chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

approximation based control of nonlinear mimo time delay systems approximation based control is presented for a class of multi input multi output mimo nonlinear systems in block triangular form with unknown state delays neural networks nns are utilized to approximate and compensate for unknown functions in the system dynamics including the unknown bounds of the functions of delayed states the use of a separation technique removes the need for any assumption on the function of delayed states and allows the handling of multiple delays in each function of delayed states by combining the use of lyapunov krasovskii functionals and adaptive nn backstepping the proposed control guarantees that all closed loop signals remain bounded while the outputs converge to a neighborhood of the desired trajectories simulation results demonstrate the effectiveness of the proposed scheme

intermittency in random optical layers at total reflection the authors show that the reflection coefficient r of a succession of n random optical layers has a well defined but sample dependent limit when n to infinity the probability distribution or r at total reflection is found to depend only on imr xi 2 3 where xi 2 is the noise variance the analytical expressions appearing in this localisation problem are found to be the same as the universal functions describing intermittency with noise

colored noise induced chaotic array synchronization the effect of a time correlated gaussian noise on one dimensional arrays consisting of diffusively coupled chaotic cells is analyzed a resonance effect between the time scale of the chaotic attractor and the colored gaussian noise has been found as well depending on the number of cells coupling and noise strength an improvement of the synchronization or a poor synchronization between cells within the array can occur for some values of the time correlation these nonlinear cooperative effects are studied in terms of a linear stability analysis around the uniform synchronized behavior

adaptive output feedback control of nonlinear systems using neural networks a direct adaptive output feedback control design procedure is developed for highly uncertain nonlinear systems that does not rely on state estimation the approach is also applicable to systems of unknown but bounded dimension in particular we consider single input single output nonlinear systems whose output has known but otherwise arbitrary relative degree this includes systems with both parameter uncertainty and unmodeled dynamics the result is achieved by extending the universal function approximation property of linearly parameterized neural networks to model unknown system dynamics from input output data the network weight adaptation rule is derived from lyapunov stability analysis and guarantees that the adapted weight errors and the tracking error are bounded numerical simulations of an output feedback controlled van der pol oscillator coupled with a linear oscillator is used to illustrate the practical potential of the theoretical results

tracking control for nonaffine systems a self organizing approximation approach this paper considers tracking control for single input single output nonaffine dynamic systems a performance dependent self organizing approximation based approach is proposed the designer specifies a positive tracking error criterion the self organizing approximation based controller then monitors the tracking performance and adds basis elements only as needed to achieve the tracking specification even though the system is not affine the approach is defined such that the approximated function is independent of the control variable u stability is proved and the self organization is derived in a lyapunov based methodology to illustrate certain novel aspects of the proposed controller a numerical example is included

stochastic dynamics of the cubic map a study of noise induced transition phenomena the effects of finite amplitude additive noise on the dynamics generated by a one dimensional two parameter cubic map are considered the underlying deterministic system exhibits bistability and hysteresis and noise induced processes associated with these phenomena are studied if a bounded noise source is applied to this system trajectories may be confined to a finite region mechanisms are given for the merging transitions between different parts of this region and the eventual escape from it as the noise level is increased the noisy dynamics is also represented by an integral evolution operator with an equilibrium density function with finite support the operator s spectrum is determined as a function of map parameters and noise amplitude such noisy one dimensional maps can provide models for the study of noise induced phenomena described by stochastic differential equations

multidimensional rovella like attractors in a joint work with a castro v pinheiro both from federal univ of bahia and m j pacifico federal university of rio de janeiro we construct a multidimensional flow exhibiting a rovella like attractor a compact transitive invariant set with an equilibrium accumulated by regular orbits and a partially hyperbolic splitting of the tangent bundle with a multidimensional non uniformly expanding direction moreover this attractor has a physical measure with full support which is a u gibbs state as in the 3 dimensional rovella attractor this example is not robust this introduces a class of multidimensional dynamics where benedicks carleson arguments can be applied to get persistent non uniform expansion

