controlling chaotic dynamical systems we review the major ideas involved in the control of chaos we present the ott grebogi yorke ogy method of controlling chaos which is a particular case of the pole placement technique but which is the one leading to the shortest time to achieve the control of chaotic systems implementation using only measured time series in experimental settings is also described

the approximate tubular neighborhood theorem skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations under dimension and compactness assumptions in general the stratified approximate fibrations cannot be replaced by bundles and the teardrops cannot be replaced by mapping cylinder neighborhoods thus this is the best possible topological tubular neighborhood theorem in the stratified setting

controlling chaotic dynamical systems using time delay coordinates abstract the ott grebogi yorke control method is analyzed in the case that the attractor is reconstructed from a time series using time delay coordinates it turns out that the control formula of ogy should be modified in order to apply to experimental systems if time dekay coordinates are used we reveal that the experimental surface of section map does not only depend on the actual parameter but also on the preceding one in order to meet this dependence two modifications are introduced which lead to a better performance of the control to compare their control abilities they are applied to simulations of a duffing oscillator the issue of measurement noise is considered

jump processes on leaves of multibranching trees the p adic numbers have found applications in a wide range of diverse fields of scientific research in some applications the algebraic properties of p adics enter as an indispensable ingredient of the theory another class of applications has to do with hierarchical treelike systems in this context the applications are based on the well known correspondence between p adics and the trees with p branches emerging from every branching point then the algebraic structure does not enter and p adics are used merely as a labeling system for the tree branches we introduce a space of sequences denoted by sb suitable for labeling the trees with varying number of branches emerging from the branching points we introduce a non archimedean metric in sb and describe the basic topological properties of sb we also demonstrate that the known constructions of the stochastic processes on p adics carry over to the stochastic processes on sb and hence on the corresponding trees

adaptive recognition and control of chaos abstract exploiting the information provided by the local variation rates of a dynamical system we show how an adaptive algorithm is able to recognize and stabilize the unstable periodic orbits embedded in a chaotic attractor through additive corrections to the dynamics the relative weight of each correction is adjusted upon the size of the local expansion rate application of this method to chaotic lorenz and four dimensional roessler systems is discussed

an algorithm to control chaotic behavior in one dimensional maps abstract an algorithm is proposed whereby any chaotic transformation i e one merely possessing a dense orbit can be modified slightly to in order to attain a desired target probability density function f the algorithm can be used in practical examples such as controlling the long term population distribution of a species whose dynamics is described by a point transformation

on the optimality of the ott grebogi yorke control scheme abstract some of the characteristics of the ott grebogi yorke ogy control technique are presented as applied to nonlinear flows as distinct from nonlinear maps specifically we consider the case where the magnitude of the control parameter varies in time within each control cycle in proportion to a given function referred to as a basis function the choice of the basis function is shown to influence the basin of convergence for a given level of parameter variation in the ogy controller an algorithm for designing the optimal basis function is presented the optimal basis function is shown to be defined by a step function with potentially several jumps thus revealing the intrinsic power of a standard ogy technique that uses a single step function as a basis function two numerical applications of the optimal design technique to a duffing oscillator are also presented to show that the standard ogy technique may be significantly improved by making an optimal choice of a basis function copyright

utilization of soma and differential evolution for robust stabilization of chaotic logistic equation this paper deals with the utilization of two evolutionary algorithms self organizing migrating algorithm soma and differential evolution de for the optimization of the control of chaos this paper is aimed at an explanation on how to use evolutionary algorithms eas and how to properly define the advanced targeting cost function cf securing fast precise and mainly robust stabilization of selected chaotic system on a desired state for any initial conditions the role of ea here is as a powerful tool for an optimal tuning of control technique input parameters as a model of deterministic chaotic system the one dimensional discrete logistic equation was used the four canonical strategies of soma and six canonical strategies of de were utilized for each ea strategy repeated simulations were conducted to outline the effectiveness and robustness of used method and targeting cf securing robust solution satisfactory results obtained by both heuristic and the two proposed cost functions are compared with previous research given by different cost function designs

renormalization approach to optimal limiter control in 1 d chaotic systems optimal limiter control of chaos in 1 d systems is described by flat topped maps when we study the properties of this control by a bifurcation analysis of the latter we find partial universal behavior the optimality of the control method is expressed by an exponentially fast control onto selected periodic orbits making targeting algorithms idle

representations of the weyl group in spaces of square integrable functions with respect to p adic valued gaussian distributions we construct a representation of the weyl group in the p adic hilbert space of functions which are square integrable with respect to a p adic valued gaussian distribution the operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p adic numbers a surprising fact is that the radii of these balls are connected by an uncertainty relation which can be considered as a p adic analogue of the heisenberg uncertainty relations the p adic hilbert space representation of the weyl group is the basis for a calculus of pseudo differential operators and for an operator quantization over p adic numbers

