strong detectability and observers conditions are given for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state the concepts of strong detectability and strong observability introduced before in the literature for discrete time systems only are defined and studied for continuous time systems it is shown that there are two different concepts of strong detectability which coincide for discrete time systems algebraic conditions for either concept are given it is shown that these concepts are intimately related to the existence of strong observers i e observers that only use the output of the plant

convergence of dynamical zeta functions i study poles and zeros of zeta functions in one dimensional maps numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function this describes convergence of the calculations of the first zero which is generally the physically interesting quantity some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow

chaotic systems counting the number of periods abstract characterization of chaotic motion may proceed both at an averaged macroscopic level using such notions as lyapunov exponents dimensions and entropies and at a microscopic level in the latter case the number of periodic orbits being a topological invariant plays an important role for various one dimensional mappings the counting problem itself has many interesting facets and may be solved more or less completely in different ways recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher dimensional dissipative chaotic systems

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

periodic orbit expansions for classical smooth flows the authors derive a generalized selberg type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow this relation is a classical analogue of the quantum trace formulae and selberg type zeta functions

curvature metric and parametrization of origami tessellations theory and application to the eggbox pattern origami tessellations are particular textured morphing shell structures their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape curvature and elongation on a global scale the existence of these global deformation modes allows for origami tessellations to fit non trivial surfaces thus inspiring applications across a wide range of domains including structural engineering architectural design and aerospace engineering the present paper suggests a homogenization type two scale asymptotic method which combined with standard tools from differential geometry of surfaces yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin jointed trusses the outcome of the method is a set of nonlinear differential equations governing the parametrization metric and curvature of surfaces that the initially discrete structure can fit the theory is presented through a case study of a fairly generic example the eggbox pattern the proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated

higher order sliding mode observers with scaled dissipative stabilisers abstractthis paper proposes a global theoretically exact finite time convergent observer for a class of nonlinear systems which are not necessarily bounded input bounded state with respect to the unknown inputs with this aim a scaled dissipative stabiliser is suggested ensuring that the highest derivative of the output estimation error is independent from the stabiliser gains after this a higher order sliding mode differentiator with the gains adjusted to the upper bound of the unknown inputs is applied achieving finite time convergence

computing periodic orbits with arbitrary precision this paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision these very high requirements are useful for example in the studies of complex pole location in many physical systems the algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation ode solver tides that uses a taylor series method nowadays this methodology is the only one capable of reaching precision up to thousands of digits for odes the method is shown to be quadratically convergent some numerical tests for the paradigmatic lorenz model and the henon heiles hamiltonian are presented giving periodic orbits up to 1000 digits

the generalized milnor thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters this paper presents an answer to an open problem in the dynamical systems of three letters the generalized milnor thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two dimensional parameter plane the concept of equal topological entropy class is introduced by the dual star product which is a generalization of the derrida gervois pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval the algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the milnor thurston characteristic polynomials moreover the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space 3 of three letters

a devil s staircase into chaotic scattering abstract we discuss analytically the transition from a regular to a chaotic classical dynamics of a sample scattering model introduced earlier the transition arises as a homoclinic bifurcation of an atypical type in phase space as a discontinuity for the grammatical complexity and as an onset of a devil s staircase for the topological entropy

observers for lipschitz nonlinear systems this paper presents some fundamental insights into observer design for the class of lipschitz nonlinear systems the stability of the nonlinear observer for such systems is not determined purely by the eigenvalues of the linear stability matrix the correct necessary and sufficient conditions on the stability matrix that ensure asymptotic stability of the observer are presented these conditions are then reformulated to obtain a sufficient condition for stability in terms of the eigenvalues and the eigenvectors of the linear stability matrix the eigenvalues have to be located sufficiently far out into the left half plane and the eigenvectors also have to be sufficiently well conditioned for ensuring asymptotic stability based on these results a systematic computational algorithm is then presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability

chaos order statistics and unstable periodic orbits we present an interesting connection between order statistics and unstable periodic orbits of chaotic maps this can be used to locate all the unstable periodic points of a given order of one dimensional chaotic maps with continuous invariant density the densities of the ordered variates of the iterates are discontinuous exactly at unstable periodic points of the map this is illustrated using the logistic map where densities corresponding to a small number of iterates have been obtained in closed form this scheme can also be applied to a class of continuous time systems where the successive maxima of the time series behave as if they were generated from a unimodal map we demonstrate this by using the lorenz model

newton method for highly unstable orbits abstract a newton method for finding unstable periodic orbits of an area preserving twist map is described the method is based on a variational principle and the execution time varies linearly with the lenghth of the orbit a number of techniques are described for obtaining good starting points for the method

indices in classical mechanics the authors discuss the linearization of the equations of motion around a periodic orbit in a conservative two degree of freedom hamiltonian system with an emphasis on computationally useful formulae indices are defined using singularities in projections conjugate points self conjugate points and the possibility of assigning a symbolic code are discussed

modelling the folding of paper into three dimensions using affine transformations abstract we model the folding of ordinary paper via piecewise isometries r 2 r 3 the collection of crease lines and vertices in the unfolded paper is called the crease pattern our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat into r 2 if the crease pattern is foldable then the product in a non intuitive order of the associated rotational matrices is the identity matrix this condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition sufficiency conditions are significantly harder and are not known except in the two dimensional single vertex case