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

average trajectories and fluctuations from noisy nonlinear maps the average trajectories and fluctuations around them resulting from an ensemble of noisy nonlinear maps are analyzed the bifurcation diagram for the average value obtained from the computer simulation of noisy maps ensemble is discussed first then a deterministic average equation of motion describing in an approximate way the time evolution of the average value and of the variance is analyzed numerically this equation predicts the existence of the bifurcation gap and of the exceptional attractors for special initial points the scaling properties of the average value and of the variance are obtained with the help of this equation

even the first iterate of a markov operator is contracting in anl2 norm a weightedl2 norm is introduced in which markov operators e g associated with noisy maps are contracting provided the kernel i e the transitional distribution is smooth enough this results in strong relaxational properties of noisy maps similar to this norm integral functionals appear useful when studying spatiotemporal chaos and random fields

a new type of intermittent transition to chaos an intermittent transition to chaos in the presence of symmetry is investigated statistical properties of intermittency are found including the case of external noise

adaptive neural control of mimo nonlinear state time varying delay systems with unknown dead zones and gain signs in this paper adaptive neural control is proposed for a class of uncertain multi input multi output mimo nonlinear state time varying delay systems in a triangular control structure with unknown nonlinear dead zones and gain signs the design is based on the principle of sliding mode control and the use of nussbaum type functions in solving the problem of the completely unknown control directions the unknown time varying delays are compensated for using appropriate lyapunov krasovskii functionals in the design the approach removes the assumption of linear functions outside the deadband as an added contribution by utilizing the integral lyapunov function and introducing an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead zone disturbance the closed loop control system is proved to be semi globally uniformly ultimately bounded simulation results demonstrate the effectiveness of the approach

stochastic sensitivity of the closed invariant curves for discrete time systems abstract discrete time nonlinear systems with invariant closed smooth curves forced by random disturbances are studied constructive approximations of the corresponding probabilistic distributions near these curves based on stochastic sensitivity functions are suggested for stochastic sensitivity matrix functions the linear equations are obtained for descriptive visualization of this technique a method of confidence domains is used the elaborated theory is illustrated on the example of the two dimensional model

on the dimensions of the spectral measure of symmetric binary substitutions substitution rules defined on the binary alphabet and invariant under the interchange of both participating symbols are considered typically the resulting symbolic sequences are weakly disordered and their fourier spectra are singular continuous exact relations are derived which express the correlation dimension of the multifractal spectral measure in terms of the entries in the substitution pattern recurrence relations are obtained for the finite length spectral sums and for the autocorrelation function

the onsager machlup theory for markov processes with discrete time parameter a characterisation of the detailed balance the onsager machlup om function is introduced for a spatially uniform system evolving in accordance with a discrete time markovian law circulation of probability flow is expressed in terms of the om function equivalence is established between the two conditions circulation identical to 0 and the fulfilment of the detailed balance

stochastic sensitivity analysis of noise induced intermittency and transition to chaos in one dimensional discrete time systems abstract we study a phenomenon of noise induced intermittency for the stochastically forced one dimensional discrete time system near tangent bifurcation in a subcritical zone where the deterministic system has a single stable equilibrium even small noises generate large amplitude chaotic oscillations and intermittency we show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium for the analysis of this system we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique an explicit formula for the value of the noise intensity threshold corresponding to the onset of noise induced intermittency is found on the basis of our approach a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given

global bifurcations to strange attractors in a model for skew varicose instability in thermal convection abstract we present a global bifurcation study of a four dimensional system of differential equations proposed by f h busse and coworkers modeling instabilities of convection rolls in the rayleigh benard experiment the rayleigh and prandtl numbers are two natural parameters on which the system depends we focus on a detailed mathematical study combining numerical pathfollowing and bifurcation analysis of chaotic dynamics and transitions to chaotic dynamics numerical continuation makes clear how homoclinic and heteroclinic bifurcations organize the bifurcation diagram in the parameter plane combined with a theoretical bifurcation analysis this explains the development of patterns and the creation of chaotic spatio temporal dynamics in the model the organizing centers such as heteroclinic cycles with resonance conditions among eigenvalues and homoclinic loops with geometric degeneracies inclination flips are identified and their unfoldings are analyzed this ties the creation of strange attractors of various geometric structures to codimension two global bifurcations

