adaptive coupling and enhanced synchronization in coupled phase oscillators we study the dynamics of an adaptive coupled array of phase oscillators the adaptive law is designed in such a way that the coupling grows stronger for the pairs which have larger phase incoherence the proposed scheme enhances the synchronization and achieves a more reasonable coupling dynamics for the network of oscillators with different intrinsic frequencies the synchronization speed and the steady state phase difference can be adjusted by the parameters of the adaptive law besides global coupling nearest neighbor ring coupling is also considered to demonstrate the generality of the method

a model reference adaptive control scheme for pure feedback nonlinear systems a model reference adaptive control mrac scheme is presented for nonlinear systems in a pure feedback canonical form with unknown parameters the present of parameter uncertainty in the system causes imperfect linearization i e it introduces nonlinear additive terms in the transformed coordinates under some mild technical assumptions global convergence of the output error is established for all initial estimates of the parameter vector lying in an open neighborhood of the true parameters in the parameter space

integral transforms and chebyshev like polynomials we use the integral representation method to derive the properties of chebyshev polynomials and of their generalized forms

structure of certain chebyshev type polynomials in onsager s algebra representation in this report we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable n state chiral potts model with a finite number of sizes the polynomials of low lying sectors are represented in two different forms one of which is directly related to the energy description of superintegrable chiral potts z n spin chain via the representation theory of onsager s algebra both two types of polynomials satisfy some n 1 term recurrence relations and nth order differential equations polynomials of one kind reveal certain chebyshev like properties here we provide a rigorous mathematical argument for cases n 2 3 and further raise some mathematical conjectures on those special polynomials for a general n

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

chaos synchronization and chaos anticontrol of a suspended track with moving loads abstract in this paper we first study the chaotic synchronization phenomenon of the suspended track with moving load system and then the transient response of chaos synchronization the lyapunov exponent is utilized to prove the chaos synchronization but under certain conditions it is more complicated next we study the synchronization of autonomous and non autonomous systems we find that slave systems cannot be synchronized with master system for certain time excitations no matter how large a is next the phase synchronization between them will be studied an application of chaos synchronization and secure communication is presented finally in order to increase the chaos phenomena we use anticontrol constant torque periodic torque periodic impulse signal time delay function and adaptive control are used successfully to control the state from order to chaos

the structure of quotients of the onsager algebra by closed ideals we study the onsager algebra from the ideal theoretic point of view a complete classification of closed ideals and the structure of quotient algebras are obtained we also discuss the solvable algebra aspect of the onsager algebra through the use of formal lie algebras

certain hypergeometric series related to the root system bc we show that the generalized hypergeometric function 2 f 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type bc in addition we prove that the jacobi polynomials of matrix argument correspond to the jacobi polynomials associated with the root system of type bc we also give a precise relation between jack polynomials and the jacobi polynomials associated with the root system of type a as a side result one obtains generalized hook length formulas which are related to harish chandra s c function and one can prove a conjecture due to macdonald relating two inner products on a space of symmetric functions

computing the moments of the density of zeros for orthogonal polynomials abstract a representation formula by means of the generalized lucas polynomials of first kind for the moments of the density of zeros of orthogonal polynomial sets defined by a three term recurrence relation is used to derive information directly from the entries of the jacobi matrix numerical computations are explicitly developed in some particular case

non linear dynamic analysis and control of chaos for a two degrees of freedom rigid body with vibrating support abstract the dynamic behaviors of a two degrees of freedom rigid body with vibrating support are studied in this paper both analytical and computational results are employed to obtain the characteristics of the system by using the lyapunov direct method the conditions of stability of the relative equilibrium position can be determined the incremental harmonic balance method ihb is used to find the stable and unstable periodic solutions for the strongly non linear system by applying various numerical results such as phase plane poincare map time history and power spectrum analysis a variety of periodic solutions and the phenomena of the chaotic motion is presented the effects of the changes of parameters in the system can be found in the bifurcation diagrams further the chaotic behavior is verified by using lyapunov exponents the modified interpolated cell mapping method micm is used to study the basins of attraction of periodic attractors and the fractal structure besides additions of a constant torque a periodic torque addition of dither signals delayed feedback control adaptive control and bang bang control are used to control the chaos phenomena effectively

