the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

a quadratically constrained quadratic optimization model for completely positive cone programming we propose a class of quadratic optimization problems which can be reformulated by completely positive cone programming with the same optimal values the objective function can be any quadratic form the constraints of each problem are described in terms of quadratic forms with no linear terms and all constraints are homogeneous equalities except one inhomogeneous equality where a quadratic form is set to be a positive constant for the equality constraints a hierarchy of copositvity condition is assumed this model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations in particular it generalizes the recent results on quadratic optimization problems by burer and the set semidefinite representation by eichfelder and povh optim lett 7 2013 pp 1373 1386

generations and mechanisms of multi stripe chaotic attractors of fractional order dynamic system abstract in this paper the generations of multi stripe chaotic attractors of fractional order system are considered the original fractional order chaotic attractors can be turned into a pattern with multiple parallel or rectangular stripes by employing certain simple periodic nonlinear functions the relationships between the parameters relate to the periodic functions and the shape of the generated attractors are analyzed theoretical investigations about the underlying mechanisms of the parallel striped attractors of fractional order system are presented with the fractional order lorenz rossler and chua s systems as examples moreover the periodic doubling striped route to chaos of fractional order rossler system and maximum lyaponov exponent calculations are also given

response of nonlinear oscillators with random frequency of excitation revisited periodically driven oscillators of low frequency random excitations are analyzed computer simulation which was carried out for the duffing equation and forced vibrations of a pendulum indicated that in these cases noise has a stabilizing effect computation of lyapunov exponents showed that by adding noise to chaotic motion the largest lyapunov exponent as a rule turns to negative and consequently the chaotic motion is annihilated

chaotic behaviors and toroidal spherical attractors generated by discontinuous dynamics abstract this paper reports a class of chaotic attractors with toroidal or spherical patterns these attractors look like spheres or tori but they are not exactly on the two dimensional toroidal spherical surfaces discontinuous structures are taken in the considered system to produce easily various chaotic tori spheres with different input functions moreover controlling the shape of these chaotic attractors can be realized by adjusting some parameters with the help of fourier series the underlying chaos generation mechanism is also explored briefly

a lagrangian dnn relaxation a fast method for computing tight lower bounds for a class of quadratic optimization problems we propose an efficient computational method for linearly constrained quadratic optimization problems qops with complementarity constraints based on their lagrangian and doubly nonnegative dnn relaxation and first order algorithms the simplified lagrangian completely positive programming cpp relaxation of such qops proposed by arima kim and kojima in 2012 takes one of the simplest forms an unconstrained conic linear optimization problem with a single lagrangian parameter in a cpp matrix variable with its upper left element fixed to 1 replacing the cpp matrix variable by a dnn matrix variable we derive the lagrangian dnn relaxation and establish the equivalence between the optimal value of the dnn relaxation of the original qop and that of the lagrangian dnn relaxation we then propose an efficient numerical method for the lagrangian dnn relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods numerical results on binary qops quadratic multiple knapsack problems maximum stable set problems and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

fundamental groups of galois closures of generic projections for the galois closure x gal of a generic projection from a surface x it is believed that 1 x gal gives rise to new invariants of x however in all examples this group is surprisingly simple in this article we offer an explanation for this phenomenon we compute a quotient of 1 x gal that depends on 1 x and data from the generic projection only in all known examples this quotient is in fact isomorphic to 1 x gal as a byproduct we simplify the computations of moishezon teicher and others

regularization methods for sdp relaxations in large scale polynomial optimization we study how to solve semidefinite programming sdp relaxations for large scale polynomial optimization when interior point methods are used typically only small or moderately large problems could be solved this paper studies regularization methods for solving polynomial optimization problems we describe these methods for semidefinite optimization with block structures and then apply them to solve large scale polynomial optimization problems the performance is tested on various numerical examples with regularization methods significantly bigger problems could be solved on a regular computer which is almost impossible with interior point methods

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

the fundamental group of a mathcal cp 2 complement of a branch curve as an extension of a solvable group by a symmetric group the main result in this paper is as follows theorem let s be the branch curve in cp 2 of a generic projection of a veronese surface then 1 cp 2 s is an extension of a solvable group by a symmetric group a group with the property mentioned in the theorem is almost solvable in the sense that it contains a solvable normal subgroup of finite index we pose the following question question for which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projec tion to cp 2 an extension of a solvable group by a symmetric group

a dynamic inequality generation scheme for polynomial programming hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs this approach rapidly becomes computationally expensive and is often tractable only for problems of small size in this paper we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs when used iteratively this scheme improves the bounds without incurring an exponential growth in the size of the relaxation as a result the proposed scheme is in principle scalable to large general polynomial programming problems when all the variables of the problem are non negative or when all the variables are binary the general algorithm is specialized to a more efficient algorithm in the case of binary polynomial programs we show special cases for which the proposed scheme converges to the global optimal solution we also present several examples illustrating the computational behavior of the scheme and provide comparisons with lasserre s approach and for the binary linear case with the lift and project method of balas ceria and cornuejols

active set algorithms for isotonic regression a unifying framework in this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods the active set approach provides a unifying framework for studying algorithms for isotonic regression simplifies the exposition of existing algorithms and leads to several new efficient algorithms we also investigate the computational complexity of several algorithms

separating doubly nonnegative and completely positive matrices the cone of completely positive cp matrices can be used to exactly formulate a variety of np hard optimization problems a tractable relaxation for cp matrices is provided by the cone of doubly nonnegative dnn matrices that is matrices that are both positive semidefinite and componentwise nonnegative a natural problem in the optimization setting is then to separate a given dnn but non cp matrix from the cone of cp matrices we describe two different constructions for such a separation that apply to 5 5 matrices that are dnn but non cp we also describe a generalization that applies to larger dnn but non cp matrices having block structure computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

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

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints we propose a doubly nonnegative dnn relaxation for polynomial optimization problems pops with binary and box constraints this work is an extension of the work by kim kojima and toh in 2016 from quadratic optimization problems to pops the dense and sparse dnn relaxations are reduced to a simple conic optimization problem cop to which an accelerated bisection and projection bp algorithm is applied the cop involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial our dnn relaxation serves as a variant of the lasserre moment sos hierarchy of relaxations for binary and box constrained pops when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values a significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the dnn relaxation and the design of efficient computation of the metric projections onto these cones in the accelerated bp algorithm numerical results on various pops are available in ito et al acm trans math softw 45 article 34 2019