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

large deformations of planar extensible beams and pantographic lattices heuristic homogenization experimental and numerical examples of equilibrium the aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call pantographic structures the interest in these materials was increased by the possibilities opened by the diffusion of technology of three dimensional printing they can be regarded once choosing a suitable length scale as families of beams also called fibres interconnected to each other by pivots and undergoing large displacements and large deformations there are however relatively few ready to use results in the literature of nonlinear beam theory in this paper we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by piola to formulate a continuum fully nonlinear beam model the homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems furthermore we consider pantographic structures find the corresponding homogenized second gradient deformation energies and study some planar problems numerical solutions for these two dimensional problems are obtained via minimization of energy and are compared with some experimental measurements in which elongation phenomena cannot be neglected

classical chaotic scattering periodic orbits symmetries multifractal invariant sets the infinite set of periodic orbits of a chaotic system is investigated their globally exact symbolic dynamics is related to birkhoff s method of iterated symmetry lines which produces symmetric periodic orbits of various classes a complete picture emerges as to what part of the periodic orbits can be obtained by means of the birkhoff lines in addition the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits

on the topology of the henon map topological invariants of the henon map are investigated by means of the pruning front first a long sequence of primary homoclinic tangencies is computed confirming the monotonicity of the front an algorithm to extract forbidden sequences is then introduced and discussed forbidden sequences of increasing lengths are used to construct a hierarchy of regular grammars represented by directed graphs which approximate the exact grammar arbitrarily well the topological entropy is estimated as the largest eigenvalue of their adjacency matrix it exhibits an exponential convergence towards the asymptotic value with an exponent in agreement with a previous conjecture based on the growth rate of the number of forbidden words

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

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

periodic orbits and spectral statistics of pseudointegrable billiards we demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than l increases as frac ensuremath pi b 0 l 2 a l where b 0 is a constant and a l is the average area occupied by these families we also find that a l increases with l before saturating finally we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies

on the existence of global tchebychev nets let s be a complete open simply connected surface suppose that the integral of the gauss curvature over arbitrary measurable sets is less than 7r 2 in magnitude we show that the surface admits a global tchebychev net

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

propagation of periodic and chaotic action potential trains along nerve fibers abstract in this paper we report the findings that the action potential trains transmitted along the nerve fibers are encoded not only periodically but also chaotically first spontaneous action potentials along a single fiber of injured sciatic nerves in the anesthetized rat were recorded then the data were divided into two groups and analyzed with different methods phase space representation spectral analysis and the calculation of correlation dimension were used for the first group of data sampled with constant frequency due to the serious influence of the measurement noise no reliable conclusion can be drawn from them for the second group of data of the interspike intervals isi which seem to convey more rich and important information nonlinear forecasting method the surrogate data and the plot of isi n 1 vs isi n were used in the analysis good results have been obtained which confirm with those from the cell model the largest lyapunov exponent lle were calculated not only to further support our findings of chaos but also to quantitatively determine the degree of chaos

technical communique global output feedback tracking control for a class of lagrangian systems the problem of global tracking control without velocity measurement of the so called euler lagrange systems has been paid a lot of attention for the past several years in this note a nice property of one degree of freedom euler lagrange systems is highlighted which in particular allows us to obtain a new solution to the problem of tracking control for this class of systems the solution is under the form of a linear like observer based controller which gives fairly good results as illustrated in the simulation the method can be generalized to any system having the same property

a semiclassical quantization of area preserving maps abstract a semiclassical path integral formalism is developed for a class of area preserving maps the quasi energy spectrum of these systems is represented as a sum over closed orbits in the manner of gutzwiller the method is illustrated with some numerical tests of the standard map

a numerical method for locating stable periodic orbits in chaotic systems abstract a numerical method aimed at locating stable periodic orbits in strongly chaotic systems is presented the method is based on the selection of trajectory segments which are characterized by a relatively low positive local lyapunov exponent once such a selection is made convergence to stable or weakly unstable periodic orbits is obtained by a newton method the algorithm is rather general and can be used for systems with more than two degrees of freedom the proposed approach is tested on the quartic oscillator model and on the potential of the hydrogen atom in a strong magnetic field in the latter case new stable periodic orbits are found in the region of strong chaotic motion the possible quantum localization on these orbits is discussed briefly

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

entropy of hidden markov processes via cycle expansion hidden markov processes hmp is one of the basic tools of the modern probabilistic modeling the characterization of their entropy remains however an open problem here the entropy of hmp is calculated via the cycle expansion of the zeta function a method adopted from the theory of dynamical systems for a class of hmp this method produces exact results both for the entropy and the moment generating function the latter allows to estimate via the chernoff bound the probabilities of large deviations for the hmp more generally the method offers a representation of the moment generating function and of the entropy via convergent series

robust exact uniformly convergent arbitrary order differentiator an arbitrary order differentiator that in the absence of noise converges to the true derivatives of the signal after a finite time independent of the initial differentiator error is presented the only assumption on a signal to be differentiated n 1 times is that its n th derivative is uniformly bounded by a known constant the proposed differentiator switches from a newly designed uniform differentiator to the classical high order sliding mode hosm differentiator the uniform part drives the differentiation error trajectories into a compact neighborhood of the origin in a time that is independent of the initial differentiation error then the hosm differentiator is used to bring the differentiation error to zero in finite time

global tchebychev nets on complete two dimensional riemannian surfaces let s be a given surface a local tchebychev net on s is a coordinate chart of s such that the coordinate vector fields have unit magnitudes it models physical surfaces which are reinforced with two sets of inextensible fibers the angle between fibers of each set at any point is allowed to change in this paper s is taken to be a two dimensional simply connected noncompact complete riemannian manifold on which the gaussian curvature is assumed to have compact support the existence of a global tchebychev net on s is investigated

design of three dimensional origami structures based on a vertex approach origami geometric design is fundamental to many engineering applications of origami structures this paper presents a new method for the design of three dimensional 3d origami structures suitable