complex macroscopic behavior in systems of phase oscillators with adaptive coupling abstract using recent dimensionality reduction techniques in large systems of coupled phase oscillators exhibiting bistability we analyze the complex macroscopic behavior arising when the coupling between oscillators is allowed to evolve slowly as a function of either macroscopic or local system properties for example we observe macroscopic excitability and intermittent synchrony in a system of time delayed kuramoto oscillators with hebbian and anti hebbian learning we demonstrate the robustness of our findings by considering systems with increasing complexity including time delayed oscillators with adaptive network structure and community interaction as well as a system with bimodally distributed frequencies

symmetric multivariate chebyshev polynomials abstract we introduce a general method for the symmetrization of univariate polynomials and use it to construct symmetric polynomials in r 1 variables that generalize the classical chebyshev polynomials of the first kind we show that on the set 1 1 r 1 such polynomials provide the best uniform approximations of the complete symmetric polynomials h x 0 x 1 x r by means of symmetric polynomials of total degree less than n where is a partition of n

discrete and continuous sine transform generalized to semisimple lie groups of rank two a new continuous group transform together with its discretization over a lattice of any density and admissible symmetry is defined and described for the four compact semisimple lie groups of rank 2 in the case of rank 1 the discrete version is the sine transform properties of the expansion functions of the transform called s functions are studied digital data processing is our motivating application

chaos synchronization and parameter identification for gyroscope system synchronization of chaos for a two degree of freedom heavy symmetric gyroscope system are studied in this paper because of the nonlinear terms of the system the system exhibits both regular and chaotic motions by lyapunov stability theory with control terms by adaptive control and by random optimization method the synchronization of two identical systems and tracking of the parameter of the systems are studied

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

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

yang lee zeros of the potts model the yang lee zeros of the three component ferromagnetic potts model in one dimension in the complex plane of an applied field are determined the phase diagram consists of a triple point where three phases coexist emerging from the triple point are three lines on which two phases coexist and which terminate at critical points yang lee edge singularity the zeros do not all lie on the imaginary axis but along the three two phase lines the model can be generalized to give rise to a tricritical point which is a new type of yang lee edge singularity gibbs phase rule is generalized to apply to coexisting phases in the complex plane

targeting chaos through adaptive control we describe adaptive control algorithms whereby a chaotic dynamical system can be steered to a target state with desired characteristics a specific implementation considered has the objective of directing the system to a state which is more chaotic or mixed than the uncontrolled one this methodology is easy to implement in discrete or continuous dynamical systems it is robust and efficient and has the additional advantage that knowledge of the detailed behavior of the system is not required

newton sum rules of polynomials defined by a three term recurrence relation abstract we derive a general formula for computing the newton sum rules of every polynomial belonging to a given polynomial set we use the following tools a recursive computation of the coefficients of the given polynomial in terms of the coefficients of the three term recurrence relation the generalized lucas polynomials of the first kind and last the newton girard formulas

learning to control dynamical behaviour abstract control of dynamical behaviour requires the transformation of one dynamical sequence into another the ability of a neural network to perform this task depends on the complexity of a time series involved and on the connectivity of the network used scaling estimates of the information extracted during learning and the resulting control error achieved are compared with simulations of period doubling maps

parametric adaptive control and parameter identification of low dimensional chaotic systems abstract the presented method of adaptive control corrects perturbations in the parameters of a system based on observations of its variables after a transient during which the desired parameter setting is reached exponentially quickly the system exhibits the goal behavior which can be a periodic quasiperiodic or chaotic solution of the equations of motion the length of the control transient scales inversely with the control strength and logarithmically with the magnitude of the original parameter perturbation it is shown that time series based models are capable of substituting for detailed knowledge of the system s nonlinear dynamics the method can be generalized to control more than one parameters to rely on observations of a single variable or to utilize a variety of control functions finally a method of parameter identification dual to the adaptive control is also discussed

