measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

the recently recognized failure of predictability in newtonian dynamics modern theories of dynamical systems have very clearly demonstrated the unexpected fact that systems governed by the equations of newtonian dynamics do not necessarily exhibit the predictability property indeed very recent researches have shown that in wide classes of very simple systems satisfying those equations predictability is impossible beyond a certain definite time horizon

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

dynamical complexity detection in short term physiological series using base scale entropy physiological systems generate complex fluctuations in their output signals that reflect the underlying dynamics the base scale entropy method was proposed as a complexity measure to investigate the complexity of time series the advantages of this method are simplicity and extremely fast calculation for very short data sets this method enables analyzing very short nonstationary and noisy data sets we employed this method for short term physiological time series for analysis of heart rate variability signals the results show that the simple and easily calculated measure can effectively detect the complexity dissimilarity of physiological time series in different physiological or pathological states which is convenient for clinical applications

a new linear convex hull algorithm for simple polygons corresp a new optimal algorithm for computing the convex hull of a simple polygon in the plane along with a proof of correctness is presented the main novelty of the proposed algorithm is its use of existing concepts such as the unimodality of the area of a triangle inscribed in a convex polygon and sldansky s scan the combination of these concepts gives us new insight into the problem

asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von karman plates with geometrically localized dissipation and critical nonlinearity abstract in this study we consider the von karman evolution equations accounting for rotational inertial forces along with a nonlinear localized damping under suitable hypotheses we prove the existence regularity and finite dimensionality of a compact attractor a r the main result of the paper is upper semicontinuity of attractors with respect to rotational inertia terms the main obstacles are criticality of the nonlinear term of the equation combined with geometrically constrained dissipation the flux multiplier used for propagation of the damping does not cooperate with the nonlinear term which is of critical exponent a new method based on uniform compactness is developed in order to handle such cases the method allows to establish uniform smoothness of the elements from the attractor which is the key ingredient in proving that the attractor is upper semicontinuous with respect to the rotational inertia 0 and conclude that it converges to the attractor a r obtained in the case 0 in geredeli et al 2013 23 when 0

lyapunov exponents for the miles spherical pendulum equations abstract lyapunov exponents and hence the lyapunov dimension have been evaluated for the equations developed by miles quart appl math 20 1962 21 physica d 11 1984 309 as an approximation to the equations of motion of a spherical pendulum forced at a frequency close to its natural frequency computations have been performed throughout the frequency range in which there are no stable fixed points and a little on either side of this range for various values of the damping parameter the results include strong dependence on the damping of both the probability of finding chaotic motion and the typical lyapunov dimension of such motion the co existence of alternative types of attractor in regions both inside and outside the range with no stable fixed points and the occurrence of metastable chaos the transitions to chaos at either end of a limit cycle window chosen as one that has also been observed in the laboratory have been explored in some detail the results are compared with other computations and with observations with laboratory pendulums and the implications of the results for future laboratory studies are discussed some results are presented for the mathematically related problem of liquid surface waves in a vibrating cylinder

non linear non planar and non periodic vibrations of a string abstract we study the vibrations of a string subject to planar harmonic excitation both experimentally and theoretically we observe multiple periodic motions planar and non planar whirling as well as quasi periodic whirling and irregularly precessing oscillations when the forcing frequency is near that of a transverse normal mode a two degree of freedom model is derived and it is shown that these motions can be partially understood in terms of the completely integrable hamiltonian system obtained when damping and forcing tend to zero in that model

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

response of nonlinear oscillators with random frequency of excitation revisited periodically driven oscillators of low frequency random excitations are analyzed computer simulation which was carried out for the duffing equation and forced vibrations of a pendulum indicated that in these cases noise has a stabilizing effect computation of lyapunov exponents showed that by adding noise to chaotic motion the largest lyapunov exponent as a rule turns to negative and consequently the chaotic motion is annihilated

bifurcations of tori and phase locking in a dissipative system of differential equations abstract in a system of ordinary differential equations obtained through a seven mode truncation of the plane incompressible navier stokes equations a two dimensional torus undergoes first two period doubling bifurcations and then a transition to a strange attractor this strange attractor of liapunov dimension larger than three in a wide parameter interval is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad jagged background as the liapunov dimension goes down towards two an interesting phenomenon of phase locking occurs which gives rise to an alternation of chaotic and periodic behavior

