nonlinear dynamics in coupled fuzzy control systems i coherence and chaos frustration in triangle configuration abstract nonlinear dynamics and chaos are studied in a system for which a complete set of equations of motion such as equations of newton navier stokes and van der pol is not available as a very general system as such we consider coupled classical spins pendulums each of which is under control by a fuzzy system that is designed to align the spin to an unstable fixed point the fuzzy system provides a deterministic procedure to control an object without use of a differential equation the positions and velocities of the spins are monitored periodically and each fuzzy control gives a momentum to its associated spin in the reverse directions if the monitoring is made with an interval short enough the spin spin interactions are overwhelmed by the fuzzy control and the system converges to a state as designed however a long interval monitoring induces dynamics of too late response and thereby results in chaos a great variety of dynamics are generated under very delicate balance between the fuzzy control and the spin spin interaction in which two independent mechanisms of creating negative and positive liapunov exponents interact with each other

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

global existence for systems of parabolic conservation laws in several space variables abstract we prove the global existence of solutions of the cauchy problem for certain systems of conservation laws with artificial viscosity terms added the system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix and the initial data is assumed to be close to a constant in l2 l in particular our result applies to the equations of compressible fluid flow in two and three space variables

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

formation of diffusion waves in a scalar conservation law with convection we study the scalar conservation law ut c x u x b u2 x uxx c 0 which is a model equation for the behavior of weak transverse waves near a viscous transitional shock solutions are shown to decay in l to a pair of diffusion waves moving apart at speeds c oo and c oo behavior that has been observed numerically in solutions of the full equations the interesting aspect of the analysis is that the asymptotic state of the solution is not known a priori in contrast to cases treated previously

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

multi order power spectrum and its application abstract a new technique was explored to investigate the time correlations in time series this is an extension of the ordinary power spectrum which is called the order q power spectrum i have extended the order q power spectrum further and made the multi order version of it this technique is able to be applied to various processes which have more than two states this approach is applied to a chaotic system which exhibits intermittent switching and shows us a surprising result appropriate choices for parameters show us various correlations that are not able to be observed in the ordinary power spectra

synchronization multistability and basin crisis in coupled pendula abstract the synchronization dynamics of two linearly coupled pendula is studied in this paper based on the lyapunov stability theory and linear matrix inequality lmi some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k th for the on set of full synchronization is obtained the numerical value of k th determined from the average energies of the systems is in good agreement with theoretical analysis prior to the on set of synchronization the boundary crisis of the chaotic attractor is identified in the bistable states where two asymmetric periodic attractors co exist it is shown that the coupled pendula can attain multistable states via a new dynamical transition the basin crisis that occur prior to the on set of stable synchronization the essential feature of basin crisis is that the two co existing attractors are destroyed while new three or more co existing attractors of the same or different periodicity are created in addition the linear perturbation technique and the routh hurwitz criteria are employed to investigate the stability of steady states and clearly identify the different types of bifurcations likely to be encountered finally two parameter phase plots show various regions of chaos hyperchaos and periodicity

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

scaling laws for noise induced superpersistent chaotic transients a super persistent chaotic transient is characterized by the following scaling law for its average lifetime tau approximately exp c p p c alpha where c 0 and alpha 0 are constants p or p c is a bifurcation parameter and p c is its critical value as p approaches p c from above the exponent in the exponential dependence diverges leading to an extremely long transient lifetime historically the possibility of such transient raised the question of whether asymptotic attractors are relevant to turbulence here we investigate the phenomenon of noise induced super persistent chaotic transients in particular we construct a prototype model based on random maps to illustrate this phenomenon we then approximate the model by stochastic differential equations and derive the scaling laws for the transient lifetime versus the noise amplitude epsilon for both the subcritical p p c cases our results are the following in the subcritical case where a chaotic attractor exists in the absence of noise noise induced transients can be more persistent in the following sense of double exponential and algebraic scaling tau approximately exp k0 exp k1epsilon gamma for small noise amplitude epsilon where k0 0 k1 0 and gamma 0 are constants the longevity of the transient lifetime in this case is striking for the supercritical case where there is already a super persistent chaotic transient noise can significantly reduce the transient lifetime these results add to the understanding of the interplay between random and deterministic chaotic dynamics with surprising physical implications