global phase synchronization in an array of time delay systems we report the identification of global phase synchronization gps in a linear array of unidirectionally coupled mackey glass time delay systems exhibiting highly non phase coherent chaotic attractors with complex topological structure in particular we show that the dynamical organization of all the coupled time delay systems in the array to form gps is achieved by sequential synchronization as a function of the coupling strength further the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster we have confirmed these results by estimating instantaneous phases including phase difference average phase average frequency frequency ratio and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non phase coherent attractors the results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non phase coherent attractors directly without explicitly introducing the measure of phase

invariants geometriques des espaces localement symetriques et theoremes de multiplicateurs on prouve un analogue du theoreme de multiplicateurs de hormander mikhlin sur les espaces localement symetriques

adaptive control in nonlinear dynamics abstract we extend an adaptive control algorithm recently suggested by huberman and lumer to multi parameter and higher dimensional nonlinear systems this control mechanism is remarkably effective in returning a system to its original dynamics after a sudden perturbation in the system parameters changes the dynamical behaviour we find that in all cases the recovery time is linearly proportional to the inverse of control stiffness for small stiffness in higher dimensions there is an additional optimization problem since increasing stiffness beyond a certain value can retard recovery the control of fixed point dynamics in systems capable of a wide variety of dynamical behaviour is demonstrated we further suggest methods by which periodic motion such as limit cycles can be adaptively controlled and demonstrate the robustness of the procedure in the presence of additive background noise

onsager s algebra and the dolan grady condition in the non self dual case in this paper it is shown that a pair of operators that satisfy the dolan grady conditions generate an onsager algebra even in the case that they are not self dual briefly considered will be the implications for the spectra of superintegrable spin chains in various sectors

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

some remarks on multivariate chebyshev polynomials abstract several generalizations of the chebyshev polynomials in one variable to chebyshev polynomials in several variables are presented

regular and chaotic dynamic analysis and control of chaos of an elliptical pendulum on a vibrating basement the dynamic behavior of an elliptical pendulum subjected to external disturbance is studied in this paper the lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system by applying numerical results phase diagrams power spectrum period t maps and lyapunov exponents are presented to observe periodic and chaotic motions the effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams for global analysis the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping micm method finally several methods the delayed feedback control the addition of constant torque the addition of periodic force adaptive control algorithm and bang bang control are used to control chaos effectively

multivariate chebyshev polynomials we study multivariate chebyshev polynomials associated with root systems using properties of specialized singular elements corresponding to a root system we construct explicitly the measure weight function the latter ensures that these polynomials are orthonormal it defines the scalar product in the function space where multivariate u type chebyshev polynomials constitute a basis the obtained results are illustrated by constructing and studying 2 variate polynomials for root systems and

onsager s algebra and superintegrability the author considers the irreducible representations of the onsager algebra and shows that for a finite system possessing such an algebra leads to an ising like structure in the spectra of associated hamiltonians and transfer matrices the chiral potts model is considered as an example for transfer in the diagonal direction it is known to be superintegrable for transfer in the principal direction a new superintegrable solution manifold is found

interfacial tension of the chiral potts model we calculate the interfacial tension of then state chiral potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions our result is valid for the general physical model with positive boltzmann weights and at all subcritical temperatures the interfacial tension has been calculated previously for the superintegrable chiral potts model with skewed boundary conditions usingz invariance baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model we show that this is indeed the case

spontaneous synchronization of coupled oscillator systems with frequency adaptation we study the synchronization of kuramoto oscillators with all to all coupling in the presence of slow noisy frequency adaptation in this paper we develop a new model for oscillators which adapt both their phases and frequencies it is found that this model naturally reproduc es some observed phenomena that are not qualitatively produced by the standard kuramoto model such as long waiting times before the synchronization of clapping audiences by assuming a self consistent steady state solution we find three stability regimes for the coupling constant k separated by critical points k1 and k2 i for k k2 the incoherent state becomes unstable and only the synchronized state exists iii for k1 k2 the average waiting time grows logarithmically with the number of oscillators

