Simple Dynamical System Research Articles

Nonobservable space dimensions and the discreteness of time☆

We present a simple dynamical system model for the effect of nonobservable space dimensions on the observable ones. There are three premises. A: Orbits consist of flows of probabilities [Ilya P. The end of certainty. NY: The Free Press; 1996] (which is the case in the setting of quantum mechanics). B: The orbits of probabilities are induced by (continuous time) differential or partial differential equations . C: The observable orbit is a flow of marginal probabilities where the nonobservable space dimensions are averaged out. A theorem is presented which proves that under certain general conditions the transfer of marginal probabilities cannot be achieved by continuous time dynamical systems acting on the space of observable variables but can be achieved by discrete time dynamical systems.

Learning and transfer from a simple dynamic system

The amount of learning gained from being assisted in completing the task of bringing a predator-prey system into equilibrium by controlling the predator population was investigated. Learning was explored both by post-task questioning and by testing for transfer to another predator-prey task. Participants were 28 undergraduate psychology students, all female. They were randomly and evenly split into an experiment group that was subjected to a learning session with the first task before being tested in the second task, and a control group that only performed the second task. What was most needed in the first task was help in sticking to analytically derived conclusions by resisting "common sense" responses. There was a significant transfer effect on performance to the second task, stemming from learning shown by half the participants in the experiment group. The other half showed hardly any learning. Learning in about half the subjects has proven a stable finding.

Trajectory correction manoeuvres in the transfer to libration point orbits

In this paper we study the manoeuvres to be done by a spacecraft in order to correct the error in the execution of the injection manoeuvre in the transfer trajectory. We will consider the case where the nominal trajectory is a halo orbit around the collinear equilibrium point L 1 . The results can be easily extended to the L 2 point and to other kinds of libration point orbits, such as Lissajous and quasi-halo orbits. For our study we use simple dynamical systems concepts related with the invariant manifolds of the target orbit, and we compare our results with those obtained by Serban et al. (Automatica, 38(2002) 571) using optimal control.

Dynamic intradigital external fixation for proximal interphalangeal joint fracture dislocations

Many skeletal traction devices have been described to treat fracture dislocations of the proximal interphalangeal (PIP) joint. Most of these techniques are technically challenging or involve cumbersome frames. We present a design modification that enhances the stability of a simple dynamic fixation system described previously and report our results with this technique. A previously described simple dynamic fixator with no rubber bands was applied to 6 patients who sustained fracture dislocations of the PIP joint. The middle finger was involved in 3 patients, the ring finger in 1 patient, and the small finger in 2 patients. The average age of the patients was 27 years (range, 21-42 y). The average involvement of the base of the middle phalanx was 48% (range, 35% to 60%). The average time from the injury to the surgery was 6 days (range, 1-14 d). The average follow-up period was 24 months (range, 7-43 mo). Immediate active flexion extension was allowed and the fixator was removed after 3 to 4 weeks. The average range of motion of the PIP joint at the final follow-up evaluation was 5 degrees to 89 degrees (range, 0 degrees to 100 degrees ). Two patients developed pin track infection that resolved with oral antibiotics. Only one patient complained of mild pain with extreme flexion. Proper reduction and congruency of the joint was noted on final anteroposterior and lateral radiographs. A simple dynamic fixator for the treatment of unstable PIP joint fracture dislocations was used successfully in 6 digits to maintain reduction and restore digital range of motion. The addition of modifications to the original technique not only improves the solidity of the construct but also provides satisfactory functional results. Based on our experience we recommend this easy technique to treat fracture dislocations of the PIP joint.