finding the convex hull of a simple polygon in linear time abstract though linear algorithms for finding the convex hull of a simply connected polygon have been reported not all are short and correct a compact version based on sklansky s original idea 7 and bykat s counter example 8 is given its complexity and correctness are also shown

computing global orbits of the forced spherical pendulum abstract orbits of the spherical pendulum with time periodic forcing are considered a numerical framework is developed which allows orbits to explore the entire globe the spherical pendulum is considered as an invariant manifold in an ambient six dimensional euclidean space the numerical integrator is the second order stormer verlet method coupled with the shake rattle algorithm the algorithm preserves numerically the phase space of the sphere which is a manifold to machine accuracy poincare sections restricted to the configuration space are used to illustrate the transition from oscillatory behavior to chaotic solutions as the amplitude of the pivot motion is changed the qualitative change in the poincare sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the lyapunov exponents les the les are also computed using a novel variant of the shake rattle algorithm the results show that irregular behavior can explore the entire sphere even at low forcing amplitudes and therefore local methods which parameterize only part of the sphere are inadequate in general and may lead to spurious dynamics the numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum

conservative numerical methods for nonlinear strings in this article a class of numerical schemes for the simulation of nonlinear coupled longitudinal transverse string vibration is presented though there are various ways of arriving at such schemes special attention is paid here to energy conservation in nonlinear model systems and its transfer to an analogous discrete quantity in a difference scheme such exact numerical energy conservation can lead to simple global stability conditions which can be otherwise difficult to ascertain for nonlinear difference schemes in particular such conditions may be arrived at without any reliance on frequency domain concepts i e fourier or laplace transforms which are of only moderate utility in the analysis of nonlinear systems implementation details are discussed and numerical results are presented

on a convex hull algorithm for polygons and its application to triangulation problems abstract a frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases due to its simplicity the algorithm is nevertheless attractive in this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons some application areas where such polygons occur are briefly discussed in addition it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0 n time

non linear beam oscillations excited by lateral force at combination resonance abstract non linear oscillations of a beam subjected to a periodic force at a combination resonance are considered using the galerkin method a partial differential equation of oscillations is reduced to a system of ordinary differential equations with a small parameter a system of three autonomous differential equations is derived the multiple scales method being used qualitative properties of trajectories are analyzed the naimark sacker bifurcations at the combination resonance are analyzed by the center manifold method almost periodic oscillations of a beam arise due to these bifurcations these oscillations are investigated qualitatively and numerically

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

recurrence plot analysis of irregularly sampled data irregularly sampled time series usually require data preprocessing before a desired time series analysis can be applied we propose an approach for distance measuring of pairs of data points which is directly applicable to irregularly sampled time series in order to apply recurrence plot analysis to irregularly sampled time series we use this approach and detect regime transitions in prototypical models and for an application from palaeoclimatatology this approach might be useful for any method that is based on distance measuring e g correlation sum or lyapunov exponent estimation

smooth attractors of finite dimension for von karman evolutions with nonlinear frictional damping localized in a boundary layer abstract in this paper dynamic von karman equations with localized interior damping supported in a boundary collar are considered hadamard well posedness for von karman plates with various types of nonlinear damping are well known and the long time behavior of nonlinear plates has been a topic of recent interest since the von karman plate system is of hyperbolic type with critical nonlinearity noncompact with respect to the phase space this latter topic is particularly challenging in the case of geometrically constrained nonlinear damping in this paper we first show the existence of a compact global attractor for finite energy solutions and we then prove that the attractor is both smooth and finite dimensional thus the hyperbolic like flow is stabilized asymptotically to a smooth and finite dimensional set

internal resonances in whirling strings involving longitudinal dynamics and material non linearities abstract internal resonance mechanisms between near commensurate longitudinal and transverse modes of a taut spatial string are identified and studied using an asymptotic method and the influence of material non linearities on the resulting solutions is considered geometrical non linearities couple longitudinal motions to in plane and out of plane transverse motions resulting in resonant and non resonant interactions between linearly orthogonal string modes past studies have included only transverse modes in the description of string motions and have predicted periodic quasi periodic and chaotic whirling motions arising from the geometrical non linearities this study considers further the inclusion of longitudinal motions and a non linear material law which are both appropriate for the study of rubber like strings an asymptotic analysis captures the aforementioned whirling motions as well as a new class of whirling motions with significant longitudinal content periodic quasi periodic and aperiodic likely chaotic responses are included among these motions their existence hardening softening characterization and stability are found to be highly dependent on the magnitude of the material non linearities

