Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation

In the present work the methods of computation of Lyapunov quantities and localization of limit cycles are demonstrated. These methods are applied to investigation of quadratic systems with small and large limit cycles. The expressions for the first five Lyapunov quantities for general Lienard system are obtained. By the transformation of quadratic system to Lienard system and the method of asymptotical integration, quadratic systems with large limit cycles are investigated. The domain of parameters of quadratic systems, for which four limit cycles can be obtained, is determined.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

A criterion for the existence of four limit cycles in quadratic systems

A method for the asymptotic integration of the trajectories is proposed for the Liénard equation. The results obtained by this method are used to prove the existence of two “large” limit cycles in quadratic systems with a weak focus. The application of standard procedures of small perturbations of the parameters of quadratic systems enables one to find additionally two “small” limit cycles. It is shown that the criterion obtained for the existence of four limit cycles generalizes the well known Shi theorem.     

Conditions for the Existence of Two Limit Cycles in a System with Hysteresis Nonlinearity

This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.     

Ricci curvature and monotonicity for harmonic functions

In this paper we generalize the monotonicity formulas of “Colding (Acta Math 209:229–263, 2012)” for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., “Almgren (Preprint)”, “Colding and Minicozzi II (PNAS, 2012)”, “Garofalo and Lin (Indiana Univ Math 35:245–267, 1986)” for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green’s function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.     

Error formulas for ideal interpolation

We study the algebraic structure of error formulas for ideal interpolation. We introduce the so-called “normal” error formulas and prove that the lexicographic order reduced Gröbner basis admits such a formula for all ideal interpolations. This formula is a generalization of the “good” error formula proposed by Carl deBoor. Finally, we discuss a Shekhtman’s example and give an explicit form of “normal” error formula for this example.     

Asymptotic computation for transient heat conduction performance of periodic porous materials in curvilinear coordinates by the second‐order two‐scale method

A novel second‐order two‐scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non‐periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non‐periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.     

Contextualism,art, and rigidity: Levinson,Currie and Davies

The topic of this paper is the role played by context in art. In this regard I examine three theories linked to the names of J. Levinson, G. Currie and D. Davies. Levinson’s arguments undermine the structural theory. He finds it objectionable because it makes the individuation of artworks independent of their histories. Secondly, such a consequence is unacceptable because it fails to recognise that works are created rather than discovered. But, if certain general features of provenance are always work-constitutive, as it seems that Levinson is willing to claim, these features must always be essential properties of works. On the other hand, consideration of our modal practice suggests that whether a given general feature of provenance is essential or non-essential depends upon the particular work in question or is “work relative”. D. Davies builds his performance theory on the basis of the critical evaluation of Currie’s action-type hypotheses (ATH). Performances, says Davies, are not to be identified with “basic actions” to which their times belong essentially, but with “doings” that permit of the sorts of variation in modal properties required by the work-relativity of modality. He is also a fierce critic of the contextualist account. Contextualism is in his view unable to reflect the fact that aspects of provenance bear upon our modal judgements with variable force.In the second part of the paper I consider Davies’s “modality principle”. Davies is inclined to defend the claim that labels used for designation of works are rigid designators. Such a view offers a ground for discussion about the historicity of art. What has been meant when people claim that art is an historical concept? I argue that any historical theory implies a two-dimensional notion of “art”. At the end of the paper I suggest that Davies should embrace the theory of contingent identity and not the colocationist view about the relationship that exists between a particular artwork and its physical bearer.     

Cauchy problem for Mathieu’s equation at parametric resonance

Mathieu’s equation is solved by an asymptotic averaging method in the fourth approximation for the first to fourth resonance domains and in the third approximation for the zero resonance domain. The general periodic and aperiodic solutions on characteristic curves are found, and the general solution is obtained in instability domains and stability-domain areas adjacent to the characteristic curves. All the solutions are explicitly found in the form of functions of an argument without using the auxiliary parameter employed in Whittaker’s method. Simple formulas depending on two parameters of the equation are derived for the characteristic exponent in instability domains and for the frequency of slow oscillations in stability domains near the characteristic curves. The theory is developed by analyzing the resonances exhibited by Mathieu’s equation.     

On Some Special Effects in Theory on Numerical Integration and Functions Recovery

We discuss two questions. First, we consider the existence of close to optimal quadrature formulas with a “bad” L²-discrepancy of their grids, and the second is the question of how much explicit quadrature formulas are preferable to sorting algorithms. Also, in the model case, we obtain the solution to the question of approximative possibilities of Smolyak’s grid in the problems of recovery of functions.     

Confirmation,Increase in Probability,and the Likelihood Ratio Measure: a Reply to Glass and McCartney

Bayesian confirmation theory is rife with confirmation measures. Zalabardo (2009) focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other three measures. Glass and McCartney (2015), hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p(H | E) and a certain other probability involving H. I then evaluate G&M’s argument (with a minor fix) in light of them.     

Updating tk−1 is significant to Caputo fractional order switching systems: A reply to Hu’s comments

This is to reply the concerns about our NAHS paper “Lyapunov and external stability of Caputo fractional order switching systems” pointed out by Hu in “Comments on ‘Lyapunov and external stability of Caputo fractional order switching systems”’ recently.     

Counting generalized Jenkins–Strebel differentials

We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $\mathbb{C }\!\mathrm{P }^1$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees.     

Sounds and pictures: dynamism and dualism in Dynamic Geometry

In this paper, we examine and evaluate several new mathematical representations developed for The Geometer’s Sketchpad v5 (GSP5) from the perspective of their dynamic mathematical and pedagogic utility or expressibility. We claim the primary contributions of Dynamic Geometry’s principle of dynamism to the emerging concept of “Dynamic Mathematics” to be twofold: first, the powerful, temporalized representation of continuity and continuous change (dynamism’s mathematical aspect), and second, the sensory immediacy of direct interaction with mathematical representations (dynamism’s pedagogic aspect). Seen from this perspective, the growth of “Dynamic Mathematics software,” beyond the initial conception of first-generation planar geometry systems, represents a tremendous diversification and expansion of the mathematical domain of the dynamic principle’s applicability (for example, to dynamic statistics, graphing and 3D geometry). But at the same time, this expansion has come at the cost of a decrease in the immediacy of sensory interactions with mathematical representations, as in so-called dynamic graphing, wherein users modify a graph “at a distance” (through slider-based manipulation of the coefficients of its symbolic equation), or in solid geometry tools, in which users’ interactions with represented solids are mediated and distanced by the inevitably-2D communication interfaces of the computer mouse and screen. Thus we focus on this second aspect–sensory interaction with mathematical representations—in evaluating how novel dynamic representations in GSP5 affect mathematical modeling opportunities, student activity and engagement.     

The Besicovitch–Federer projection theorem is false in every infinite-dimensional Banach space

The theme of the article is the study of the unipotent part of Arthur’s trace formula for general linear groups. The case of regular (or “regular by blocks”) unipotent orbits has been treated in another paper (cf. [10]). Here we are interested in the contribution of Richardson orbits that are induced by Levi subgroups with two-by-two distinct blocks. In this case, the contribution is remarkably given by a global unipotent weighted orbital integral. As a corollary, we get integral formulas for some of Arthur’s global coefficients. We also present a new construction of Arthur’s local unipotent weighted orbital integral and an explicit computation of some of them.     

A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities

Given the importance of understanding and using indeterminate quantities in algebraic thinking, the development of learning trajectories about how Kindergarten and first grade students understand variable and use variable notation in the context of algebraic expressions is critical. Based on an empirically developed learning trajectory, we analyzed children’s responses at three different points in a classroom teaching experiment. Our purpose was to describe levels of thinking among 16 students (eight in Kindergarten and eight in first grade). Our results revealed qualitative changes in the thinking about indeterminate quantities of most student participants. As students progressed through the experiment, we found that they advanced from what we characterized as a “Pre-variable” Level to a “Letters as representing indeterminate quantities as varying unknowns; explicit operations on indeterminate quantities” Level. Learning trajectories such as that developed here hold promise for informing the design of interventions that support young children’s early algebraic thinking.     

The loop quantities and bifurcations of homoclinic loops

The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.     

Correction to: Fiber Path Optimization in a Variable-Stifness Cylinder to Maximize Its Buckling Load Under External Hydrostatic Pressure

Mechanics of Composite Materials - In Table 1 on page 769, the second and third columns’ title should be: “Lay-up corresponding to the maximum buckling pressure”     

Artworks,context and ontology

Horgan believes that the truth of the statement “Beethoven’s fifth symphony has four movements” does not require that there be some “dedicated object” answering to the term “Beethoven’s fifth simphony”. To the contrary, the relevant language/world correspondence relation is less direct than this. Especially appropriate is the behavior by Beethoven that we would call “composing his fifth symphony”.Our objections go along two directions: (1) is the process ontology (a) really a right kind of ontology for artworks (symphonies, novels) and, (b) more important, is this kind of ontology compatible with Parmenidian approach to ontology? The answer to (a) is negative. With reference to point (1b) we might say that Parmenides was a typical staunch advocate of substance metaphysics rather than process metaphysics. Traditionally, substances are individuated by their properties, namely, there are assumed two sorts of properties: primary and secondary. Primary properties describe the substance as it is in and by itself; secondary properties underlie the impact of substances upon others and the responses they elicit from them. (2) The claim that it is most unlikely to suppose that we can have some kind of cognitive contact “with an entity which has no spatio-temporal location and does not causally interact with anything” does not hold. We underpin the claim that there is some cognitive access to entity such as Beethoven’s fifth symphony with Bender’s theory of realization relation between musical work and performance.