brief paper novel adaptive neural control design for nonlinear mimo time delay systems in this paper we address the problem of adaptive neural control for a class of multi input multi output mimo nonlinear time delay systems in block triangular form based on a neural network nn online approximation model a novel adaptive neural controller is obtained by constructing a novel quadratic type lyapunov krasovskii functional which not only efficiently avoids the controller singularity but also relaxes the restriction on unknown virtual control coefficients the merit of the suggested controller design scheme is that the number of online adapted parameters is independent of the number of nodes of the neural networks which reduces the number of the online adaptive learning laws considerably the proposed controller guarantees that all closed loop signals remain bounded while the output tracking error dynamics converges to a neighborhood of the origin a simulation example is given to illustrate the design procedure and performance of the proposed method

the influence of noise on the logistic model the behavior of the logistic system which is generated by the functionf x ax 1 x changes in an interesting way if it is perturbed by external noise it turns out that the chaotic behavior which was predicted by li and yorke for orbits of period 3 becomes visible and that a sequence of mergence transitions occurs at the critical parameter the change of the invariant probability density and the lyapunov exponents are examined numerically the power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete time analog of a chapman kolmogorov equation

a method for the numerical inversion of laplace transforms abstract in this paper a numerical inversion method for laplace transforms based on a fourier series expansion developed by durbin 5 is presented the disadvantage of the inversion methods of that type the encountered dependence of discretization and truncation error on the free parameters is removed by the simultaneous application of a procedure for the reduction of the discretization error a method for accelerating the convergence of the fourier series and a procedure that computes approximately the best choice of the free parameters suitable for a given problem the inversion method allows the adequate application of these procedures therefore in a big range of applications a high accuracy can be achieved with only a few function evaluations of the laplace transform the inversion method is implemented as a fortran subroutine

symbolic dynamics behind the singular continuous power spectra of continuous flows abstract we investigate a class of dissipative flows which possess singular continuous fractal power spectra the attractor for these flows contains a saddle point so that the times between the returns of an orbit onto a secant plane are unbounded as a result the poincare map does not reflect precisely the qualitative dynamics of the underlying flow the power spectrum of this map can be discrete we introduce the construction which takes into account that the time intervals between the intersections of the poincare plane can vary and construct a symbolic representation for the continuous process with logarithmic divergence in the distribution of such intervals spectral properties of the resulting symbolic sequences are shown to reproduce rather well those of the original flow

intermittency in the presence of noise we analyse iterations of maps on an interval with an added noise term in the neighbourhood of an intermittency threshold we rigorously derive a universal scaling function for the laminar time expressed as a function of the distance from the threshold and the variance of the noise

stochastic shilnikov maps fluctuations in dynamical systems may give rise naturally to stochastic maps which in the absence of fluctuations reduce to maps similar to those considered by shilnikov 1970 noise appears in a highly non trivial manner the effect of the noise on the chaotic dynamics is studied it is shown that the different noise terms can give rise to radically different behaviour

asymptotic theory of multidimensional chaos a delay differential equation u t u t f u t 1 0 t and its generalization are investigated in the limit 0 when the attractor s dimension increases infinitely it is shown that a number of statistical characteristics are asymptotically independent of as for the attractor it can be regarded as a direct product ofo 1 equivalent subattractors their statistical characteristics being asymptotically independent of the results enable one to predict some characteristics of the attractor with fractal dimensiond 1 for the case 1 when they are inaccessible numerically the approach developed seems to be applicable for a wide class of spatiotemporal systems

influence of noise on dynamics with fractal fourier spectra abstract in deterministic dynamics the processes with fractal power spectra occupy the intermediate position between ordered and chaotic temporal behavior we study numerically the influence of different kinds of noise upon these processes in general under the action of noise the spectral measure becomes absolutely continuous and the correlations decay however during the certain intermediate period the autocorrelation retains the pattern which is typical for dynamics with fractal spectra duration of this period is related to the noise intensity by the power law

