus wrinkling breakup of a torus and toroidal chaos the organization of these phenomena with respect to each other is well described by studying a partition of the control parameter plane of the curry yorke map

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

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

oscillation for a class of neutral parabolic differential equations some sufficient conditions are established for the oscillation of a class of neutral parabolic differential equations of the form n u x t email protected email protected ku x email protected k t n a t email protected i 1np i x t u x email protected i email protected j 1mq j x t u x email protected j h t f u x t r 1 u x t r l 0 x t email protected t 0 g t email protected r where n is an odd number is a bounded domain in r m with a smooth boundary email protected and d is the laplacian operation with three different boundary conditions we obtained some new oscillatory conditions for the odd order neutral parabolic differential equation to some extent our results are new oscillatory conditions and extended some oscillatory results of some references

some symmetric two dimensional dissipative maps abstract a class of simple symmetric two dimensional maps have been studied theoretically and one finds periode n doubling at values of a parameter of the form cos k n a computer simulation confirms this and it is also found that in general there are only a finite number of successive period doubling generations finite jump first order transitions are shown to take place

an exact formula for the number of spiral self avoiding walks an exact closed form expression is obtained for the number sn of spiral self avoiding walks with n steps on the square lattice this result is used to derive a complete asymptotic expansion for sn which is valid as n to infinity

an exactly solvable self avoiding walks model a self avoiding walks model is studied in which the walker is in addition to the usual self avoiding condition restricted not to make any turn which will put the walker in a direction rotated more than phi max from any of the directions previously taken here phi max phi r where phi r 2 pi the smallest exterior angle of the unit cell of the lattice under consideration the generating function for the square lattice is obtained and various properties such as the number of n step walks the total number of steps to given directions and the mean square end to end distance are evaluated analytically the model exhibits characteristics reminiscent of one dimensional self avoiding walks in all directions while retaining the anisotropic effect of the direction of the first step

asymptotic properties of spiral self avoiding walks the authors consider the spiral self avoiding walk model recently introduced by privman 1983 on the basis of an analogy with the partitioning of the integers they argue that the number of n step spiral walks should increase asymptotically as rho square root n with rho a constant leading to an essential singularity in the generating function enumeration data to 65 terms indicates however that cn apparently varies as rho n alpha with alpha approximately 0 55 they also study the n dependence of the mean square end to end distance rn2 and of the mean rotation angle theta n for n step walks from series extrapolations they estimate that rn2 approximately n1 2 and theta n approximately n0 55

noise scaling of phase synchronization of chaos we investigate the effect of noise on phase synchronization of coupled chaotic oscillators it is found that additive white noise can induce phase slips in integer multiples of 2p s in parameter regimes where phase synchronization is observed in the absence of noise the average time duration of the temporal phase synchronization scales with the noise amplitude in a way that can be described as superpersistent transient we give two independent heuristic derivations that yield the same numerically observed scaling law pacs number s 05 45 xt the phenomenon of synchronous chaos has attracted much attention since the work of pecora and carroll in 1990 1 typically when two chaotic oscillators are coupled together synchronization between them can occur when the coupling strength is large enough recently a more delicate type of synchronization phenomenon was discovered by rosenblum pikovsky and kurths 2 this is the phase synchronization of chaotic oscillators which occurs at smaller coupling strength than that required for complete synchronization briefly if trajectories in each chaotic oscillator can be regarded as a rotation then the phase angle of the rotation increases steadily with time u t 5vt1f t where v is the average rotation frequency and f t is a term characterizing chaotic fluctuations as such the rate of increase of phase can be modeled as a drift v plus a zero mean chaotic process in the absence of coupling the phase angles of the two oscillators u 1 t and u 2 t are uncorrelated that is if one measures the difference du t uu 1 t 2u 2 t u one finds that du t increases steadily with time however when a small amount of coupling is present du t can be confined within 2p while the amplitudes of the rotations are still completely uncorrelated the bifurcation that leads to this phase synchronization was subsequently investigated 3 5 the ability of chaotic systems to have phase synchronization has implications on digital communication with chaos using the natural chaotic symbolic dynamics 6 in such a case it is highly desirable to suppress phase diffusions between chaotic communication channels to ensure proper timing for decoding in this brief report we address to what extent phase synchronization can be observed in laboratory experiments by investigating the effect of noise on phase synchronization our principal results are 1 additive white noise a type of noise encountered commonly in experimental situations can induce phase slips in units of 2p between the coupled oscillators which would otherwise be synchronized in phase in the absence of noise and 2 the average time duration between successive phase slips appears to obey a scaling law with the noise amplitude e