global attractors for a third order in time nonlinear dynamics abstract long time behavior of a third order in time nonlinear pde equation is considered this type of equations arises in the context of nonlinear acoustics 12 20 22 24 where modeling accounts for a finite speed of propagation paradox the latter results in hyperbolic nature of the dynamics it will be proved that the underlying pde generates a well posed dynamical system which admits a global and finite dimensional attractor the main difficulty associated with the problem studied is the lack of lyapunov function along with the lack of compactness of trajectories which fact prevents applicability of standard tools in the area of dynamical systems

non periodic driving of coupled oscillators a spherical swing abstract nonlinearly coupled damped oscillators at 1 1 frequency ratio one oscillator being driven coherently for efficient excitation are exemplified by a spherical swing with some phase mismatch between drive and response for certain damping range excitation is found to succeed if it lags behind but to produce a chaotic attractor if it leads the response although a period doubling sequence for damping increasing leads to the attractor this is actually born as a hard as regards amplitude bifurcation at a zero growth rate parametric line as damping decreases an unstable fixed point crosses an invariant plane to enter as saddle focus a phase space domain of physical solutions a second hard bifurcation occurs at the zero mismatch line the saddle focus leaving that domain times on the attractor diverge when approaching either line leading to exactly one dimensional and noninvertible limit maps which are analytically determined

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

long time dynamics of von karman semi flows with non linear boundary interior damping abstract we study von karman evolution equations with non linear dissipation and with partially clamped and partially free boundary conditions two distinctive mechanisms of dissipation are considered i internal dissipation generated by non linear operator and ii boundary dissipation generated by shear forces friction acting on a free part of the boundary the main emphasis is given to the effects of boundary dissipation under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension we also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which in turn depends on the behaviour at the origin of the functions describing the dissipation

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

breakdown to chaotic motion of a forced damped spherical pendulum abstract the investigation here is of a pendulum forced by a horizontal sinusoidal coplanar oscillation of its pivot moving in nonplanar oscillations near the downward vertical from the pivot single or double fourier expansions of the different forms of periodic solution of the governing differential equations are calculated by newton s method applied in fourier space as the forcing frequency is changed a typical sequence begins with stable symmetric nonplanar oscillations followed by bifurcation to a stable periodic modulation of the oscillations then weakly chaotic modulation merging into fully chaotic modulation of the oscillations the weakly chaotic oscillations at a given forcing frequency are shown to lie in the neighbourhood of the unstable strictly periodic oscillatory solutions at the same frequency changes in averaged measures such as the lyapunov exponents are explained in terms of changes in the amount of order in the chaotic oscillations

birth and death of the double scroll abstract the double scroll is a chaotic attractor observed from an extremely simple electronic circuit made of 4 linear circuit elements 2 capacitors 1 inductor and 1 resistor and a 2 terminal nonlinear resistor characterized by a 3 segment piecewise linear v i characteristic this nonlinear resistor can be built in the laboratory using 2 transistors and diodes the qualitative behavior of the double scroll attractor has been described elsewhere where each cross section of the double scroll attractor is shown to consist of 2 tightly wound spirals our objective in this paper is to present a bifurcation analysis of this circuit in terms of 2 basic parameters various bifurcation phenomena such as hopf bifurcation period doubling cascades rossler s spiral and screw type attractors periodic windows shilnikov phenomenon double scroll and boundary crisis are described in particular special emphasis is given to the mechanisms leading to the birth and the death of the double scroll attractor all bifurcation phenomena described in this paper are observed not only experimentally but also confirmed by digital computer simulations moreover an analytical foundation of the double scroll family of equations has been developed and used to explain in a rigorous way all observed bifurcation phenomena in addition rossler s spiral type and screw type attractors have been observed from the same circuit where the nonlinear resistor has only one break point i e it is described by a 2 segment piecewise linear v i characteristic this means that extremely complicated non periodic chaotic waveforms can arise in the simplest third order uncoupled electrical circuit in which all elements except one a resistor are linear and passive and in which the constitutive relation of the nonlinear resistor is made of the simplest conceivable nonlinearity namely 2 straight line segments