a generic intermittency model and its 1 d meta map power laws invariants and the succession of laminar sequences abstract intermittent behaviour has been found in many systems able to switch between two different dynamic states e g between long laminar phases and short chaotic bursts despite the apparently high dimensional complexity certain one dimensional 1 d maps are known to mimic properties of such dynamics to these belongs the iterative map x n 1 i x n i x n i z mod 1 giving rise to long laminar lengths the statistics of the laminar lengths are of special interest starting from this map we are interested in the values of x 0 i which arise after passing through the modulo operation these determine the laminar lengths uniquely a 1 d meta map x 0 i f x 0 i 1 is derived heuristically it is used to calculate statistical properties of the laminar phases our results show an improvement in the behaviour of short and very long laminar phases as compared to earlier analytical results introducing the concept of the generic starting value we find laminar phases not to be strictly independent of their predecessors

computing the topological entropy of maps we give an algorithm for determining the topological entropy of a unimodal map of the interval given its kneading sequence we also show that this algorithm converges exponentially in the number of letters of the kneading sequence

critical behaviour and intermittency in sinai s billiard abstract we study numerically the transition that occurs in the dynamics of sinai s billard when the circles come into contact singular behaviours are predicted by a local analysis of the region of phase space where the motion is nearly integrable the link with intermittency is discussed we use a renormalization transformation and deduce the scaling laws we recover transposed in the language of critical phenomena certain predictions of the ergodic theory

tracking control for a class of unknown nonsquare mimo nonaffine systems a deep rooted information based robust adaptive approach the tracking control problem of multi input multi output mimo nonaffine dynamic systems is studied in this note a globally uniformly ultimately bounded guub stable tracking control solution for non square mimo nonaffine systems with unknown dynamics and disturbances is established different from most existing methods the proposed control design is based on the readily computable deep rooted information on the system and does not involve linearization or approximation and the resultant control algorithms are simple in structure inexpensive in computation user friendly in design and undemanding in implementation both theoretical analysis and numerical simulation verify the effectiveness and benefits of the proposed method

noise and bifurcations the influence of while noise on bifurcating dynamical systems is investigated using both fokker planck and functional integral methods noise leads to fuzzy bifurcations where physically relevant quantities become smooth functions of the bifurcation parameters we study dynamical and probabilistic quantities such as invariant measures liapunov exponents correlation functions and exit times the behavior of these quantities near the deterministic bifurcation point changes for distinct values of the control parameter therefore the very concept of bifurcation point becomes meaningless and must be replaced by the notion of bifurcation region

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

state space approach of two temperature generalized thermoelasticity of one dimensional problem abstract in this paper we will consider a half space filled with an elastic material which has constant elastic parameters the governing equations are taken in the context of the two temperature generalized thermoelasticity theory youssef h 2005a the dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity j appl math mech 26 4 4827 youssef h 2005b theory of two temperature generalized thermoelasticity ima j appl math 1 8 the medium is assumed initially quiescent laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions the general solution obtained is applied to a specific problem of a half space subjected to thermal shock and traction free the inverse laplace transforms are computed numerically using a method based on fourier expansion techniques some comparisons have been shown in figures to estimate the effect of the two temperature parameter

an experimental observation of a new type of intermittency abstract a novel hysteretic form of regular intermittency has been observed experimentally in a variant of the taylor couette problem where the fluid flow is confined between a stationary outer cylinder and a rotating inner cylinder and end plates the results cannot all be explained within the standard pomeau manneville classification of intermittency and an extension to this theory is proposed to account for the observed phenomena a satisfactory agreement is found between experiment and theory both for the reinjection burst frequency scaling law and the influence of noise on the regularity of the bursts

noisy precursors of nonlinear instabilities this paper studies the effect of external noise on systems displaying nonlinear instabilities of periodic orbits each class of instability is found to have its own characteristic signature as displayed by the power spectrum results are derived for each of the codimension one instabilities familiar from bifurcation theory