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

renormalization in a circle map with two inflection points abstract an extended version of the sine circle map becomes non invertible by developing simultaneously two critical points in part of parameter space universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits four operators are needed to generate all such orbits the existence of non trivial equivalence relations for operation sequences makes the present theory different from other multi operator renormalisation approaches possible implications of the theory in higher dimensional systems are discussed

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

torus breakdown in noninvertible maps we propose a criterion for the destruction of a two dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus this mechanism is specific to noninvertible maps the cusp points arise when the tangent to the torus at the point of intersection with the critical curve l 0 coincides with the eigendirection corresponding to vanishing eigenvalue for the noninvertible map further parameter changes lead typically to the generation of loops self intersections of the invariant manifolds followed by the transformation of the torus into a complex chaotic set

the asymptotic form for the number of spiral self avoiding walks the author investigates the asymptotic behaviour of the number sn of spiral self avoiding walks on the square lattice and show that sn approximately rho square root n with exp pi 312 or rho or exp 2 pi 312

characterizing bifurcations and classes of motion in the transition to chaos through 3d tori of a continuous experimental system in solid mechanics abstract the dynamics of a continuous experimental system exhibiting quasiperiodic transition to chaos has been investigated an elastic cable mass system hanging at in phase vertically moving supports is considered with vibrational parameters in the neighbourhood of external and internal resonance conditions adjusted to be able to introduce three torus dynamics measurements of the nonregular response are made in frequency ranges including primary resonance of the first symmetric in plane mode of the cable quantitative characterisation of regular and nonregular attractors and of the configuration variables involved in the motion is made by means of delay embedding technique and proper orthogonal decomposition of spatio temporal flow attention is devoted to analyse in depth the scenario of transition to chaos exhibited by the model steady quasiperiodic motions on three tori partial and full phase locking of the flow and chaotic attractors are observed in spite of the dynamics variedness the systematic characterisation of the response allows us to set the bifurcation behaviour of the experimental system in the framework of theoretic numeric transition to chaos through three dimensional tori

global convergence method for singularly perturbed boundary value problems we give uniformly convergent splines difference scheme for singularly perturbed boundary value problems eu p x u q x u f x u a 0 u b 1 1 by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration we prove the o min h2 e2 order of uniform convergence with respect to small parameter e at nodes on uniform mesh and o min h e order of uniform global convergence with respect to the approximate solution given by s x i 1n s i x h xi x where h is the heaviside function which is the approximation for the closed form of the exact solution

torus maps and the problem of a one dimensional optical resonator with a quasiperiodically moving wall abstract we study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically with d 2 incommensurate frequencies we show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d dimensional torus that preserves a foliation by irrational straight lines in particular the jacobian of this map has d 1 eigenvalues equal to 1 we present rigorous and numerical results about several dynamical features of such maps we also show how these dynamical features translate into properties for the field in the cavity in particular we show that when the torus map satisfies a kam theorem which happens for a cantor set of positive measure of parameters the energy of the electromagnetic field remains bounded when the torus map is in a resonant region which happens in open sets of parameters inside the gaps of the previous cantor set the energy grows exponentially