analysis adaptive control and circuit simulation of a novel nonlinear finance system in the last three decades a growing interest in developing nonlinear dynamical systems for economic models displaying chaotic behavior has been developed to this direction a novel 3 d nonlinear finance chaotic system consisting of two nonlinearities is presented the dynamical analysis of the proposed system confirms its complex dynamic behavior which is studied by using well known simulation tools of nonlinear theory such as the bifurcation diagram lyapunov exponents and phase portraits also some interesting phenomena related with nonlinear theory are observed such as route to chaos through a period doubling sequence antimonotonicity and crisis phenomena in addition an interesting scheme of adaptive control of finance system s behavior is presented furthermore the novel nonlinear finance system is emulated by an electronic circuit and its dynamical behavior is studied by using the electronic simulation package cadence orcad in order to confirm the feasibility of the theoretical model

low frequency behavior in a frictionally excited beam abstract a cantilevered beam excited by a periodically reciprocating friction contact surface exhibited extremely low frequency responses with frequencies as low as 100 times lower than the driver example responses near 1 1 and 1 2 resonances between the excitation and the degree of freedom normal to the direction of sliding show two frequency quasiperiodicity and in one case three frequency quasiperiodicity underlying circle maps were extracted and winding numbers were matched to response frequencies quantified in the fast fourier transforms of the responses a torus doubling bifurcation was documented

a counter example to a convex hull algorithm for polygons abstract a linear time algorithm was recently published pattern recognition22 561 565 1989 for computing the convex hull of a simple polygon in this note we present a counter example to that algorithm

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

an analysis of two dimensional water waves based on o 2 the classic problem of inviscid irrotational water waves in a plane is analysed from a group theoretic point of view

measurement and analysis of complex modulated motions in a weakly nonlinear system abstract the two timing dynamics of a forced weakly nonlinear system are considered experimental results on a roll forced spherical pendulum verify the occurrence of slow time subharmonically and chaotically modulated oscillations in a frequency band near resonance the modulatory behavior follows the occurrence of a hopf bifurcation in the slow time coordinates a phase sensitive detection method for experimentally isolating the slow time behavior is described a lagrangian model for the pendulum is developed and slow time equations are presented a two thirds power scalling law relating roll angle to the appearance of the modulations is derived from the slow time equations and tested with experimental results

resonant motion of a spherical pendulum abstract the weakly nonlinear resonant response of a damped spherical pendulum length l damping ratio natural frequency 0 to the planar displacement e l cos t e 1 of its point of suspension is examined in a four dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion the loci of equilibrium points and the corresponding bifurcation points in this space are determined the control parameters are 2 e 2 3 and v 2 2 2 0 e 2 3 2 if v within which no stable equilibrium points exist as v decreases through the upper bound a hopf bifurcation point of this interval the motion in the phase space becomes periodic and then following a period doubling cascade chaotic there may be alternating sub intervals of chaotic and periodic motion the chaotic trajectories in the phase space appear to lie on fractal attractors

towards long term prediction abstract this paper describes a simple method of obtaining longer term predictions from a nonlinear time series assuming one already has a reasonably good short term predictor the usefulness of the technique is that it eliminates to some extent the systematic errors of the iterated short term predictor the technique we describe also provides an indication of the prediction horizon we consider systems with both observational and dynamic noise and analyse a number of artificial and experimental systems obtaining consistent results we also compare this method of longer term prediction with ensemble prediction

amplitude modulated and chaotic dynamics in resonant motion of strings abstract the weakly non linear resonant motions of a stretched string to a harmonic planar excitation are investigated one mode truncation of the in plane and out of plane equation results in a two degree to freedom system which is examined by using the method of averaging and the method of integral manifolds it is shown that the averaged equations for small enough damping possess non planar constant solutions which become unstable by a hopf bifurcation the resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string there are two limit cycle branches for the averaged equations one arising due to the hopf bifurcation and the other due to a global saddle node bifurcation the two branches merge as the damping is further reduced also with variation in detuning the isolated branch exhibits period doubling bifurcations chaotic attractors and merging of attractors giving rise to rossler as well as lorentz type solutions the truncated string equations are also directly integrated it is shown that there are non planar periodic responses that bifurcate into amplitude modulated motions on a 2 torus changes in detuning result in torus doubling merging of tori and then destruction of the torus leading to chaotic amplitude modulations the results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response