simple adaptive feedback controller for identical chaos synchronization based on the invariance principle of differential equations a simple adaptive feedback scheme is proposed to strictly synchronize almost all chaotic systems unlike the usual linear feedback the variable feedback strength is automatically adapted to completely synchronize two almost arbitrary identical chaotic systems so this scheme is analytical and simple to implement in practice moreover it is quite robust against the effect of noise the famous lorenz and rossler hyperchaos systems are used as illustrative examples

adaptive control of spatially extended systems targeting spatiotemporal patterns and chaos we describe adaptive control algorithms whereby a spatially extended nonlinear system can be steered to a target state with desired spatiotemporal characteristics specifically we implement our control on a two dimensional coupled map lattice and successfully direct the system to desired targets ranging from spatiotemporal fixed points and regular spatial patterns to spatiotemporal chaos the proposed methodology entails monitoring the local neighborhood of only one arbitrary site in order to regulate the entire lattice further knowledge of the system s governing equations is not required we also demonstrate the success of this method in controlling an unstable elastic array a system of interest in engineering applications

chaos chaos control and synchronization of a gyrostat system abstract the dynamic behavior of a gyrostat system subjected to external disturbance is studied in this paper by applying numerical results phase diagrams power spectrum period t maps and lyapunov exponents are presented to observe periodic and choatic motions the effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams for global analysis the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping micm method several methods the delayed feedback control the addition of constant torque the addition of periodic force the addition of periodic impulse torque injection of dither signal control adaptive control algorithm aca control and bang bang control are used to control chaos effectively finally synchronization of chaos in the gyrostat system is studied

globally coupled noisy oscillators with inhomogeneous periodic forcing we study the collective properties of an array of nonlinear noisy oscillators driven by nonidentical periodic signals we consider the case of a globally coupled array of harmonically forced weakly nonlinear oscillators where there is a constant difference between the phases of the forcing signals applied to adjacent oscillators this system is a prototypical model of a nonlinear phased array receiver we derive analytical results for the array output in the limit of a large number of oscillators for the noise free and noisy cases numerical simulations show good agreement with the theoretical analysis

chiral potts model with skewed boundary conditions we obtain the transfer matrix functional relations for the chiral potts model with skewed boundary conditions and find that they are the same as for periodic boundary conditions but with modified selection rules as a start toward calculating the interfacial tension in general we here evaluate it in a low temperature limit performing a bethe ansatz type calculation finally we specialize the relations to the superintegrable case verifying the ansatz proposed by albertiniet al

controlling transient chaos the method of ott grebogi and yorke 1990 is extended to control transient chaos the controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor the time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime the number of controlled trajectories however decreases the maximum perturbation according to a power law applicability to experimental situations and comparison with permanent chaotic cases are discussed

hr bounds for spectral multipliers on graphs we study the boundedness on the hardy spaces h p of spectral multiplier operators associated with the discrete laplacian on a weighted graph we assume that the graph satisfies the doubling volume property and a poincare inequality we prove that there is p 0 0 1 depending on the geometry of the graph such that if the multiplier satisfies a condition similar to the one we have in the classical hormander multiplier theorem then the corresponding operator is bounded on h p p p 0 1

bounds for spectral multipliers on nilpotent groups a criterion is given for the lp boundedness of a class of spec tral multiplier operators associated to left invariant homogeneous subelliptic second order differential operators on nilpotent lie groups generalizing a theo rem of hormander for radial fourier multipliers on euclidean space the order of differentiability required is half the homogeneous dimension of the group improving previous results in the same direction