chaotic motion on hamiltonian tori kam tori are well known to be that finite part of phase space in which the motion of a weakly perturbed classical hamiltonian system remains integrable moreover they are barriers in the phase space of systems with two degrees of freedom the authors show that certain hamiltonian systems contain invariant non regular tori with compressible flow using as an example an electron moving in electromagnetic fields which are periodic in space they demonstrate i that strange attractors and repellers of well known autonomous or quasi periodically time driven systems may occur on such strange tori ii that one may find barriers consisting of non regular hamiltonian tori in systems with any number of degrees of freedom the flow on such barriers is non chaotic though and iii that certain kam tori transform into non regular tori rather than breaking up when the perturbation becomes strong

a turbulence model with stochastic soliton motion a dissipative benjamin ono equation is used to study fluid and plasma turbulence the system is studied by an exact nonlinear mode truncation method in which a finite number of poles are used to present the solution the justification of the pole expansion approach is discussed with the proof of a completeness theorem the stability and spectrum analysis show that asymptotic behavior of the system is completely represented by a finite number of nonlinear modes the behavior of those nonlinear modes resembles solitons and exhibits a wide range of bifurcation phenomena and routes to turbulence

a route to computational chaos revisited noninvertibility and the breakup of an invariant circle abstract in a one parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme lorenz studied a sequence of transitions from an attracting fixed point to computational chaos as part of the transition sequence he proposed the following as a possible scenario for the breakup of an invariant circle ic the ic develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps beyond this parameter value the ic fails to persist and the system exhibits chaotic behavior on an invariant set with loops computational chaos a prelude to computational instability physica d 35 1989 299 we investigate this problem in more detail and show that the ic is actually destroyed in a global bifurcation before it has a chance to develop cusps instead the global unstable manifolds of saddle type periodic points are the objects which develop cusps and subsequently loops or antennae the one parameter study is better understood when embedded in the full two parameter space and viewed in the context of the two parameter arnold horn structure certain elements of the interplay of noninvertibility with this structure the associated ics periodic points and global bifurcations are examined

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

type ii intermittency from markov binary block visibility graph perspective in this work we study the type ii intermittency based on asymptotic modes and the optimized markov binary visibility graphs perspective in fact we investigate the behavior of a dynamical system in the vicinity of subcritical hopf bifurcations of pre fixed point fixed point and post fixed point using networks language we use self maps in order to generate asymptotic modes in the type ii intermittency we find their properties based on statistical tools such as the length between reinjection points and the mean length and also length distributions numerical results show that asymptotic modes affect on the trajectory and the length between reinjection points of type ii intermittency in situations of pre fixed point fixed point and post fixed point however their mean length are approximately similar to each other for further illustration we compute the degree distribution of the complex network generated by type ii intermittency experimental results are found to agree well with the analytical results derived from the optimized markov binary visibility graph

spiralling directed self avoiding walks a self avoiding walk model is proposed in which the walker performs two choice directed self avoiding walks while it may also make spiralling turns and thus visiting every quarter of the plane unlimited times the generating function for the square lattice is obtained and an asymptotic form for the number of n step walks an is derived the an of the model belongs to the same universality class with the two choice directed self avoiding walk however the model achieves a higher number of n step walks than the two choice directed self avoiding walk by a factor of c approximately 4 23609

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

chaotic attractors on a 3 torus and torus break up abstract two coupled driven van der pol oscillators can have three frequency quasiperiodic attractors which lie on a 3 torus the evidence presented in this paper indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent furthermore no chaotic orbits lying on a torus were observed suggesting that in most cases at least in the case of this system orbits do not become chaotic before their tori are destroyed to expedite the calculations a method was developed which can be used to determine if an orbit is on a torus without actually displaying that orbit the method also described in this paper was designed specifically for our system the basic idea however could be used for studying attractors of other systems very few modifications of the method if any would be necessary when studying systems with the number of degrees of freedom equal to that of our van der pol system

oscillation of parabolic neutral delay difference equations with several positive and negative coefficients we obtain some oscillation criteria for the parabolic neutral delay difference equations with several positive and negative coefficients