superlinearly scalable noise robustness of redundant coupled dynamical systems we illustrate through theory and numerical simulations that redundant coupled dynamical systems can be extremely robust against local noise in comparison to uncoupled dynamical systems evolving in the same noisy environment previous studies have shown that the noise robustness of redundant coupled dynamical systems is linearly scalable and deviations due to noise can be minimized by increasing the number of coupled units here we demonstrate that the noise robustness can actually be scaled superlinearly if some conditions are met and very high noise robustness can be realized with very few coupled units we discuss these conditions and show that this superlinear scalability depends on the nonlinearity of the individual dynamical units the phenomenon is demonstrated in discrete as well as continuous dynamical systems this superlinear scalability not only provides us an opportunity to exploit the nonlinearity of physical systems without being bogged down by noise but may also help us in understanding the functional role of coupled redundancy found in many biological systems moreover engineers can exploit superlinear noise suppression by starting a coupled system near not necessarily at the appropriate initial condition

homoclinic tangencies and non normal jacobians effects of noise in nonhyperbolic chaotic systems abstract a general study of the effects of noise interacting with the deterministic dynamics of nonhyperbolic chaotic 2 d maps and 3 flows is presented noise in these systems can have dramatic effects on the invariant measure of the system in regions of homoclinic tangencies perturbations are transported away from the neighborhood of the attractor leading to deformations which can be one to two orders of magnitude larger than the noise level a qualitative understanding is attained by a study of the universal properties of the unstable and stable foliation of the phase space in the vicinity of homoclinic tangencies through the investigation of nontrivial structures of the tangent space at homoclinic tangencies related to the non normal jacobians we obtain a quantitative description of these noise induced deformations of the attractor local expansion rates which are much larger than the maximal lyapunov exponent can be used as a measure of the system s structural instability under perturbations we exemplify our general results on the henon ikeda and duffing system a new and effective algorithm to calculate the homoclinic tangencies in the entire phase space based on the results of this paper is presented

a two dimensional thermoelasticity problem for thermomechanical shock with two relaxation times in this work a two dimensional problem of thermoelasticity with two relaxation times is introduced laplace and fourier transform techniques are used the resulting formulation is applied to a thermomechanical shock half space problem the solution in the transformed domain obtained by using a direct approach numerical inversion of both transforms is carried out to obtain the temperature stress and displacement distributions in the physical domain numerical results are represented graphically

reduced order observer in the sliding mode control of nonlinear nonaffine systems the note considers the variable structure control of nonlinear known nonaffine systems when the state vector is not completely available and the use of observers is required the strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems a new reduced order observer is proposed and conditions are found under which it is proven the convergence to the unique ideal solution of both system and observer the control problem is solved by forcing a sliding regime for the observer while satisfying an exponential stability criterion for the observation error state equation

adaptive fuzzy control of a class of siso nonaffine nonlinear systems this paper presents a direct adaptive fuzzy control scheme for a class of uncertain continuous time single input single output siso nonaffine nonlinear dynamic systems based on the implicit function theory the existence of an ideal controller that can achieve control objectives is firstly shown since the implicit function theory guarantees only the existence of the ideal controller and does not provide a way for constructing it a fuzzy system is employed to approximate this unknown ideal control law the adjustable parameters in the used fuzzy system are updated using a gradient descent adaptation algorithm this algorithm is designed in order to minimize a quadratic cost function of the error between the unknown ideal implicit controller and the used fuzzy control law the stability analysis of the closed loop system is performed using a lyapunov approach in particular it is shown that the tracking error converges to a neighborhood of zero the effectiveness of the proposed adaptive control scheme is demonstrated through the simulation of a simple nonaffine nonlinear system

internal fluctuations period doubling and chemical chaos the effects of internal molecular fluctuations on period doubling bifurcations and chaotic band merging are studied in a well stirred reactive chemical system where spatial degrees of freedom play no role the calculations are carried out using a stochastic model based on reactive lattice gas cellular automata molecular fluctuations arising from reactive collisions are incorporated in the model and the mass action rate equations are recovered in the mean field limit large system sizes can be studied and permit the investigation of the internal noise scaling structure in the periodic and chaotic regimes for the willamowski r ossler system the noise scaling exponent is consistent with that found for one dimensional quadratic maps subject to external noise the model provides a means to study small non equilibrium reacting systems where fluctuations may not be neglected