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

on calculation of generating functions of chebyshev polynomials in several variables we propose a new method of calculation of generating functions of chebyshev polynomials in several variables associated with root systems of the simple lie algebras we obtain the generating functions of the polynomials in two variables corresponding to the lie algebra c2 as an illustration

discrete transforms and orthogonal polynomials of anti symmetric multivariate cosine functions the discrete cosine transforms of types v viii are generalized to the antisymmetric and symmetric multivariate discrete cosine transforms four families of discretely and continuously orthogonal chebyshev like polynomials corresponding to the antisymmetric and symmetric generalizations of cosine functions are introduced each family forms an orthogonal basis of the space of all polynomials with respect to some weighted integral cubature formulas which correspond to these families of polynomials and which stem from the developed discrete cosine transforms are derived examples of three dimensional interpolation formulas and three dimensional explicit forms of the polynomials are presented

hypersurfaces with many aj singularities explicit constructions a construction of algebraic surfaces based on two types of simple arrangements of lines containing the prototiles of substitution tilings has been proposed recently the surfaces are derived with the help of polynomials obtained from the lines generating the simple arrangements one of the arrangements gives the generalizations of the chebyshev polynomials known as folding polynomials the other produces a family of polynomials which generates surfaces having more real nodes and they can also be used in combination with belyi polynomials to derive hypersurfaces in the complex projective space with many a j singularities in some cases explicit expressions can be obtained from the classical jacobi polynomials the lower bounds for the maximum possible number of a j singularities in certain hypersurfaces of degree d are improved for several values of d and j

the algebra of two dimensional generalized chebyshev koornwinder oscillator in the previous works of borzov and damaskinsky chebyshev koornwinder oscillator theor math phys 175 3 765 772 2013 and ladder operators for chebyshev koornwinder oscillator in proceedings of the days on diffraction 2013 the authors have defined the oscillator like system that is associated with the two variable chebyshev koornwinder polynomials we call this system the generalized chebyshev koornwinder oscillator in this paper we study the properties of infinite dimensional lie algebra that is analogous to the heisenberg algebra for the chebyshev koornwinder oscillator we construct the exact irreducible representation of this algebra in a hilbert space of functions that are defined on a region which is bounded by the steiner hypocycloid the functions are square integrable with respect to the orthogonality measure for the chebyshev koornwinder polynomials and these polynomials form an orthonormalized basis in the space the generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators h h

non linear dynamics and control of chaos for a tachometer abstract the dynamic behaviors of a rotational tachometer with vibrating support are studied in the paper both analytical and computational results are used to obtain the characteristics of the system the lyapunov direct method is applied to obtain the conditions of stability of the equilibrium position of the system the center manifold theorem determines the conditions of stability for the system in a critical case by applying various numerical analyses such as phase plane poincare map and power spectrum analysis a variety of periodic solutions and phenomena of the chaotic motion are observed the effects of the changes of parameters in the system can be found in the bifurcation diagrams and parametric diagrams by using lyapunov exponents and lyapunov dimensions the periodic and chaotic behaviors are verified finally various methods such as the addition of a constant torque the addition of a periodic torque delayed feedback control adaptive control bang bang control optimal control and the addition of a periodic impulse are used to control chaos effectively

synchronization in the kuramoto model a dynamical gradient network approach we propose a dynamical gradient network approach to consider the synchronization in the kuramoto model our scheme to adaptively adjust couplings is based on the dynamical gradient networks where the number of links in each time interval is the same as the number of oscillators but the links in different time intervals are also different the gradient network in the n 1 th time interval is decided by the oscillator dynamics in the nth time interval according to the gradient network in the n 1 th time interval only one inlink s coupling for each oscillator is adjusted by a small incremental coupling during the transition to synchronization the intensities for all oscillators are identical direct numerical simulations fully verify that the synchronization in the kuramoto model is realized effectively even if there exist delayed couplings and external noise

