Anosov C-systems and random number generators

We further develop our previous proposal to use hyperbolic Anosov C-systems to generate pseudorandom numbers and to use them for efficient Monte Carlo calculations in high energy particle physics. All trajectories of hyperbolic dynamical systems are exponentially unstable, and C-systems therefore have mixing of all orders, a countable Lebesgue spectrum, and a positive Kolmogorov entropy. These exceptional ergodic properties follow from the C-condition introduced by Anosov. This condition defines a rich class of dynamical systems forming an open set in the space of all dynamical systems. An important property of C-systems is that they have a countable set of everywhere dense periodic trajectories and their density increases exponentially with entropy. Of special interest are the C-systems defined on higher-dimensional tori. Such C-systems are excellent candidates for generating pseudorandom numbers that can be used in Monte Carlo calculations. An efficient algorithm was recently constructed that allows generating long C-system trajectories very rapidly. These trajectories have good statistical properties and can be used for calculations in quantum chromodynamics and in high energy particle physics.     

Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows

We investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the framework of small amplitude waves, we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case, and we describe the possible particle trajectories. Depending on the relation between the initial data and the constant vorticity, some particle trajectories are undulating curves to the right, or to the left, others are loops with forward drift, or with backward drift, others can follow some peculiar shapes.     

KmL: k-means for longitudinal data

Cohort studies are becoming essential tools in epidemiological research. In these studies, measurements are not restricted to single variables but can be seen as trajectories. Statistical methods used to determine homogeneous patient trajectories can be separated into two families: model-based methods (like Proc Traj) and partitional clustering (non-parametric algorithms like k-means). KmL is a new implementation of k-means designed to work specifically on longitudinal data. It provides scope for dealing with missing values and runs the algorithm several times, varying the starting conditions and/or the number of clusters sought; its graphical interface helps the user to choose the appropriate number of clusters when the classic criterion is not efficient. To check KmL efficiency, we compare its performances to Proc Traj both on artificial and real data. The two techniques give very close clustering when trajectories follow polynomial curves. KmL gives much better results on non-polynomial trajectories.     

Chaos in a continuous stirred tank reactor

A model is derived for a nonisothermal, continuous, stirred tank reactor in which two, exothermic, first order, irreversible, reactions are occuring in parallel. A numerical examination of this model reveals that complex periodic and chaotic oscillations are possible. This behavior is studied using linear stability analysis, autocorrelation analysis, and next-amplitude plots. The next-amplitude plots are found to be fractal curves. An examination of the system trajectories shows that in some circumstances the trajectories follow a Möbius-band attractor in state space.     

Expanding Notions of “Learning Trajectories” in Mathematics Education

Over the past 20 years learning trajectories and learning progressions have gained prominence in mathematics and science education research. However, use of these representations ranges widely in breadth and depth, often depending on from what discipline they emerge and the type of learning they intend to characterize. Learning trajectories research has spanned from studies of individual student learning of a single concept to trajectories covering a full set of content standards across grade bands. In this article, we discuss important theoretical assumptions that implicitly guide the development and use of learning trajectories and progressions in mathematics education. We argue that diverse theoretical conceptualizations of what it means for a student to “learn” mathematics necessarily both constrains and amplifies what a particular learning trajectory can capture about the development of students’ knowledge.     

Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.     

Dynamic spanning without probabilities

The paper presents a non-probabilistic approach to continuous-time trading where, in analogy to the binomial option-pricing model, terminal payoffs resulting from a given trading strategy are meaningful ‘state-by-state’, i.e., path-by-path. In particular, we obtain results of the form: “If a certain trading strategy is applied and if the realized price trajectory satisfies a certain analytical property, then the terminal payoff is.…” This way, derivation of the Black and Scholes formula and its extension become an exercise in the analysis of a certain class of real functions. While results of the above forms are of great interest if the analytical property in question is believed to be satisfied for almost all realized price trajectories (for example, if the price is believed to follow a certain stochastic process which has this property with probability 1), they are valid regardless of the stochastic process which presumably generates the possible price trajectories or the probability assigned to the set of all paths having this analytical property.     