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

two mechanisms of the transition to chaos in finite dimensional models of convection abstract the onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation a connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point this connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem two routes are considered a well known mechanism suggested by lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle point the asymptotical properties of the latter route prove to be entirely determined by the above ratio

noisy one dimensional maps near a crisis i weak gaussian white and colored noise we study one dimensional single humped maps near the boundary crisis at fully developed chaos in the presence of additive weak gaussian white noise by means of a new perturbation like method the quasi invariant density is calculated from the invariant density at the crisis in the absence of noise in the precritical regime where the deterministic map may show periodic windows a necessary and sufficient condition for the validity of this method is derived from the quasi invariant density we determine the escape rate which has the form of a scaling law and compares excellently with results from numerical simulations we find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz 1 finally we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak gaussian noise within this extended class of noises and for single humped maps with any fixed orderz 0 of the maximum in the scaling law for the escape rate both the critical exponents and the scaling function are universal

noisy one dimensional maps near a crisis ii general uncorrelated weak noise the escape rate for one dimensional noisy maps near a crisis is investigated a previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise including multiplicative white noise as a special case for single humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived it predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order a simple heuristic explanation of this effect is given the escape rate is discussed in detail for noise distributions of levy dichotomous and exponential type in the latter case the rate is dominated by an exponentially leading arrhenius factor in the deep precritical regime however the preexponential factor may still depend more strongly than any power law on the noise strength

a stochastic galerkin approach to uncertainty quantification in poroelastic media recent concerns over the safety of oil and natural gas extraction fracking and carbon sequestration have driven the need to develop methods for uncertainty quantification for coupled subsurface flow and deformation processes traditional monte carlo methods are versatile but exhibit prohibitively slow convergence in this work we develop an intrusive polynomial chaos expansion method for biot s poroelasticity equations based on the galerkin projection with uniform and log normally distributed material parameters we analyze accuracy and efficiency of our method and compare it to the monte carlo method we verify exponential convergence of the stochastic galerkin approach

stochastic sensitivity analysis of noise induced order chaos transitions in discrete time systems with tangent and crisis bifurcations abstract we study noise induced order chaos transitions in discrete time systems with tangent and crisis bifurcations to study these transitions parametrically we suggest a generalized mathematical technique using stochastic sensitivity functions and confidence domains for randomly forced equilibria cycles and chaotic attractors this technique is demonstrated in detail for the simple one dimensional stochastic system in which points of crisis and tangent bifurcations are borders of the order window lying between two chaotic parametric zones a stochastic phenomenon of the extension and shift of this window towards crisis bifurcation point under increasing noise is presented and analyzed shifts of borders of this order window are found as functions of the noise intensity by our analytical approach based on stochastic sensitivity functions we construct a parametric diagram of chaotic and regular regimes for the stochastically forced system

chaos noise and synchronization reconsidered in a recent paper maritan and banavar phys rev lett 72 1451 1994 reported the synchronization of identical chaotic systems by additive noise we relate such a synchronization to the maximum lyapunov exponent of a single system and discuss the underlying mechanisms of the effect in the case of the lorenz equations the nonvanishing mean of the noise mimics a parameter change leading to synchronization for the logistic map a state dependence of the fluctuations is induced by the boundary conditions

on the correlation dimension of the spectral measure for the thue morse sequence using the relationship between the decay rate of autocorrelation and the characteristics of singular fourier spectra we show that the correlation dimension of the spectral measure for the infinite thue morse symbolic sequence equals 3 log 1 17 log 2 0 64298

the theory of individual based discrete time processes a general theory is developed to study individual based models which are discrete in time we begin by constructing a markov chain model that converges to a one dimensional map in the infinite population limit stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map from the chapman kolmogorov equation for the discrete time markov process we derive the analogues of the fokker planck equation and the langevin equation which are routinely employed for continuous time processes in particular a stochastic difference equation is derived which accurately reproduces the results found from the markov chain model stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map the proposed scheme is tested on stochastic models which have the logistic and ricker maps as their deterministic limits