on group fourier analysis and symmetry preserving discretizations of pdes in this paper we review some group theoretic techniques applied to discretizations of pdes inspired by the recent years active research in lie group and exponential time integrators for differential equations we will in the first part of the paper present algorithms for computing matrix exponentials based on fourier transforms on finite groups as an example we consider spherically symmetric pdes where the discretization preserves the 120 symmetries of the icosahedral group this motivates the study of spectral element discretizations based on triangular subdivisions in the second part of the paper we introduce novel applications of multivariate non separable chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub domains these generalized chebyshev polynomials are intimately connected to the theory of root systems and weyl groups used in the classification of semi simple lie algebras and these polynomials share most of the remarkable properties of the classical chebyshev polynomials such as near optimal lebesgue constants for the interpolation error the existence of fft based algorithms for computing interpolants and pseudo spectral differentiation and existence of gaussian integration rules the two parts of the paper can be read independently

on the control of complex dynamic systems a method is described for the limited control of the dynamics of systems which generally have several dynamic attractors associated either with maps or first order ordinary differential equations ode in rn the control is based on the existence of convergent regions cc k 1 2 in the phase space of such systems where there is local convergence of all nearby orbits the character of the control involves the entrainment and subsequent possible migration of the experimental system from one attractor to another entrainment means that limt x t g t 0 where x t rn is the system s controlled dynamics and the goal dynamics g t gk has any topological form but is limited dynamically and to regions of phase space gk contained in some ck gk ck the control process is initiated only when the system enters a basin of entrainment bek gk associated with the goal region gk aside from this macroscopic initial state information about the system no further feedback of dynamic information concerning the response of the system is required the experimental reliability of the control requires that the regions bek be convex regions in the phase space which can apparently be assured if gk ck simple illustrations of these concepts are given using a general linear and a piecewise linear ode in rn in addition to these entrainment goals migration goal dynamics is introduced which intersects two convergent regions g cj g cj i j and permits transferring the dynamics of a system from one attractor to another or from one convergent region to another in the present study these concepts are illustrated with various one dimensional maps involving one or more attractors and convergent regions several theorems concerning entrainment are derived for very general continuous one dimensional maps sufficient conditions are also established which ensure near entrainment for a system when the dynamic model of the system is not exactly known the applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies

chebyshev polynomials in several variables and the radial part of the laplace beltrami operator chebyshev polynomials of the first and the second kind in n variables 1 z25 n zn are introduced the yariables n z2 zn are the characters of the representations of sl n 1 c corresponding to the fundamental weights the chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the laplace beltrami operator on certain symmetric spaces we give an explicit expression of this operator in the coordinates z1 ff z2 s zn and then show how many results in the literature on differential equations satisfied by chebyshev polynomials in several variables follow immediately from well known results on the radial part of the laplace beltrami operator related topics like orthogonality symmetry relations generating functions and recurrence relations are also discussed finally we note that the chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables

extended affine weyl groups of bcd type frobenius manifolds and their landau ginzburg superpotentials abstract for the root systems of type b l c l and d l we generalize the result of 7 by showing the existence of frobenius manifold structures on the orbit spaces of the extended affine weyl groups that correspond to any vertex of the dynkin diagram instead of a particular choice made in 7 it also depends on certain additional data we also construct landau ginzburg superpotentials for these frobenius manifold structures

generalized chebyshev polynomials associated with affine weyl groups we begin with a compact figure that can be folded into smaller replicas of itself such as the interval or equilateral triangle such figures are in one to one correspondence with affine weyl groups for each such figure in n dimensional euclidean space we construct a sequence of polynomials pk rn rn so that the mapping pk is conjugate to stretching the figure by a factor k and folding it back onto itself if n 1 and the figure is the interval this construction yields the chebyshev polynomials up to conjugation the polynomials pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials

algorithms for controlling chaotic motion application for the bvp oscillator the bvp oscillator is a biologically and physically important dynamical system exhibiting chaotic motion phaselocking phenomenon and so on in this paper we investigate the controlling of chaos in this system both when periodic and constant currents are present we use the various control algorithms and investigate their effectiveness in particular we study the controlling by the following mechanisms i adaptive control algorithm ii weak periodic perturbations iii ott grebogi yorke method iv weak feedback control and v addition of noise also we show the suppression of chaos by numerical models