bifurcations of the horizontally forced spherical pendulum abstract various planar motions of the horizontally forced damped spherical pendulum are considered and in particular their stability to non planar perturbations by making a careful choice of coordinates all solutions of the planar pendulum can be considered including small amplitude periodic solutions running oscillations and chaotic solutions the full nonlinear equations in the chosen coordinates are derived and the symmetries of the system are described bifurcation diagrams for various types of solutions are presented stability of the chaotic solutions is determined by considering a normal lyapunov exponent

small scale turbulence in the plankton low order deterministic chaos or high order stochasticity abstract almost all regions of the oceans are heavily influenced by the effects of physical processes such as turbulence which in turn influence the distribution and ecology of organisms that occupy these regions there is a real need for additional focus placed on the precise knowledge of both physical and biological processes which is often difficult using basic time series analysis in that way we applied nonlinear analysis techniques to high frequency time series of temperature salinity and phytoplankton concentrations recorded in different hydrodynamical regimes related to tidal forcing in a tidally mixed coastal ecosystem techniques devoted to the identification of low order deterministic chaos cannot find evidence of chaos while a lower dimensionality was identified in low hydrodynamic conditions the results rather suggest stochastic time series with many degrees of freedom no obvious attractor in phase space trajectory lle indistinguishable from zero absence of convergence of the correlation integral we then applied to these data specific multifractal analysis techniques and showed that these time series clearly exhibit high order stochasticity in addition the stochastic structure of purely passive scalars i e temperature and salinity remained invariant while the one of phytoplankton biomass must be regarded as highly structured in time by both hydrodynamic and advective processes

tsallis like information measures and the analysis of complex signals abstract a tsallis like information measure that elaborates upon those of the tsallis type is shown to be a rather powerful tool for the analysis of complex non stationary signals as exemplified by the henon map and eegs the ensuing results could be of importance concerning the automatic detection of epileptic spikes

resonance and symmetry breaking for the pendulum abstract periodic solutions of the differential equation for periodic forcing of a lightly damped pendulum are obtained on the alternative hypotheses that i the contributions of second and higher harmonics to the average lagrangian are negligible or ii the solution is close to that for free oscillations the resonance curves amplitude or root mean energy vs driving frequency and stability boundaries for symmetric swinging oscillations and their asymmetric descendents following symmetry breaking are determined for 1 and o where is the ratio of actual to critical damping and is the ratio of the maximum external moment to the maximum gravitational moment resonance as defined by synchronism between the external moment and the damping moment is found to be impossible and the conventional resonance curve separates into two branches if 3 28 o 3 which condition is necessary for normal symmetry breaking a numerical fourier series determination of the resonance curve and bifurcation points for 1 8 and 1 2 is presented in an appendix by p j bryant

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

computing the convex hull of a simple polygon abstract an algorithm for computing the convex hull of a simple polygon is presented its correctness and complexity are also shown

the necessity for a time local dimension in systems with time varying attractors abstract we show that a simple non linear system for ordinary differential equations may possess a time varying attractor dimension this indicates that it is infeasible to characterize eeg and meg time series with a single time global dimension we suggest another measure for the description of non stationary attractors

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

forcing induced asymmetry on dynamical systems with cubic non linearities abstract the present work investigates the dynamics of a class of two degree of freedom oscillators with cubic non linearity in the restoring forces these oscillators are under the action of an external load including constant and harmonic components initially a perturbation analysis is applied to the equations of motion demonstrating the effect of the asymmetry induced by the constant loading component on the classical 1 1 and 1 3 internal resonances as well as on the possibility of the appearance of a first order 1 2 internal resonance next sets of slow flow equations governing the amplitudes and phases of vibration are derived for the special case of no internal resonance and for the most complicated case corresponding to 1 1 internal resonance the analytical findings are then complemented by numerical results obtained by examining the dynamics of a two degree of freedom mechanical system first the effect of certain system parameters on the existence and stability of constant and periodic solutions of the slow flow equations is illustrated by presenting a sequence of response diagrams finally the dynamics of the system used as an example is investigated further by direct integration of the slow flow equations this shows the existence of a period doubling sequence culminating into a continual interchange between quasiperiodic and chaotic response it also demonstrates a new transition scenario from phase locked to phase entrained and drift response