a stochastic return map for stochastic differential equations a method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small amplitude o e additive gaussian colored noise the construction is valid provided the correlation time iso e oro 1 the effective noise in the return map has nonzeroo e 2 mean and is state dependent the method is applied to a model dynamical system illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics

a chaotic map with a flat segment can produce a noise induced order matsumoto and tsuda studied the effects of noise on chaos in a one dimensional belousov zhabotinsky bz map and found noise induced order that is an external noise destroys a chaotic behavior and produces some kind of order periodicities this phenomenon is very interesting in understanding the relation between chaos and natural phenomena the present paper proposes a unimodal piecewise linear map which has a flat segment it is shown numerically that the noise induced order can be observed in this simple map in the same way as the bz map these numerical results clarify the mechanism of noise induced order

positive lyapunov exponents in the kramers oscillator the maximum lyapunov exponent is computed numerically for the double well oscillator in a heat bath positive exponents are found in a wide range of friction coefficients in the low damping regime

attractors of convex maps with positive schwarzian derivative in the presence of noise abstract one dimensional maps have proved to be useful models for understanding the transition to turbulence we investigate a smooth perturbation of the well known logistic system in order to examine numerically the change in the bifurcation behavior which is observed when the schwarzian derivative is allowed to become positive we find coexistence of a fixed point attractor and a periodic or chaotic two band attractor the chaotic two band attractor can disappear by yielding a preturbulent state which will asymptotically settle down to a fixed point the chaotic behavior of some systems can be destroyed by arbitrarily small amounts of external noise the concept of diffusions is used to describe the sensitivity of attractors against external noise we also observe a direct transition from a fixed point to a chaotic one band attractor this can be interpreted as type iii intermittency of pomeau and manneville but with an almost linear scaling behavior of the lyapunov exponent

state space approach to generalized thermoelastic problem with thermomechanical shock in this work the state space formulation for one dimensional problem of generalized thermoelasticity with one relaxation time is introduced the resulting formulation together with the laplace transform technique is applied to a thermomechanical shock half space problem the solution in the transformed domain is obtained the inverse laplace transforms is evaluated numerically the results are obtained and represented graphically

noise induced order a new noise effect on chaos in one dimensional mappings is reported the transition from chaotic behavior to ordered behavior induced by external noise is observed in a certain class of one dimensional mappings this transition is clearly shown in terms of the lyapunov number entropy power spectrum and the nature of orbits

non symmetric forms of non linear vibrations of flexible cylindrical panels and plates under longitudinal load and additive white noise abstract parametric non linear vibrations of flexible cylindrical panels subjected to additive white noise are studied the governing marguerre equations are investigated using the finite difference method fdm of the second order accuracy and the runge kutta method the considered mechanical structural member is treated as a system of many infinite number of degrees of freedom dof the dependence of chaotic vibrations on the number of dofs is investigated reliability of results is guaranteed by comparing the results obtained using two qualitatively different methods to reduce the problem of pdes partial differential equations to odes ordinary differential equations i e the faedo galerkin method in higher approximations and the 4th and 6th order fdm the cauchy problem obtained by the fdm is eventually solved using the 4th order runge kutta methods the numerical experiment yielded for a certain set of parameters the non symmetric vibration modes forms with and without white noise in particular it has been illustrated and discussed that action of white noise on chaotic vibrations implies quasi periodicity whereas the previously non symmetric vibration modes are closer to symmetric ones

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

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

adaptive neural control of high order uncertain nonaffine systems a transformation to affine systems approach abstract this brief investigates the adaptive neural network nn control of a class of high order nonaffine nonlinear systems with completely unknown dynamics since the control terms appear within the unknown nonlinearity traditional control schemes and stability analysis are usually rendered extremely complicated our main contribution includes a novel system transformation that converts the nonaffine system into an affine system through a combination of a low pass filter and state transformation as a result the state feedback control of the nonaffine system can be viewed as the output feedback control of an affine system in normal form the transformed system becomes linear with respect to the new input while the traditional backstepping approach is not needed thus allowing the synthesis to be extremely simplified it is theoretically proven that all the signals in the closed loop system are uniformly ultimately bounded uub simulation results are provided to demonstrate the performance of the developed controller