enhancing chaoticity of spatiotemporal chaos in some practical situations strong chaos is needed this introduces the task of chaos control with enhancing chaoticity rather than suppressing chaoticity in this paper a simple method of linear amplifications incorporating modulo operations is suggested to make spatiotemporal systems which may be originally chaotic or nonchaotic strongly chaotic specifically this control can eliminate periodic windows increase the values and the number of positive lyapunov exponents make the probability distributions of the output chaotic sequences more homogeneous and reduce the correlations of chaotic outputs for different times and different space units the applicability of the method to practical tasks in particular to random number generators and secure communications is briefly discussed

orthogonal polynomials on the sphere with octahedral symmetry for any finite reflection group g acting on rn there is a family of g invariant measures h2 dw where h is a certain product of linear functions whose zero sets are the reflecting hyperplanes for g on the unit sphere and an associated partial differential operator lh f a fh fa h a is the laplacian analogously to spherical harmonics there is an orthogonal with respect to h2 do decomposition of homogeneous polynomials that is if p is of degree n then i n 21 p x j ix12jpn 2 x 0 where lh p 0 and deg p i for each i but with the restriction that p and p must all be g invariant the main topic is the hyperoctahedral group with h x x ix2 i n u x xj i j the special case n 2 leads to jacobi polynomials a detailed study of the case n 3 is made an important result is the construction of a third order differential operator that maps polynomials associated to h with indices a i to those associated with a 2 fi 1 introduction regular figures have long been objects of fascination we study here the interplay between them and the theory of orthogonal polynomials this will also touch on finite groups generated by reflections and their polynomial invariants as well as a generalization of spherical harmonics the underlying situation throughout will consist of spaces of homogeneous polynomials on rn that satisfy a second order differential equation and whose restrictions to the unit sphere have an orthogonality relation with respect to a weight function and of a finite reflection group for which all these objects are invariant the weight measure h2 x dw x on the unit sphere s in r3 where h x xlx2x3 a x 23 3 a 2 dw is surface area and the associated polynomials will be studied most intensely some of the highlights of the theory are as follows define the differential operator lhf f fah received by the editors april 19 1982 and in revised form december 15 1982 1980 mathematics subject classification primary 33a45 33a65 secondary 20h15 this research was partially supported by nsf grant mcs 81 02581 d 984 american mathematical society 0002 9947 84 1 00 25 per page

non linear dynamics and chaos control of a physical pendulum with vibrating and rotating support the dynamic behavior of a physical pendulum system of which the support is subjected to both rotation and vertical vibration are studied in this paper both analytical and computational results are employed to obtain the characteristics of the system by using lyapunov s direct method the conditions of stability of the relative equilibrium position can be determined melnikov s method is applied to identify the existence of chaotic motion the incremental harmonic balance method is used to find the stable and unstable periodic solutions for the strong non linear system by applying various numerical results such as phase portrait pioncare map time history and power spectrum analysis a variety of the periodic solutions and the phenomena of the chaotic motion can be presented the effects of the changes of parameters in the system could be found in the bifurcation and parametric diagrams further chaotic motion can be verified by using lyapunov exponent and lyapunov dimension the global analysis of basin boundary and fractal structure are observed by the modified interpolated cell mapping method besides non feedback control delayed feedback control adaptive control and variable structure control are used to control the chaos effectively

enhancing chaos in chaotic maps and flows we propose a mechanism by which the degree of chaoticity as measured by the average lyapunov exponent in chaotic flows can be increased our mechanism consists of introducing small changes in the system parameters in regions of phase space where the local lyapunov exponent falls substantially below its average value we have applied our mechanism to several typical chaotic flows and maps dissipative as well as area preserving an interesting consequence of this increase in chaoticity is an enhancement of the rate of mixing of the system we find that our method is quite efficient as it gives a substantial enhancement of chaos as measured by the average lyapunov exponent and also the rate of mixing for small changes in system parameters