parameter estimation of delay dynamical system from a scalar time series under external noise we introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series sufficient condition for synchronization is derived using krasovskii lyapunov theory this scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations one of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available we present numerical examples for mackey glass system with periodic delay time which are used to illustrate the validity of this scheme the corresponding numerical results and the effect of external noise are also studied

stochastic stabilization and destabilization of nonlinear differential equations abstract this paper is concerned with the stochastic stabilization and destabilization of nonlinear differential equations which more precisely further develops the general theory of stochastic stabilization and destabilization that was studied by mao 1994 4 and appleby et al 2008 1 the proposed results are of substantial improvements and reveal the more fundamental principle for stochastic stabilization and destabilization of dynamical systems which is also verified with examples of application to mass action systems

statistical and stabilizability properties of equivalent discrete time systems with stochastic parameters this paper considers the influence of the statistics of a scalar continuous time system with white stochastic parameters on the statistics the correlation and the stabilizability of the corresponding scalar equivalent discrete time system by means of numerical examples some insight into the behaviour of the equivalent discrete time system is given furthermore these examples illustrate the crucial influence of the correlation between the state and control dependent noise of the continuous time system on the stabilizability of the equivalent discrete time system

stabilization of a plane periodic channel flow by noise wall normal controllers the steady state flows in a 2 d channel with periodic conditions along the axis are stabilized by a boundary feedback stochastic controller with vertical velocity observation which acts on the normal component of velocity only

lyapunov exponents and rotation number of two dimensional systems with telegraphic noise exact expressions for the exponential growth rates lyapunov exponents and rotation numbers of two dimensional systems perturbed by telegraphic noise are derived asymptotic expansions for small large slow and fast noise and the white noise limit are constructed then results are applied to the random oscillator stochastic stability in particular stabilization by noise and stochastic bifurcation

suppression and stabilisation of noise in this article we investigate the stochastic suppression and stabilisation of nonlinear systems given an unstable differential equation in which f satisfies the one sided polynomial growth condition we introduce two brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation this article shows that appropriate may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable

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

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

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 complete list of invariants for defective crystals the classical theory of continuous distributions of dislocations has traditionally focused on the burgers vectors and the dislocation density tensor as descriptions of defectiveness we prove that generally there is an infinite number of tensor densities with similarly descriptive properties and that there is a functional basis for this list which strictly includes the burgers vectors and dislocation density moreover the changes of state which preserve these densities turn out to represent slip in certain surfaces associated with crystal geometry so that the basic mechanism of plasticity emerges naturally from abstract ideas which neither anticipate nor involve the kinematics of particular types of crystal defects

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

almost sure exponential stability of hybrid stochastic functional differential equations abstract this paper is concerned with the almost sure exponential stability of the n dimensional nonlinear hybrid stochastic functional differential equation sfde d x t f 1 x t t r t t d t g 2 x t t r t t d b t where x t x t u u 0 is a c 0 r n valued process b t is an m dimensional brownian motion while r t is a markov chain we show that if the corresponding hybrid stochastic differential equation sde d y t f y t r t t d t g y t r t t d b t is almost surely exponentially stable then there exists a positive number such that the sfde is also almost surely exponentially stable as long as we also describe a method to determine which can be computed numerically in practice

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

correlation free forms for nonlinear stochastic systems abstract an asymptotic method for analysis of nonlinear systems with wide band and high frequency stochastic perturbation is developed correlation free ito limiting equations are derived and a generalization of the wong zakai correction for the drift term is obtained a correction for the diffusion term is also shown to exist several illustrative examples are considered

correlation anti correlation transition by state dependent poisson noise noise may induce some degrees of order in many biological and complex systems we present an example of a noise dependent transition from correlation to anti correlation of the states of a bivariate system coupled by state dependent poisson noises by varying the degree of dependence of the rate of jumps on the other variable the system undergoes different degrees of correlation from independence to perfect correlation which in this case coincides with perfect synchronization in the limit of the deterministic periodic case

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

configurational stress yield and flow in rate independent plasticity the role of configurational stress in yield and plastic flow is discussed for a macroscopic model of rate independent finite strain plasticity the model is based on the traditional elastic plastic decomposition of the deformation gradient on integral balance laws and on thermodynamically restricted rate independent constitutive relations its formulation emphasizes the intermediate configuration in both the development of constitutive relations and the expression of balance laws in addition to the usual balance laws a couple balance is included to represent the action of plastic couples in the intermediate configuration in particular it is shown that the internal couple decomposes into a non dissipative configurational stress and a dissipative couple that resists plastic flow the couple balance thus determines a relation between the configurational stress and the plastic flow resistance a relation that can be interpreted as a generalized yield condition a dissipation function is introduced and a maximum dissipation criterion is used to obtain additional constitutive restrictions which lead to a counterpart in the intermediate configuration of the classical normality conditions the versatility of the framework is illustrated by applying it to rigid plastic flow in which case a nonlinear generalization of the classical levy von mises theory is obtained