adaptive t s fuzzy neural modeling and control for general mimo unknown nonaffine nonlinear systems using projection update laws this paper describes a novel design of an on line takagi sugeno t s fuzzy neural controller for a class of general multiple input multiple output mimo systems with unknown nonlinear functions and external disturbances instead of modeling the unknown systems directly the t s fuzzy neural model approximates a virtual linearized system vls of a real system with modeling errors and external disturbances compared with previous approaches the main contribution of this paper is an investigation of more general mimo unknown systems using on line adaptive t s fuzzy neural controllers in this paper we also use projection update laws which generalize the projection algorithm to tune the adjustable parameters this prevents parameter drift and ensures that the parameter matrix is bounded away from singularity we prove that the closed loop system controlled by the proposed controller is robust stable and the effect of all the modeling errors and external disturbances on the tracking error can be attenuated finally two examples covering four cases are simulated in order to confirm the effectiveness and applicability of the proposed approach in this paper

direct adaptive self structuring fuzzy controller for nonaffine nonlinear system a direct adaptive state feedback controller for highly nonlinear systems is proposed this paper considers uncertain or ill defined nonaffine nonlinear systems and employ a static fuzzy logic system fls with an on line structuring algorithm the fls approximates and adaptively cancels an unknown plant nonlinearity a control law and adaptive laws for unknown fuzzy parameters and bounding constant are established so that the whole closed loop system is stable in the sense of lyapunov the tracking error is guaranteed to be uniformly asymptotically stable rather than uniformly ultimately bounded with the aid of an additional robustifying control term no a priori knowledge of an upper bound on an lumped uncertainty is required

intermittent diffusion in the presence of noise abstract the diffusion coefficient for intermittent diffusion in the presence of weak additive gaussian white noise is calculated the recent result by tsimring physica d 63 1993 41 is shown to be correct only in a certain limit of the parameter values

proportional integral controllers for minimum phase nonaffine in control systems we show that stabilizing tracking proportional integral pi controllers can be constructed for minimum phase nonaffine in control systems the constructed pi controller is an equivalent realization of an approximate dynamic inversion controller this equivalence holds only for the time response when applied to the unperturbed system even when restricted to unperturbed minimum phase linear time invariant systems their closed loop robustness properties differ this shows that in general properties that do not define the equivalence relation for systems controllers are not preserved under such equivalence transformations

quantum fluctuations and the lorenz equations the quantisation of the lorenz equations is shown to take the form of a set of two complex and one real stochastic differential equations with multiplicative noise phase diffusion is the dominant feature for small values of the noise quantities such as the probability of the modulus of the variables are unchanged from those in the classical lorenz equations moreover a fractal dimension can be associated with the stochastic process for sufficiently small noise for large noises there is a radical breakdown of this picture

spectral structure of intermittent chaos a statistical physical theory of power spectra of intermittent chaos near it onset point is developed and is applied to the type i intermittency generated by the collapse of a stable and an unstable cycle with a eigenfrequency 0 when the variance of phase jumps by turbulent bursts is small the power spectra around 0 are shown to consist of equally spaced sharp lines if the variance of durations of laminar motions between two consecutive bursts is small compared to the mean duration their envelope obeys an inverse power law 0 where 3 if the mean value of phase jumps is nonzero whereas 1 if 0 the relative fluctuation depends on the type of the reinjection into laminar motions by bursts as increase the lines become broad and overlap each other when the power spectra become the 1 spectra irrespective of

a simplified adaptive neural network prescribed performance controller for uncertain mimo feedback linearizable systems in this paper the problem of deriving a continuous state feedback controller for a class of multiinput multioutput feedback linearizable systems is considered with special emphasis on controller simplification and reduction of the overall design complexity with respect to the current state of the art the proposed scheme achieves prescribed bounds on the transient and steady state performance of the output tracking errors despite the uncertainty in system nonlinearities contrary to the current state of the art however only a single neural network is utilized to approximate a scalar function that partly incorporates the system nonlinearities furthermore the loss of model controllability problem typically introduced owing to approximation model singularities is avoided without attaching additional complexity to the control or adaptive law simulations are performed to verify and clarify the theoretical findings