Batch mode reinforcement learning based on the synthesis of artificial trajectories

In this paper, we consider the batch mode reinforcement learning setting, where the central problem is to learn from a sample of trajectories a policy that satisfies or optimizes a performance criterion. We focus on the continuous state space case for which usual resolution schemes rely on function approximators either to represent the underlying control problem or to represent its value function. As an alternative to the use of function approximators, we rely on the synthesis of “artificial trajectories” from the given sample of trajectories, and show that this idea opens new avenues for designing and analyzing algorithms for batch mode reinforcement learning.     

Determination of Periodic Trajectories of Dynamic Systems Subject to Switching Input Constraints

The present paper is concerned with the determination of periodic state trajectories of discrete-time dynamic systems resulting from the application of bidirectional on/off input sequences subject to switching constraints. The main contribution consists of characterizing the feasible input sequences and associated state trajectories in terms of a set of linear constraints involving integer variables. For practical purposes, the resulting solutions obtained by an integer linear solver can be evaluated in terms of engineering criteria such as amplitude or fuel expense in order to choose an appropriate trajectory to be used as target in a control law. A numerical example involving a double-integrator system is presented to illustrate the use of the proposed constraint formulation for the determination of feasible periodic state trajectories.     

Piecewise-maximum closed Geodesic trajectories on bounded Toroidal manifolds

We use toroidal coordinates for the investigation of maximum geodesic trajectories for arbitrary parameters of a torus. Conditions under which trajectories are located in a bounded part of the toroidal manifold are considered. Using global invariants, we construct closed piecewise-maximum geodesic trajectories.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1139–1142, August, 2004.     

Conditional targeting for communication

In this work we propose the use of a targeting method applied to chaotic systems in order to reach special trajectories that encode arbitrary sources of messages. One advantage of this procedure is to overcome dynamical constraints which impose limits in the amount of information that the chaotic trajectories can encode. Another advantage is the message decoding, practically instantaneous and independent of any special technique or algorithm. Furthermore, with this procedure, information can be transmitted with no errors due to bounded noise.     

Limit Cycles of a Planar Vector Field

This paper presents a solution to the problem to find isolated closed trajectories of two-dimensional dynamic systems. In contrast to the method of Bendixsons ring regions, the new method is constructive. It allows the determination of the location of closed trajectories and therefore gives an upper bound for their number. The method is based on the idea to use inherent geometrical and physical extremal properties of these trajectories to transform the problem into an optimization task (isoperimetric problem of variational calculus) that can be solved by numerical algorithms, e.g., by hillclimbing.     

A new proof of a theorem on optimal control of switched systems

Recently Sorin C. Bengea and Raymond A. DeCarlo [Sorin C. Bengea, Raymond A. DeCarlo, Optimal control of switching systems, Automatica J. IFAC 41 (2005) 11-27] have offered a key result that the set of trajectories of the two-switched system is dense in the set of trajectories of the embedded system. This result was proven by means of relaxed controls and the Chattering Lemma. In this paper we use the Lyapunov theorem to give a new simple proof.     

Connections Between Classical and Generalised Trajectories of a State/Signal System

In our earlier article “Well-posed state/signal systems in continuous time”, we originally defined the notion of a trajectory of a state/signal system by means of a generating subspace. However, it was left as an open problem whether the generating subspace is uniquely determined by a given family of all generalised trajectories of a well-posed state/signal system. In this article we give a positive answer to this question and show how this insight simplifies some formulations in the theory of well-posed state/signal systems. The main contribution of the article is an explicit convolution scheme for constructing classical trajectories approximating an arbitrary generalised trajectory. We apply this scheme by studying relationships between classical and generalised trajectories of continuous-time state/signal systems under very weak assumptions. Among others, we show that there exists a space of classical trajectories that is invariant under differentiation and dense in the space of generalised trajectories. Some of our results generalise known results for strongly continuous semigroups and input/state/output systems, but we make no use of decompositions of the signal space into an input space and an output space, and in particular, none of our results depend on well-posedness.     

The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory     

Periodic Image Trajectories in Earth–Moon Space

The problem of identifying orbits that enclose both the Earth and the Moon in a predictable way has theoretical relevance as well as practical implications. In the context of the restricted three-body problem with primaries in circular orbits, periodic trajectories exist and have the property that a third body (e.g. a spacecraft) can describe them indefinitely. Several approaches have been employed in the past for the purpose of identifying similar orbits. In this work the theorem of image trajectories, proven five decades ago, is employed for determining periodic image trajectories in Earth–Moon space. These trajectories exhibit two fundamental features: (i) counterclockwise departure from a perigee on the far side of the Earth, and (ii) counterclockwise arrival to a periselenum on the far side of the Moon. An extensive, systematic numerical search is performed, with the use of a modified Poincaré map, in conjunction with a numerical refinement process, and leads to a variety of periodic orbits, with various interesting features for possible future lunar missions.     

Scaling,shifting and weighting in interior-point methods

We examine certain questions related to the choice of scaling, shifting and weighting strategies for interior-point methods for linear programming. One theme is the desire to make trajectories to be followed by algorithms into straight lines if possible to encourage fast convergence. While interior-point methods in general follow curves, this occurrence of straight lines seems appropriate to honor George Dantzig's contributions to linear programming, since his simplex method can be seen as following either a piecewise-linear path inn-space or a straight line inm-space (the simplex interpretation).Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.Research supported in part by NSF, AFOSR, and ONR through NSF Grant DMS-8920550.     

Stabilizing chaotic system on periodic orbits using multi-interval and modern optimal control strategies

In this paper, optimal approaches for controlling chaos is studied. The unstable periodic orbits (UPOs) of chaotic system are selected as desired trajectories, which the optimal control strategy should keep the system states on it. Classical gradient-based optimal control methods as well as modern optimization algorithm Particle Swarm Optimization (PSO) are utilized to force the chaotic system to follow the desired UPOs. For better performance, gradient-based is applied in multi-intervals and the results are promising. The Duffing system is selected for examining the proposed approaches. Multi-interval gradient-based approach can put the states on UPOs very fast and keep tracking UPOs with negligible control effort. The maximum control in PSO method is also low. However, due to its inherent random behavior, its control signal is oscillatory.     

Algorithm for the numerical proof of the existence of periodic trajectories in two-dimensional nonautonomous systems of ordinary differential equations

We suggest an algorithm that permits one to prove the existence of limit periodic trajectories (cycles) in two-dimensional nonautonomous dissipative systems with periodic coefficients with the use of computational methods alone. We prove a fixed point theorem for two-dimensional mappings and describe methods of its application to two-dimensional nonautonomous systems with the use of the Poincaré mapping and interval arithmetics.     

A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems

There has been significant progress in the development of numerical methods for the determination of optimal trajectories for continuous dynamic systems, especially in the last 20 years. In the 1980s, the principal contribution was new methods for discretizing the continuous system and converting the optimization problem into a nonlinear programming problem. This has been a successful approach that has yielded optimal trajectories for very sophisticated problems. In the last 15–20 years, researchers have applied a qualitatively different approach, using evolutionary algorithms or metaheuristics, to solve similar parameter optimization problems. Evolutionary algorithms use the principle of “survival of the fittest” applied to a population of individuals representing candidate solutions for the optimal trajectories. Metaheuristics optimize by iteratively acting to improve candidate solutions, often using stochastic methods. In this paper, the advantages and disadvantages of these recently developed methods are described and an attempt is made to answer the question of what is now the best extant numerical solution method.