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

brief paper exponential stabilization of the linearized navier stokes equation by pointwise feedback noise controllers we address the problem of exponential stabilization in probability of the linearized navier stokes equations in an equilibrium point this is done by designing a linear stochastic feedback controller with support in a point or on a discrete set of points of the domain this controller consists of a steady state impulse component with support in a finite set of points modulated by an unsteady feedback noise controller

the uniform convergence of thin plate spline interpolation in two dimensions let f be a function from cal r 2 to cal r that has square integrable second derivatives and let s be the thin plate spline interpolant to f at the points underline v i i 1 2 ldots n in cal r 2 we seek bounds on the error f underline x s underline x when underline x is in the convex hull of the interpolation points or when underline x is close to at least one of the interpolation points but need not be in the convex hull we find for example that if underline x is inside a triangle whose vertices are any three of the interpolation points then f underline x s underline x is bounded above by a multiple of h where h is the length of the longest side of the triangle and where the multiplier is independent of the interpolation points further if cal d is any bounded set in cal r 2 that is not a subset of a single straight line then we prove that a sequence of thin plate spline interpolants converges to f uniformly on cal d specifically we require h rightarrow 0 where h is now the least upper bound on the numbers d underline x cal v underline x in cal d and where d underline x cal v underline x in cal r 2 is the least euclidean distance from underline x to an interpolation point our method of analysis applies integration by parts and the cauchy schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation

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

stability of a stochastic two dimensional non hamiltonian system we study the top lyapunov exponent of the response of a two dimensional non hamiltonian system driven by additive white noise the origin is not a fixed point for the system however there is an invariant measure for the one point motion of the system in this paper we consider the stability of the two point motion by khasminskii s method of linearization along trajectories the specific system we consider is the third order truncated normal form of the unfolding of a hopf bifurcation we show that in the small noise limit the top lyapunov exponent always approaches zero from below and is thus negative for noise sufficiently small we also show that there exist open sets of parameters for which the top lyapunov exponent is positive thus the two point motion can be either stable or unstable while the stationary density that describes the one point motion always exists

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

striping of nematic elastomers abstract we consider a recent experiment of kundler and finkelmann macromol rapid commun 16 1995 679 who subjected an aligned specimen of nematic elastomer to uniaxial extension and observed the formation of striped domains in so doing we apply the general theory for nematic elastomers developed by anderson et al j elast 56 1999 33 and work with an energy density that combines the effects included in the molecular statistical theory of nematic rubber elasticity with the oseen zocher frank theory of nematic curvature elasticity assuming that the deformation and orientation fields remain in plane we arrive at a boundary value problem which admits solutions corresponding to striped states we use elementary aspects of bifurcation theory to explore the local stability of these solutions we also obtain analytical estimates for the energy and thickness of interstripe domain walls as functions of imposed extension and compare these with numerical predictions

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

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

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

improved accuracy of lp approximation to derivatives by radial basis function interpolation the present paper considers the approximation to a function and its derivatives by radial basis function interpolation and its derivatives respectively on the sobolev space it is known that due to edge effects we lose some accuracy near the boundary thus the goal of this paper is to show that the convergence rate of the approximation error can be at least doubled when a certain boundary condition is met

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

noise suppresses explosive solutions of differential systems a new general polynomial growth condition abstract recently liu and shen 2012 have discussed the problem of suppression explosive solutions by noise for nonlinear deterministic differential system with coefficients satisfying a general one sided polynomial growth condition liu and shen introduced the brownian noise feedback x t x t d b t to the nonlinear differential system and they showed that the corresponding stochastic system has a unique global solution however liu and shen s condition is still very restrictive and many more general nonlinear differential systems fail this paper aims to introduce a new more general polynomial growth condition which can be applied to more nonlinear differential systems under the new polynomial growth condition we show that the considered stochastic system has a unique global solution although the corresponding deterministic system may explode in a finite time moreover we show that the flexible stochastic feedback g x t can guarantee the solution not only be limited in the sense of moment but also grows at most polynomially finally we provide two examples to illustrate superiority of the new general polynomial growth condition

stability measures in metastable states with gaussian colored noise we present a study of the escape time from a metastable state of an overdamped brownian particle in the presence of colored noise generated by ornstein uhlenbeck process we analyze the role of the correlation time on the enhancement of the mean first passage time through a potential barrier and on the behavior of the mean growth rate coefficient as a function of the noise intensity we observe the noise enhanced stability effect for all the initial unstable states used and for all values of the correlation time tau c investigated we can distinguish two dynamical regimes characterized by weak and strong correlated noises depending on the value of tau c with respect to the relaxation time of the 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

modified kramers formulas for the decay rate in agreement with dynamical modeling abstract accuracy of the kramers approximate formulas for the thermal decay rate of the metastable state is studied for the anharmonic shapes of the potential pocket and the barrier this is done by the comparison with the quasistationary rate resulting from the dynamical modeling disagreement between the kramers rate and the dynamical one is shown to reach 15 in the cases when much better agreement is expected corrections to the kramers formulas accounting for the higher derivatives of the potential are obtained the small parameters are the ratios of the thermal energy to the stiffnesses at the extremes of the potential the distance between the potential barrier and the absorptive border is accounted for as well this corrected kramers rate is demonstrated to agree with the dynamical rate typically within 2 probably the most interesting result is that despite the corrections are derived in the case of the overdamped brownian motion the above 2 agreement holds even in the case of medium friction

regularization and stabilization of randomly switching dynamic systems this work concerns regularization or explosion suppression and stabilization of randomly switching dynamic systems by selecting suitable input functions starting with a randomly switching system modulated by a continuous time markov chain our aim is to find a feedback control input function so that the system becomes stable but before addressing the stabilization issue another problem needs to be resolved first because of the fast growth in the continuous state variable the solution of the system has a finite explosion time with probability one to ensure the existence of global solutions a feedback control a noise perturbation is added to make the system regular then another feedback control a second perturbing noise is added to ensure the stability of the resulting systems owing to the nonlinearity closed form solutions are often virtually impossible to obtain thus a discrete time approximation algorithm is constructed under simple conditions a suitably interpolated sequence of the

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

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

synchronization of random linear maps abstract we study synchronization of random one dimensional linear maps for which the lyapunov exponent can be calculated exactly certain aspects of the dynamics of these maps are explained using their relation with a random walk we confirm that the lyapunov exponent changes sign at the complete synchronization transition we also consider partial synchronization of nonidentical systems it turns out that the way partial synchronization manifests depends on the type of differences in lyapunov exponent or in contraction points between the systems the crossover from partial to complete synchronization is also examined

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 stability study of a new explicit numerical scheme for a system of differential equations with a large skew symmetric component explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions implicit schemes allow a larger time step but require more computations when the differential equations are dominated by a skew symmetric component the problem is not stiffness in the sense that the size of the eigenvalues are unequal rather that the real eigenvalues are dominated by imaginary eigenvalues a skew dominated system of this type may be seen in magnetohydrodynamics and in control problems we present and compare analytical results for stable time step limits for several explicit methods including the super time stepping method of alexiades amiez and gremaud which is an explicit runge kutta method for parabolic partial differential equations and a new method modeled on a predictor corrector scheme with multiplicative operator splitting this new explicit method presented in regular and super time stepping form increases stability without forcing the step size to zero

stochastic bifurcations and coherencelike resonance in a self sustained bistable noisy oscillator we investigate the influence of additive gaussian white noise on two different bistable self sustained oscillators duffing van der pol oscillator with hard excitation and a model of a synthetic genetic oscillator in the deterministic case both oscillators are characterized with a coexistence of a stable limit cycle and a stable equilibrium state we find that under the influence of noise their dynamics can be well characterized through the concept of stochastic bifurcation consisting in a qualitative change of the stationary amplitude distribution for the duffing van der pol oscillator analytical results obtained for a quasiharmonic approach are compared with the result of direct computer simulations in particular we show that the dynamics is different for isochronous and anisochronous systems moreover we find that the increase of noise intensity in the isochronous regime leads to a narrowing of the spectral line this effect is similar to coherence resonance however in the case of anisochronous systems this effect breaks down and a new phenomenon anisochronous based stochastic bifurcation occurs

the asymptotic stability of a noisy non linear oscillator abstract the purpose of this work is to obtain an approximation for the top lyapunov exponent the exponential growth rate of the response of a single well kramers oscillator driven by either a multiplicative or an additive white noise process to this end we consider the equations of motion as dissipative and noisy perturbations of a two dimensional hamiltonian system a perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation we show analytically that the top lyapunov exponent is positive and for small values of noise intensity e and dissipation e the exponent grows in proportion with e1 3

constructive effects of noise in homoclinic chaotic systems many chaotic oscillators have coherent phase dynamics but strong fluctuations in the amplitudes at variance with such a behavior homoclinic chaos is characterized by quite regular spikes but strong fluctuation in their time intervals due to the chaotic recurrence to a saddle point we study influences of noise on homoclinic chaos we demonstrate both numerically and experimentally on a co2 laser various constructive effects of noise including coherence resonance noise induced synchronization in uncoupled systems and noiseenhanced phase synchronization deterministic resonance with respect to signal frequency and stochastic resonance versus noise intensity in response to weak signals the peculiar sensitivity of the system along the weak unstable manifold of the saddle point underlines the unified mechanism of these nontrivial and constructive noise induced phenomena

stochastic stabilization of differential systems with general decay rate some sufficient conditions concerning stability of solutions of stochastic differential evolution equations with general decay rate are first proved then these results are interpreted as suitable stabilization ones for deterministic and stochastic systems also they permit us to construct appropriate linear stabilizers in some particular situations

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

stabilization of linear systems by rotation we introduce the concept of stabilization by rotation for deterministic linear systems with negative trace this concept encompasses the well known concept of vibrational stabilization introduced by meerkov in the 1970s and is a deterministic version of stabilization by noise for stochastic systems as introduced by arnold and coworkers in the 1980s it is shown that a linear system with negative trace can be stabilized by adding a skew symmetric matrix multiplied by a suitable scalar so called gain function possibly a constant which is su ciently large to overcome the problem of what is su ciently large we also present a servo mechanism which which tunes the gain function by learning from the trajectory until flnally the trajectory tends to zero this approach allows to show that one of meerkov s assumptions for vibrational stabilization is super uous moreover while meerkov as well as arnold and coworkers assume that a stabilizing periodic function or the noise has su ciently large frequency and amplitude we also provide a servo mechanism to determine this function dynamically in a deterministic setup

proximity to periodic windows in bifurcation diagrams as a gateway to coherence resonance in chaotic systems we show that chaotic states situated in the proximity of periodic windows in bifurcation diagrams are eligible for the observation of coherence resonance in particular additive gaussian noise of appropriate intensity can enhance the temporal order in such chaotic states in a resonant manner results obtained for the logistic map and the lorenz equations suggest that the presented mechanism of coherence resonance is valid beyond particularities of individual systems we attribute the findings to the increasing attraction of imminent periodic orbits and the ability of noise to anticipate their existence and use a modified wavelet analysis to support our arguments

relationship between directed percolation and the synchronization transition in spatially extended systems we study the nature of the synchronization transition in spatially extended systems by discussing a simple stochastic model an analytic argument is put forward showing that in the limit of discontinuous processes the transition belongs to the directed percolation dp universality class the analysis is complemented by a detailed investigation of the dependence of the first passage time for the amplitude of the difference field on the adopted threshold we find the existence of a critical threshold separating the regime controlled by linear mechanisms from that controlled by collective phenomena as a result of this analysis we conclude that the synchronization transition belongs to the dp class also in continuous models the conclusions are supported by numerical checks on coupled map lattices too

noise induced unstable dimension variability and transition to chaos in random dynamical systems results are reported concerning the transition to chaos in random dynamical systems in particular situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle which can be expected in any periodic window of a nonlinear dynamical system under noise the asymptotic attractor of the system can become chaotic as characterized by the appearance of a positive lyapunov exponent generic features of the transition include the following 1 the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions unstable dimension variability and this nonhyperbolicity develops as soon as the attractor becomes chaotic 2 for systems described by differential equations the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero lyapunov exponent after the noisy attractor becomes chaotic and 3 the largest lyapunov exponent becomes positive from zero in a continuous manner and its scaling with the variation of the noise amplitude is algebraic formulas for the scaling exponent are derived in all dimensions numerical support using both low and high dimensional systems is provided

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 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

l p error estimates for shifted surface spline interpolation on sobolev space the accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant f is reasonably smooth however for functions which have smoothness below a certain order associated with the basis function no approximation power has yet been established hence the purpose of this study is to discuss the lp approximation order 1 p of interpolation to functions in the sobolev space wpk with k max 0 d 2 d p we are particularly interested in using the shifted surface spline which actually includes the cases of the multiquadric and the surface spline moreover we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met

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

regularization by noise for stochastic hamilton jacobi equations we study regularizing effects of nonlinear stochastic perturbations for fully nonlinear pde more precisely path by path l infty bounds for the second derivative of solutions to such pde are shown these bounds are expressed as solutions to reflected sde and are shown to be optimal

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

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

phase diagram of the random frequency oscillator the case of ornstein uhlenbeck noise abstract we study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an ornstein uhlenbeck noise this system undergoes a noise induced bifurcation when the amplitude of the noise grows the critical curve that separates the absorbing phase from an extended non equilibrium steady state corresponds to the vanishing of the lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy we derive various expressions for this lyapunov exponent by using different approximation schemes this allows us to study quantitatively the phase diagram of the random parametric oscillator

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

brief paper almost sure exponential stabilisation of stochastic systems by state feedback control so far a major part of the literature on the stabilisation issues of stochastic systems has been dedicated to mean square stability this paper develops a new class of criteria for designing a controller to stabilise a stochastic system almost surely which is unable to be stabilised in mean square sense the results are expressed in terms of linear matrix inequalities lmis which are easy to be checked in practice by using matlab toolbox moreover the control structure in this paper appears not only in the drift part but also in the diffusion part of the underlying stochastic system

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

internal stabilization of the oseen stokes equations by stratonovich noise abstract the design of a stratonovich noise feedback controller with support in an arbitrary open subset o 0 of o is described this exponentially stabilizes in probability that is with probability one the oseen stokes systems in a domain o r d d 2 3 this completes the stabilization results from the author s work barbu 2011 3 which is concerned with design of an ito noise stabilizing controller

on the stationary lp approximation power to derivatives by radial basis function interpolation in this paper we consider approximation to derivatives of a function by using radial basis function interpolation most of well known theories for this problem provide error analysis in terms of the so called native space say c f however if a basis function f is smooth the space c f is extremely small thus the purpose of this study is to extend this result to functions in the homogenous sobolev space

stationary energy probability density of oscillators driven by a random external force we derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive gaussian noise our study focuses on two cases i a harmonic oscillator subjected to gaussian colored noise with an arbitrary correlation function and ii nonlinear oscillators with a general potential driven by gaussian white noise we also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy to illustrate our general results we consider specifically the case of exponentially correlated noise for i and power law and bistable potentials for ii our theoretical results are substantiated by langevin simulations between the kinetic energy and the potential energy of the oscillator our studies focus on two cases first we consider a harmonically bound particle driven by general additive gaussian colored noise second we consider a particle in a general potential driven by gaussian white noise in both cases we develop a general theory and study specific illustrative examples the paper is organized as follows in sec ii we formulate the langevin equations for a particle with inertia in a general potential driven by an additive random force we also derive the general relation between the stationary energy pdf and the stationary pdf for the particle s position and velocity section iii is devoted to the motion of a particle in a harmonic potential driven by colored gaussian noise with an arbitrary correlation function the stationary energy pdf as well as the first and second moments of the energy is derived analytically we apply these results to the case of exponentially correlated gaussian noise and to gaussian white noise for these specific cases we also determine the partition of the energy between kinetic and potential energy we analyze the effect of the potential shape on the stationary energy pdf for oscillators driven by an additive gaussian white noise in sec iv w e apply our analytical results to two specific cases a general power law potential and a bistable potential the first type of oscillator without friction and with additive noise 36 and multiplicative noise 37 has recently been studied in the context of anomalous diffusion the second type the duffing oscillator has been investigated as an alternative to the linear oscillator for harvesting energy from external random sources 28 29 35 38 39

a stochastic model for the evolution of bone metastasis persistence and recovery abstract bone metastasis is a complex disease that modifies the natural dynamics of the bone remodeling process which is vital for maintaining a healthy bone structure nowadays there are deterministic differential models that describe the bone metastasis evolution by considering interaction mechanisms between cancer cells and basic bone cells such as osteoclasts and osteoblasts based on these deterministic models a stochastic differential model is proposed to describe the bone metastasis evolution in order to analyze the effect of environmental fluctuations inclusions conditions for the existence and uniqueness of a nonnegative solution of the proposed stochastic model are given to characterize whether the invasion of cancer persists or is eradicated finally we illustrate different dynamical behaviors of the stochastic solution by means of numerical simulations

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

convergence rate determined for gradient based adaptive estimators abstract a convergence rate estimate is derived for the homogeneous gradient based adaptive linear estimator algorithm this estimate involves the eigenvalues of the regression vector covariance matrix yielding a useful measure for the choice of input signals for adaptive parameter estimation the connection between this criterion and those more familiar from nonadaptive system identification is made and comparisons are drawn between the two areas