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

coexisting tori and torus bubbling in non smooth systems abstract pulse modulated power electronic converters represent an important class of piecewise smooth dynamical systems with a broad range of applications in modern power supply systems the paper presents a detailed investigation of a number of unusual bifurcation phenomena that can occur in power converters with multilevel control in the first example a closed invariant curve arises in a border collision bifurcation as a period 6 saddle cycle collides with a stable fixed point of focus type and transforms it into an unstable focus point the second example involves the formation of a structure of coexisting tori through the interplay between border collision and global bifurcations we examine the behavior of the system in the presence of two coexisting stable resonance tori and finally show how an existing torus can develop heteroclinic bubbles that connect the points of a stable resonance cycle with an external pair of saddle and focus cycles the appearance of these structures is explained in terms of a sequence torus birth bifurcations with pairs of stable and unstable tori folding one over the other

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

attractors on an n torus quasiperiodicity versus chaos the occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance however current understanding concerning the question of how prevalent such motions should be is incomplete with this in mind the types of attractors which can exist for flows on an n torus are studied numerically for n 3 and 4 specifically nonlinear perturbations are applied to maps representing n frequency quasiperiodic attractors these perturbations can cause the original n frequency quasiperiodic attractors to bifurcate to other types of attractors our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows n frequency quasiperiodic attractors are the most common followed by n 1 frequency quasiperiodic attractors followed by periodic attractors however as the nonlinearity is further increased n frequency quasiperiodicity becomes less common ceasing to occur when the map becomes noninvertible chaotic attractors are very rare for n 3 for small to moderate nonlinearity but are somewhat more common for n 4 examination of the types of chaotic attractors that occur for n 3 reveals a rich variety of structure and dynamics in particular we see that there are chaotic attractors which apparently fill the entire n torus i e limit sets of orbits on these attractors are the entire torus furthermore these are the most common types of chaotic attractors at moderate nonlinearities

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

spiral self avoiding walks on the triangular lattice the authors study the behaviour of spiral self avoiding walks on the triangular lattice the spiral constraint simply says that no step in an anticlockwise direction may be taken imposing the additional constraint that steps may not deviate from the straight ahead direction by pi 3 defines model i previously solved by joyce and brak 1985 as well as by lin 1985 if deviations of 2pi 3 are forbidden this model is referred to as model ii while model iii allows deviations of both pi 3 and 2pi 3 one finds for both model ii and model iii that the number of n step self avoiding spirals is sn approximately c exp 2pi square root n log n 12 n13 4 where c phi2 768gamma5 phi model ii approximately 0 009 phi model iii approximately 0 16 and gamma 1 12 log 2 pi 2 the confluent logarithm is an additional feature not present in the simpler case of the square lattice and model i triangular spiral self avoiding walks use of two new results in the theory of partitions are made

novel routes to chaos through torus breakdown in non invertible maps the paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non invertible maps by means of detailed numerically calculated phase portraits we first describe how three and five layered tori arise through period doubling and or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus we then describe several different mechanisms for the destruction of five layered tori in a system of two linearly coupled logistic maps one of these scenarios involves the destruction of the two intermediate layers of the five layered torus through the transformation of two unstable node cycles into unstable focus cycles followed by a saddle node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set other scenarios involve different combinations of local and global bifurcations including bifurcations that lead to various forms of homoclinic and heteroclinic tangles we finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two dimensional non invertible maps that have previously been used to study the properties of invariant sets

non transitive maps in phase synchronization abstract concepts from ergodic theory are used to describe the existence of special non transitive maps in attractors of phase synchronous chaotic oscillators in particular it is shown that for a class of phase coherent oscillators these special maps imply phase synchronization we illustrate these ideas in the sinusoidally forced chua s circuit and two coupled rossler oscillators furthermore these results are extended to other coupled chaotic systems in addition a phase for a chaotic attractor is defined from the tangent vector of the flow finally it is discussed how these maps can be used for the real time detection of phase synchronization in experimental systems

phase synchronization in the perturbed chua circuit we show experimental and numerical results of phase synchronization between the chaotic chua circuit and a small sinusoidal perturbation experimental real time phase synchronized states can be detected with oscilloscope visualization of the attractor using specific sampling rates arnold tongues demonstrate robust phase synchronized states for perturbation frequencies close to the characteristic frequency of the unperturbed chua