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

double precision errors in the logistic map statistical study and dynamical interpretation the nature of the round off errors that occur in the usual double precision computation of the logistic map is studied in detail different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined histograms of errors in trajectories given and for the case of fully developed chaos an explicit formula is found it is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram both jumps and locations seem to present geometric convergence characterized by the two first feigenbaum constants eventually a comparison with benford s law for the distribution of the leading digit of compilation of numbers is discussed

the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator abstract using recently developed techniques of wiggins 1988 we study the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator we show how homoclinic tori give rise to chaotic dynamics for single degree of freedom quasiperiodically forced oscillators in much the same way as smale horseshoes in the periodically forced case we give a complete description of the bifurcation set in the five dimensional parameter space for the two frequency forced duffing oscillator and then show how the two frequency results can be used to understand the effects of an arbitrary but finite number of frequencies

unfolding a chaotic bifurcation the differential equation of the sinusoidally forced pendulum is studied by digital simulation in a regime where two simple symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased by introducing a small constant bias in the forcing to break the symmetry two discontinuous bifurcations unfold from the single merging event considering both forcing amplitude and bias together as controls a codimension two bifurcation of chaotic attractors is defined whose geometric structure in control phase space is closely related to the elementary cusp catastrophe the chaotic bifurcations are explained in terms of homoclinic structures smale cycles in the poincare map

role of invariant and minimal absorbing areas in chaos synchronization in this paper the method of critical curves a tool for the study of the global dynamical properties of two dimensional noninvertible maps is applied to the study of chaos synchronization and related phenomena of riddling blowout and on off intermittency a general procedure is suggested in order to obtain the boundary of a particular two dimensional compact trapping region called absorbing area containing the onedimensional chaotic set on which synchronized dynamics occur the main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a milnor attractor with locally riddled basin or a chaotic saddle exists and may strongly influence the effects of riddling and blowout bifurcations some examples are given for a system of two coupled logistic maps and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area some general considerations are given concerning the transition from locally riddled to globally riddled basins and the role of the absorbing area in the occurrence of such transition is discussed s1063 651x 98 11611 2

two state on off intermittency caused by unstable dimension variability in periodically forced drift waves certain high dimensional dynamical systems present two or more attractors characterized by different energy branches for some parameter values the dynamics oscillates between these two branches in a seemingly random fashion a phenomenon called two state on off intermittency in this work we show that the dynamical mechanism underlying this intermittency involves the severe breakdown of hyperbolicity of the attractors through a mechanism known as unstable dimension variability we characterize the parametric evolution of this variability using statistical properties of the finite time lyapunov exponents as a model system that exhibits this behavior we consider periodically forced and damped drift waves in this spatiotemporal example there is a low dimensional chaotic attractor that is created by an interior crisis already presenting unstable dimension variability

explosions of chaotic sets abstract large scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied these bifurcations include sudden changes in the size and or type of the set an explosion is a bifurcation in which new recurrent points suddenly appear at a non zero distance from any pre existing recurrent points we discuss the following in a generic one parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur 1 a saddle node bifurcation isolated from other recurrent points 2 a saddle node bifurcation embedded in the set of recurrent points 3 outer homoclinic tangencies and 4 outer heteroclinic tangencies the term outer tangency refers to a particular configuration of the stable and unstable manifolds at tangency in particular we examine different types of tangencies of stable and unstable manifolds from orbits of pre existing invariant sets this leads to a general theory that unites phenomena such as crises basin boundary metamorphoses explosions of chaotic saddles etc we illustrate this theory with numerical examples

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

flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus we present evidence for a sequence of bifurcations in simulations of the differentially heated baroclinic rotating annulus similar to bifurcations of the logistic map the met office oxford rotating annulus laboratory simulation morals code is used to construct a detailed numerical regime diagram for the annulus and the distribution of regimes in parameter space is discussed the bifurcations are observed in a sequence of runs at high temperature forcing identified by poincare sections of the dominant temperature mode amplitude time series higher order return maps and predictions using quadratic fits to the data are used to verify this result and lyapunov exponents are calculated to identify and quantify the chaotic parts of the sequence

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

bayesian semiparametric multiple shrinkage high dimensional and highly correlated data leading to non or weakly identified effects are commonplace maximum likelihood will typically fail in such situations and a variety of shrinkage methods have been proposed standard techniques such as ridge regression or the lasso shrink estimates toward zero with some approaches allowing coefficients to be selected out of the model by achieving a value of zero when substantive information is available estimates can be shrunk to nonnull values however such information may not be available we propose a bayesian semiparametric approach that allows shrinkage to multiple locations coefficients are given a mixture of heavy tailed double exponential priors with location and scale parameters assigned dirichlet process hyperpriors to allow groups of coefficients to be shrunk toward the same possibly nonzero mean our approach favors sparse but flexible structure by shrinking toward a small number of random locations the methods are illustrated using a study of genetic polymorphisms and parkinson s disease

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

bifurcation behavior of the generalized lorenz equations at large rotation numbers the bifurcation structure and periodic orbits of the lorenz stenflo equations at large rotation numbers are given it is shown that rotation can lead to a much richer dynamical behavior than that of the original lorenz system and can be used to control or modify the latter s chaos behavior orbits with new topology arising from the merging and splitting of different periodic windows are observed abrupt changes in the one dimensional map are pointed out and studied in terms of the interaction of the interior and exterior boundaries