optimization of local control of chaos by an evolutionary algorithm an evolutionary algorithm for optimizing local control of chaos is presented based on a lyapunov approach a linear control law and the state space region in which this control law is activated are determined in addition we study a relation between certain adjustable design parameters and a particular measure of the uncontrolled chaotic attractor in the state space region of control ssrc from this relation the objective function to be optimized is derived in that context we assume a linear control law to be given and optimize size and shape of the ssrc using an evolutionary algorithm it is shown by examples how the algorithm can also be applied to higher dimensional systems with possibly more than one positive lyapunov exponent 2000 elsevier science b v all rights reserved

asymptotical behavior of one class of p adic singular fourier integrals abstract we study the asymptotical behavior of the p adic singular fourier integrals j m t f m x p x t x f f m t t p t q p where f m d q p is a quasi associated homogeneous distribution generalized function of degree x x p 1 1 x and order m x 1 x and p x are a multiplicative a normed multiplicative and an additive characters of the field q p of p adic numbers respectively d q p is a test function m 0 1 2 c if re 0 the constructed asymptotics constitute a p adic version of the well known erdelyi lemma theorems which give asymptotic expansions of singular fourier integrals are the abelian type theorems in contrast to the real case all constructed asymptotics have the stabilization property

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

optimal control of nonlinear systems to given orbits abstract using optimal control techniques we derive and demonstrate the use of an explicit single step control method for directing a nonlinear system to a target orbit and keeping it there we require that control values remain near the uncontrolled settings the full nonlinearity of the problem in state space variables is retained the one step of the control is typically a composition of known or learned maps over a the time required to learn the state b the time to compute the control and c the time to apply the control no special targeting is required yet the time to control is quite rapid working with the dynamics of a well studied nonlinear electrical circuit we show how this method works efficiently and accurately in two situations when the known circuit equations are used and when control is performed only on a poincare section of the reconstructed phase space in each case because the control rule is known analytically the control strategy is computationally efficient while retaining high accuracy the target locations on the selected target trajectory at each control stage are determined dynamically by the initial conditions and the system dynamics and the target trajectory is an approximation to an unstable periodic orbit of the uncontrolled system a linear stability analysis shows that dissipation in the dynamical system is essential for reaching a controllable state

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

higher dimensional targeting this paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations such a procedure is called targeting previous work on targeting for chaotic attractors has been in the context of one and two dimensional maps here it is shown that targeting can also be done in higher dimensional cases the method is demonstrated with a mechanical system described by a four dimensional mapping whose attractor has two positive lyapunov exponents and a lyapunov dimension of 2 8 the target is reached by making very small successive changes in a single control parameter in one typical case 35 iterates on average are required to reach a target region of diameter 10 sup minus 4 as compared to roughly 10 sup 11 iterates without the use of the targeting procedure

optimal multi dimensional ogy controller abstract a technique which generalizes the standard ogy control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented the proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with ogy type controllers an optimal ogy control scheme designed for larger dimensional systems having improved convergence properties compared to the standard ogy controller is also presented numerical examples for the well known van der pol duffing oscillator are presented to illustrate the proposed control schemes

a new approach to controlling chaotic systems abstract given a piecewise monotonic transformation which is strongly transitive and preserves an invariant probability density function f a construction is described which modifies very slightly but which changes f significantly in order to possess desired properties this construction permits control of the global long term behavior of chaotic dynamical systems

associated and quasi associated homogeneous distributions generalized functions in this paper analysis of the concept of associated homogeneous distributions generalized functions is given and some problems related to these distributions are solved it is proved that in the one dimensional case there exist only associated homogeneous distributions of order k 1 we introduce a definition of quasi associated homogeneous distributions which is a natural generalization of the notion of associated homogeneous distributions and provide a mathematical description of all quasi associated homogeneous distributions and their fourier transforms it is proved that the class of quasi associated homogeneous distributions coincides with the class of distributions introduced in i m gel fand g e shilov generalized functions vol 1 properties and operations academic press new york 1964 chapter i 4 as the class of associated homogeneous distributions for the multidimensional case it is proved that f is a quasi associated homogeneous distribution if and only if it satisfies an euler type system of differential equations a new type of functions generated by quasi associated homogeneous distributions is defined

controlling chaotic transport through recurrence abstract transport times for a chaotic system are highly sensitive to initial conditions and parameter values we present a technique to find rough orbits epsilon chains that achieve a desired transport rapidly and which can be stabilized with small parameter perturbations the strategy is to build the epsilon chain from segments of a long orbit the point is that long orbits have recurrences in neighborhoods where faster orbits must also pass the recurrences are used as the switching points between segments the resulting epsilon chain can be refined by gluing orbit segments over the switching points providing that a local hyperbolicity condition is satisfied as an example we show that transport times for the standard map can be reduced by factors of 10 4 the techniques presented here can be easily generalized to higher dimensions and to systems where the dynamics is known only as a time series