dynamics of the two frequency torus breakdown in the driven double scroll circuit in this work we numerically identify three scenarios for the two frequency torus breakdown to chaos using the driven double scroll circuit with varying driving parameters two of these scenarios follow the curryyorke route to chaos for one scenario we identify the transition to chaos through the onset of a heteroclinic tangle and its heteroclinic points in the other scenario chaos appears via period doubling bifurcations the third scenario is through the type ii intermittency for which a quasiperiodic torus grows in size breaks by touching external saddle points and forms a heteroclinic saddle connection these dynamic scenarios have distinct structure evolutions for the curry yorke route chaos appears softly and alternates with phase locking while for the type ii intermittency chaos appears abruptly and is preserved for a large range of the varying driving parameter s1063 651x 98 05210 6

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 asymptotic behaviour of directed self avoiding walks chakrabarti and manna 1983 have made a numerical study of directed self avoiding walks on a square lattice in which the random walker is forbidden to move upward extrapolating from an enumeration of n step walks up to n 14 they claimed that the mean end to end displacement rn behaves asymptotically for large n as nnu with nu 0 86 or 0 02 it is shown that a simple application of the generating function method yields the behaviour of n step walks of this type exactly for all n and the result nu 1 is rigorously proved

transitions from phase locked dynamics to chaos in a piecewise linear map recent work has shown that torus formation in piecewise smooth maps can take place through a special type of border collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle transitions from an ergodic to a resonant torus take place via border collision fold bifurcations we examine the transition to chaos through torus destruction in such maps considering a piecewise linear normal form map we show that this transition by virtue of the interplay of border collision bifurcations with period doubling and homoclinic bifurcations can involve mechanisms that differ qualitatively from those described by afraimovich and shilnikov

spiralling self avoiding walks self avoiding walks which only step straight ahead or to the left are studied heuristic arguments generating function studies and extrapolation of few step enumerations are described exact asymptotic results are obtained it is found that these spiralling walks are in new universality class wherein the number of n step walks does not behave in a size extensive manner

type ii intermittency in the driven double scroll circuit abstract in this work we show experimental evidences confirmed by numerical results from type ii intermittency in the driven double scroll circuit numerically we found a new scaling power law dependence on the critical parameter this result is a consequence of the new global bifurcation scenario for the t 2 torus breakdown observed in this system a homoclinic saddle connection is the nonlinear mechanism responsible for the reinjection of the trajectory around a repelling focus in fact in this global scenario the total laminar phase is the spiraling laminar period usually considered plus the time the trajectory spends in the vicinity of the saddle points

type i intermittency in nonstationary systems models and human heart rate variability abstract type i intermittency occurs whenever a discrete dynamical system in a chaotic state is about to undergo a tangential bifurcation then long stretches of almost periodic behavior laminar phases occur these laminar phases are interspersed with chaotic behavior in type i intermittency simple models show that a characteristic u shaped probability distribution is obtained for the laminar phase length we have shown elsewhere that in some cases of pathology the laminar phase length distribution characteristic for type i intermittency may be obtained in human heart rate variability data the heart and its regulatory systems are usually presumed to be noisy although the effect of additive noise on the laminar phase distribution in type i intermittency is well known the effect of multiplicative noise has not been studied in this presentation we discuss the properties of classes of models of type i intermittency in which the control parameter is changed randomly within the specified parameter range we show that the properties of these models are importantly different from those obtained for type i intermittency in the presence of additive noise we also show how the two models help explain some of the features seen in the intermittency in human heart rate variability some technical problems with measuring laminar phase lengths in time series will also be discussed

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

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

oscillation of delay parabolic differential equations with several coefficients some new sufficient conditions are established for the oscillation of delay parabolic differential equations of the form u x t t a t u i 1n pi x t u x t i j 1m qj x t u x t j x t t0 g where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn with three different boundary conditions our results improve the well known oscillation result of e when n m 1 an example is considered to illustrate our main results

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

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

an exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases