Academic literature on the topic 'Linked twist maps'

AbstractLinked-twist maps are area-preserving, piecewise diffeomorphisms, defined on a subset of the torus. They are non-uniformly hyperbolic generalizations of the well-known Arnold cat map. We show that a class of canonical examples have polynomial decay of correlations for$\alpha $-Hölder observables, of order$1/ n$.

Abstract Rigorous, elementary upper and lower bounds upon the Lyapunov exponents of a parametrised family of linked twist maps are given, and obtained explicitly for a specific range of parameter values. The method used to obtain the bounds utilises the existence of invariant cones for specific products of the underlying family of shear maps, and the return time partition of the overlap region of the two annuli. Improvements upon the accuracy of this method are then obtained by considering preceding sequences of matrices on the orbits.

We consider a two-parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and discussed. It is proved that there exists a one-parameter family of twists identical to a simple Jordanian twist. The twisted coalgebra, star product and coordinate realizations of the [Formula: see text]-Minkowski noncommutative space–time are presented. Real forms of Jordanian deformations are also discussed. The method of similarity transformations is applied to the Poincaré–Weyl Hopf algebra and two types of one-parameter families of dispersion relations are constructed. Mathematically equivalent deformations, that are related to nonlinear changes of symmetry generators and linked with similarity maps, may lead to differences in the description of physical phenomena.

AbstractWe construct linked twist maps of the two-dimensional torus which are Bernoulli and possess infinite entropy. Inparticular, we construct a Bernoulli toral linked twist map B* of infinite entropy which has smooth, absolutely continuous local (un)stable manifolds and positive Lyapunov exponents defined almost everywhere. This map is continuous at each point save those on two line segments.B* is shown to be stochastically stable under the following random perturbation: apply the map to a point p and then jump (all points move the same distance and in the same direction) according to a B-process (not necessarily an independent process) such that the expected distance moved is equal to r. Stochastic stability means that given α > 0 if r > 0 is sufficiently small then the perturbed and unperturbed systems are α-congruent. We prove a similar stability result for B* under a perturbation in which the random jump described above is distributed according to a general stochastic process. These stability results are also shown to hold (in a slightly modified form) for a general class of finite-entropy toral linked twist maps under the same perturbations.

We study the mixing properties of two systems: (i) a half-filled quasi-two-dimensional circular drum whose rotation rate is switched between two values and which can be analysed in terms of the existing mathematical formalism of linked twist maps; and (ii) a half-filled three-dimensional spherical tumbler rotated about two orthogonal axes bisecting the equator and with a rotational protocol switching between two rates on each axis, a system which we call a three-dimensional linked twist map, and for which there is no existing mathematical formalism. The mathematics of the three-dimensional case is considerably more involved. Moreover, as opposed to the two-dimensional case where the mathematical foundations are firm, most of the necessary mathematical results for the case of three-dimensional linked twist maps remain to be developed though some analytical results, some expressible as theorems, are possible and are presented in this work. Companion experiments in two-dimensional and three-dimensional systems are presented to demonstrate the validity of the flow used to construct the maps. In the quasi-two-dimensional circular drum, bidisperse (size-varying or density-varying) mixtures segregate to form lobes of small or dense particles that coincide with the locations of islands in computational Poincaré sections generated from the flow model. In the 3d spherical tumbler, patterns formed by tracer particles reveal the dynamics predicted by the flow model.

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

This study maps the media discourses surrounding Ben Johnson’s life “in the fast lane” to further understand one particular aspect of a contemporary crisis of identity (or, more accurately, identities) in Canada. Specifically, this study provides: (a) a context within which to locate Johnson’s rise and fall from hero to scapegoat as articulated to the 1988 crisis of Canadian identity; (b) a chronology of the twist of race, or changing racial discourses which serve to define and redefine Ben Johnson’s racial and national identities; and (c) a discussion of the politics of identity in relation to multiculturalism and the representation of Ben Johnson as the “other” in Canada. The results reveal that Ben Johnson’s identity was the subject of a range of representations including those linked to racist stereotypes. Moreover, the results suggest that the discourses defining Ben Johnson are constituted by, and constitutive of, broader debates about identity in Canada.

In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Such theory was developed in the planar case by Papini and Zanolin in [11,12] and it has been extended to the N-dimensional framework by the author and Zanolin in [16]. In the bidimensional setting, elementary theorems from plane topology suffice, while in the higher dimension some results from degree theory are needed, leading to the study of the so-called "Cutting Surfaces" [16]. Our method is also significant from a dynamical point of view, as it allows to detect complex dynamics. As it is well-known, a prototypical example of chaotic system is represented by the Smale horseshoe. However, in order to prove conjugacy with the shift map, it requires the verification of hyperbolicity conditions, which are difficult or impossible to prove in practical cases. For such reason more general and less stringent definitions of horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe while discarding the hyperbolicity hypotheses. This led to the study of the so-called "topological (or geometrical) horseshoes" [2,5]. In particular, different characterizations have been proposed by various authors in order to establish the presence of complex dynamics for continuous maps defined on subsets of the N-dimensional Euclidean space (see, for instance, [10,21,23] and the references therein). The tools employed in these and related works range from the Conley index [10] to the Lefschetz fixed point theory [20]. On the other hand, our approach, although mathematically rigorous, avoids the use of more advanced topological theories and it is relatively easy to apply to specific models arising in applications. For example we have employed such method to study discrete and continuous-time models arising from economics and biology [9,18]. In more details, the topics considered along the thesis can be summarized as follows. The description of the Stretching Along the Paths method and suitable variants of it can be found in Chapter 1. In Chapter 2 we discuss which are the chaotic features that can be obtained for a given map when our technique applies. In particular, we are able to prove semi-conjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover we show the mutual relationships among various classical notions of chaos (such as those by Devaney, Li-Yorke, etc.). We also introduce an alternative geometrical framework related to the so-called "Linked Twist Maps" [3,4,22], where it is possible to employ our method in order to detect complex dynamics. The theoretical results obtained so far find an application to discrete and continuous-time systems in Chapters 3 and 4. As regards the former, in Chapter 3 we deal with some one-dimensional and planar discrete economic models, both of the Overlapping Generation and of the Duopoly Game classes. The bidimensional models are taken from [8,19] and [1], respectively. On the other hand, in Chapter 4, with respect to continuous-time models, we study some nonlinear ODEs with periodic coefficients through a combination of a careful but elementary phase-plane analysis with the results on chaotic dynamics for Linked Twist Maps from Chapter 2. In more details, we consider a modified version of the Volterra predator-prey model, in which a periodic harvesting is included, as well as a simplification of the Lazer-McKenna suspension bridges model [6,7] from [13,14]. When dealing with ODEs with periodic coefficients, our method is applied to the associated Poincaré map. The contents of the present thesis are based on the papers [9,13,16,17,18] and partially on [14], where maps expansive along several directions were considered. [1] H.N. Agiza and A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004), 843-860. [2] K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), 95-118. [3] R. Burton and R.W. Easton, Ergodicity of linked twist maps, In: Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 35-49, Lecture Notes in Math., 819, Springer, Berlin, 1980. [4] R.L. Devaney, Subshifts of finite type in linked twist mappings, Proc. Amer. Math. Soc. 71 (1978), 334-338. [5] J. Kennedy and J.A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001), 2513-2530. [6] A.C. Lazer and P.J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. Henry Poincar e, Analyse non lineaire 4 (1987), 244-274. [7] A.C. Lazer and P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537-578. [8] A. Medio, Chaotic dynamics. Theory and applications to economics, Cambridge University Press, Cambridge, 1992. [9] A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions. A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 3283-3309. [10] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205-236. [11] D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud. 4 (2004), 71-91. [12] D. Papini and F. Zanolin, Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells, Fixed Point Theory Appl. 2004 (2004), 113-134. [13] A. Pascoletti, M. Pireddu and F. Zanolin, Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, Electron. J. Qual. Theory Differ. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ. 14 (2008), 1-32. [14] A. Pascoletti and F. Zanolin, Example of a suspension bridge ODE model exhibiting chaotic dynamics: a topological approach, J. Math. Anal. Appl. 339 (2008), 1179-1198. [15] M. Pireddu and F. Zanolin, Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on N-dimensional cells, Adv. Nonlinear Stud. 5 (2005), 411-440. [16] M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics, Topol. Methods Nonlinear Anal. 30 (2007), 279-319. [17] M. Pireddu and F. Zanolin, Some remarks on fixed points for maps which are expansive along one direction, Rend. Istit. Mat. Univ. Trieste 39 (2007), 245-274. [18] M. Pireddu and F. Zanolin, Chaotic dynamics in the Volterra predator-prey model via linked twist maps, Opuscula Math. 28/4 (2008), 567-592. [19] P. Reichlin, Equilibrium cycles in an overlapping generations economy with production, J. Econom. Theory 40 (1986), 89-102. [20] R. Srzednicki, A generalization of the Lefschetz fixed point theorem and detection of chaos, Proc. Amer. Math. Soc. 128 (2000), 1231-1239. [21] R. Srzednicki and K. Wojcik, A geometric method for detecting chaotic dynamics, J. Differential Equations 135 (1997), 66-82. [22] S. Wiggins, Chaos in the dynamics generated by sequence of maps, with application to chaotic advection in flows with aperiodic time dependence, Z. angew. Math. Phys. 50 (1999), 585-616. [23] P. Zgliczy nski and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 32-58.

Compressor maps of aero engines show the relation between corrected mass flow, corrected shaft speed, pressure ratio, and efficiency, where different operating conditions of the compressor are represented by different speed lines. These speed lines are an important information for the compressor design process, since they show important operation bounds like surge and choke. Typically, 3D CFD compressor maps are computed with the so called hot geometry given by the aerodynamic design point. But in reality aerofoil shapes change depending on engine speeds and gas loads resulting in twist of the blades and changes of tip clearance. In order to obtain a higher quality compressor map, all these effects must be taken into account. Therefore, a process is utilized which uses coupled CFD and FE analyses to account for load adjusted geometries aside the design point. For transformation of FE results into the CFD model a cold-to-hot blade morphing technique is used. The studies are performed for a 4.5 stage high speed axial compressor, where effects of varying tip clearance and geometry deformation are considered separately from each other. Finally, their combined effects are studied.

In this investigation, a three dimensional gradient deficient beam element (BEAM9) using the absolute nodal coordinate formulation (ANCF) is introduced. This element has nine coordinates per node, this includes the position vector and the two gradient vectors rx and ry. Like most of the ANCF elements, this element has constant mass matrix and zero centrifugal and Coriolis inertia forces. The plane strain elastic force model and the elastic line approach are two elastic force models presented in this paper in order to simulate the element internal resistance. Both models support resistance to the general bending and twist moments. The possibilities of employing nonlinear material models will be discussed in future work. Furthermore, the proposed element has the advantage of easy integration over general cross section area that is not easy to perform using the fully parameterized ANCF beam element (BEAM12). Comparing to the ANCF cable element (BEAM6), the proposed element can resist general bending and twist loads. Moreover, shear deformations in the xy plane due to shear force and in the yz plane due to twist moment are considered with the gradient deficient beam element proposed in this work. However, no shear deformations are considered with the ANCF cable element. Comparing to the fully parameterized ANCF beam element, the gradient deficient beam element (BEAM9) avoids some locking issues, shows better computational efficiency and offers better convergency characteristics. Numerical examples are presented in order to validate the proposed gradient deficient beam element and to compare with other ANCF beam elements.

High resolution RANS CFD analysis is performed to support the design and development of the Ocean Renewable Power Company (ORPC) TidGen™ multi-directional tidal turbine. Two-dimensional and three-dimensional unsteady, moving-mesh CFD is utilized to parameterize the device performance and to provide guidance for device efficiency improvements. The unsteady CFD analysis was performed using a well validated, naval hydrodynamic CFD solver and implementing dynamic overset meshes to perform the relative motion between geometric components. This dynamic capability along with the turbulence model for the expected massively separated flows was validated against published data of a high angle of attack pitching airfoil. Two-dimensional analyses were performed to assess both blade shape and operating conditions. The blade shape performance was parameterized on both blade camber and trailing edge thickness. The blades shapes were found to produce nearly the same power generation at the peak efficiency tip speed ratio (TSR), however off-design conditions were found to exhibit a strong dependency on blade shape. Turbine blades with the camber pointing outward radially were found to perform best over the widest range of TSR’s. In addition, a thickened blade trailing edge was found to be superior at the highest TSR’s with little performance degradation at low TSR’s. Three-dimensional moving mesh analyses were performed on the rotating portion of the full TidGen™ geometry and on a turbine blade stack-up. Partitioning the 3D blades axially showed that no sections reached the idealized 2D performance. The 3D efficiency dropped by approximately 12 percentage points at the peak efficiency TSR. A blade stack-up analysis was performed on the complex 3D/barreled/twisted turbine blade. The analysis first assessed the infinite length blade performance, next end effects were introduced by extruding the 2D foil to the nominal 5.6m length, next barreling was added to the straight foils, and finally twist was added to the foils to reproduce the TidGen™ geometry. The study showed that making the blades a finite length had a large negative impact on the performance, whereas barreling and twisting the foils had only minor impacts. Based on the 3D simulations the largest factor impacting performance in the 3D turbine was a reduction in mass flow through the turbine due to the streamlines being forces outward in the horizontal plane due to the turbine flow resistance. Strategies to mitigate these 3D losses were investigated, including adding flow deflectors on the turbine support structure and stacking multiple turbines in-line.