Use of Response Surface Metamodels for Identification of Stiffness and Damping Coefficients in a Simple Dynamic System

Metamodels have been used with success in many areas of engineering for decades but only recently in the field of structural dynamics. A metamodel is a fast running surrogate that is typically used to aid an analyst or test engineer in the fast and efficient exploration of the design space. Response surface metamodels are used in this work to perform parameter identification of a simple five degree of freedom system, motivated by their low training requirements and ease of use. In structural dynamics applications, response surface metamodels have been utilized in a forward sense, for activities such as sensitivity analysis or uncertainty quantification. In this study a polynomial response surface model is developed, relating system parameters to measurable output features. Once this relationship is established, the response surface is used in an inverse sense to identify system parameters from measured output features.A design of experiments is utilized to choose points, representing a fraction of the full design space of interest, for fitting the response surface metamodel. Two parameters commonly used to characterize damage in a structural system, stiffness and damping, are identified. First changes are identified and located with success in a linear 5DOF system. Then parameter identification is attempted with a nonlinear 5DOF system and limited success is achieved. This work will demonstrate that use of response surface metamodels in an inverse sense shows promise for use in system parameter identification for both linear and weakly nonlinear systems and that the method has potential for use in damage identification applications.

On the complexity of algebraic numbers, II. Continued fractions

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $\alpha$ is, in some sense, simple, then $\alpha$ is either quadratic or transcendental. The term can of course lead to many interpretations. It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity/complexity for the continued fraction expansion of algebraic real numbers of degree at least three.

Bifurcation and chaos in simple jerk dynamical systems

In recent years, it is observed that the third-order explicit autonomous differential equation, named as jerk equation, represents an interesting sub-class of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we investigate the global dynamics of a special family of jerk systems {ie075-01}, whereG(x) is a non-linear function, which are known to exhibit chaotic behaviour at some parameter values. We particularly identify the regions of parameter space with different asymptotic dynamics using some analytical methods as well as extensive Lyapunov spectra calculation in complete parameter space. We also investigate the effect of weakening as well as strengthening of the non-linearity in theG(x) function on the global dynamics of these jerk dynamical systems. As a result, we reach to an important conclusion for these jerk dynamical systems that a certain amount of non-linearity is sufficient for exhibiting chaotic behaviour but increasing the non-linearity does not lead to larger regions of parameter space exhibiting chaos.

Nonobservable space dimensions and the discreteness of time☆

We present a simple dynamical system model for the effect of nonobservable space dimensions on the observable ones. There are three premises. A: Orbits consist of flows of probabilities [Ilya P. The end of certainty. NY: The Free Press; 1996] (which is the case in the setting of quantum mechanics). B: The orbits of probabilities are induced by (continuous time) differential or partial differential equations. C: The observable orbit is a flow of marginal probabilities where the nonobservable space dimensions are averaged out. A theorem is presented which proves that under certain general conditions the transfer of marginal probabilities cannot be achieved by continuous time dynamical systems acting on the space of observable variables but can be achieved by discrete time dynamical systems.

Feedback processes in cellulose thermal decomposition: implications for fire-retarding strategies and treatments

A simple dynamical system that models the competitive thermokinetics and chemistry of cellulose decomposition is examined, with reference to evidence from experimental studies indicating that char formation is a low activation energy exothermal process and volatilization is a high activation energy endothermal process. The thermohydrolysis chemistry at the core of the primary competition is described. Essentially, the competition is between two nucleophiles, a molecule of water and an −OH group on C6 of an end glucosyl cation, to form either a reducing chain fragment with the propensity to undergo the bond-forming reactions that ultimately form char, or a levoglucosan end-fragment that depolymerizes to volatile products. The results of this analysis suggest that promotion of char formation under thermal stress can actually increase the production of flammable volatiles. Thus, we would like to convey an important safety message in this paper: in some situations where heat and mass transfer is restricted in cellulosic materials, such as furnishings, insulation, and stockpiles, the use of char-promoting treatments for fire retardation may have the effect of increasing the risk of flaming combustion.

The influence of laser field noise on controlled quantum dynamics.

The influence of laser noise on the dynamics of simple quantum systems is analyzed. An anharmonic ladder is chosen for illustration and several pulses are obtained that optimize the yield of a quantum transition by constraining the laser parameters. The following models of laser noise are introduced: Amplitude white noise, phase white noise, frequency white noise and shot-to-shot static noise in the different pulse parameters. It is shown that the optimal pulses are robust to white amplitude noise, since the system acts as a dynamical filter. White phase noise affects the optimal pulses in a similar way by reducing the pulse area. This effect can be easily compensated for by pulse amplitude rescaling, up to a high level of noise. White frequency noise reduces the pulse area and induces spectral broadening, more strongly affecting the high frequency components. It can be partially compensated for by amplitude rescaling. The effects of static noise in the parameters cannot be easily corrected. It is shown that optimal pulses that drive n-photon transitions become more sensitive to noise in the amplitude and less sensitive to noise in the frequency as n increases. The effects of noise in the relative phase rapidly become constant for a large number of interfering pathways.

Stochastic representation of a class of non-Markovian completely positive evolutions

By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong non-exponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects [S.M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)] is clarified in this context.

Spinning deformations of rational maps

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as spinning. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges to infinity in the moduli space of rational maps of a fixed degree. When the limit exists, it has either a parabolic fixed point, or a pre-periodic critical point in the Julia set, depending on the combinatorics of the data defining the deformation. The proofs are soft and rely on two ingredients: the construction of a Riemann surface containing the closure of the family, and an analysis of the geometric limits of some simple dynamical systems. An interpretation in terms of Teichmüller theory is presented as well.

Aeroelastic behaviour of a solar-powered high-altitude long endurance unmanned air vehicle (HALE-UAV) slender wing

A preliminary dynamic and aeroelastic investigation is performed in a simplified equivalent configuration derived from the HELIPLAT HAVE-UAV (high-altitude very-long endurance unmanned air vehicle) wing structure. The presence of slender wing structures introduces in the structural design a certain degree of non-linearity, forcing the designer to deal with specific phenomena, one of which is described. A simple dynamic system is used to point out the important effect due to the inclusion of the deformed equilibrium configuration into the dynamic behaviour of the system. A modified wing structure configuration is used to point out this effect in the flutter behaviour of slender structures. Some interesting design indications are presented.

Spinodal decomposition and the deconfining phase transition

We study the Glauber dynamics of simple spin systems to identify dynamical scenarios which may be of relevance for the deconfining phase transition in heavy ion collisions.

Cell cycle progression

In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model. To cite this article: J. Tyrcha, C. R. Biologies 327 (2004).

Landau–Lifshitz Hierarchy and Infinite-Dimensional Grassmann Variety

The Landau—Lifshitz equation is an example of soliton equations with a zero-curvature representation defined on an elliptic curve. This equation can be embedded into an integrable hierarchy of evolution equations called the Landau—Lifshitz hierarchy. This Letter elucidates its status in Sato, Segal and Wilson's universal description of soliton equations in the language of an infinite-dimensional Grassmann variety. To this end, a Grassmann variety is constructed from a vector space of 2 × 2 matrices of Laurent series of the spectral parameter z. A special base point W 0, called ‘vacuum,’ of this Grassmann variety is chosen. This vacuum is ‘dressed’ by a Laurent series φ(z) to become a point of the Grassmann variety that corresponds to a general solution of the Landau—Lifshitz hierarchy. The Landau—Lifshitz hierarchy is thereby mapped to a simple dynamical system on the set of these dressed vacua. A higher-dimensional analogue of this hierarchy (an elliptic analogue of the Bogomolny hierarchy) is also presented.

Analysis Still Matters: A Surprising Instance of Failure of Runge--Kutta--Felberg ODE Solvers

This paper provides a nice example to illustrate that without supporting analysis, a numerical simulation may lead to incorrect conclusions. We explore a pedagogical example of failure of Runge--Kutta--Felberg (RKF) algorithms for a simple dynamical system that models the coupling of two oscillators. Although the system appears to be well-behaved, the explicit RKF solvers provide erratic numerical solutions. The mode of failure is based in a period-doubling route to chaos due to the existence of stable linear solutions in the problem.

Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincare recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite ‘minimum recurrence time’. We find the minimum recurrence time as a function of the energy.

Education

This edition of the Education section contains two wonderfully written papers that will be very accessible to students and useful to instructors. They come from two quite different areas: complex analysis, and dynamical systems/numerical solution of differential equations. Both, in quite different ways, combine computation and mathematics. The first paper may possess one of the longer, more technical titles in the short history of the Education section, but that title serves as an excellent short abstract of the paper. The paper is "Recovering Holomorphic Functions from Their Real and Imaginary Parts without the Cauchy--Riemann Equations" by William Shaw. Written in a clear and pleasant tutorial style with exercises and examples, the paper provides a simple alternative to a fundamental component of complex analysis. It presents a purely algebraic method to recover a holomorphic (also known as analytic, or regular) complex function given just its real, or just its imaginary, part. This contrasts with the standard approach based on partial differential equations through direct application of the Cauchy--Riemann equations. The author acknowledges that the method is not entirely original, having appeared previously at least in simplified form. But apparently it is neither well known nor commonly cited, and this paper seeks to remedy that. Based upon its clarity, accessibility, and elegance, it should succeed! The paper fits any introductory course in complex analysis. It even contains a discussion and examples of a symbolic computation implementation of the approach, in Mathematica. "'The computer is always right'---wrong" and "mathematical analysis helps" could be the mantras of the second paper, "Analysis Still Matters: A Surprising Instance of Failure of Runge--Kutta--Felberg ODE Solvers." Author Joseph Skufca presents a simple dynamical system where the popular and reliable Runge--Kutta--Felberg method fails, providing an erratic solution after a certain time. Rather than revealing an interesting, perhaps chaotic solution to the system, the numerical solution simply is wrong. The culprit is a stepsize that has gotten too large and does not recover, but the real question is "why?" The paper uses simple analysis to show why the code's stepsize estimation method fails on this problem, and then shows that while typically the stepsize and solution would recover, on certain problems like this one, that is not the case. It also explains the relation between this problem and phenomena of solving stiff ordinary differential equations, and indeed, a stiff solver handles this problem well. The basic premise of the paper is that understanding of numerical simulation and the underlying mathematical analysis must go hand in hand. This is an excellent lesson that can fit in courses in basic numerical analysis, numerical differential equations, or dynamical systems, and the very clear exposition of this paper makes it ideal for teaching this lesson.

Time's square: deriving cultural theory from rubbish theory

Starting with the hypothesis that duration (the standard idea of time that we associate with the steady tick-tock of a clock) can only exist if it is framed by something that is not duration—eternity or oblivion—it is possible to make sense of the fusing of time and space that is going on, both in acephalous societies in West Africa and in the more familiar dynamic categorizations of modern Western societies: secondhand, rubbish and antique, for instance. The same hypothesis also sheds some helpful light on problems over space and time in comtemporary physics. Forgetting, we now see, is a prerequisite for remembering, and worthlessness a prerequisite for value: insights that suggest we take a closer look at what is involved in the deceptively simple three-category dynamical system in which objects, once produced, have only two possible destinations—the museum or the rubbish dump. Such a dynamical system, it turns out, can give rise to shifts within the social totality across two fundamental dimensions—status and power—thereby generating many of the ‘grand types’ that have been discerned by social scientists: class and caste, for instance. But, if it is to do this, there must be at least four contending ways of organizing and justifying within that totality. Cultural Theory (the four ways of organizing and justifying) is thus revealed to be inherent in the seemingly much simpler rubbish theory the three-category dynamical system): a surprising outcome that has some intriguing implications.