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31 
An Algorithmic Approach to The Lattice Structures of Attractors and Lyapunov functionsUnknown Date (has links)
Ban and Kalies [3] proposed an algorithmic approach to compute attractor
repeller pairs and weak Lyapunov functions based on a combinatorial multivalued
mapping derived from an underlying dynamical system generated by a continuous
map. We propose a more e cient way of computing a Lyapunov function for a Morse
decomposition. This combined work with other authors, including Shaun Harker,
Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes
the process of nding a global Lyapunov function for Morse decomposition very e 
cient. One of the them is to utilize highly memorye cient data structures: succinct
grid data structure and pointer grid data structures. Another technique is to utilize
Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is
an essential step to compute a Lyapunov function. Finally, another major technique
in achieving a signi cant improvement in e ciency is the utilization of the lattice
structures of the attractors and attracting neighborhoods, as explained in [32]. The
lattice structures have made it possible to let us incorporate only the joinirreducible
attractorrepeller pairs in computing a Lyapunov function, rather than having to use
all possible attractorrepeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical
system allow for general algebraic treatment of global gradientlike dynamics. The
separation of these algebraic structures from underlying topological structure is the
basis for the development of algorithms to manipulate those structures, [32, 31].
There has been much recent work on developing and implementing general compu
tational algorithms for global dynamics which are capable of computing attracting
neighborhoods e ciently. We describe the lifting of sublattices of attractors, which
are computationally less accessible, to lattices of forward invariant sets and attract
ing neighborhoods, which are computationally accessible. We provide necessary and
su cient conditions for such a lift to exist, in a general setting. We also provide
the algorithms to check whether such conditions are met or not and to construct the
lift when they met. We illustrate the algorithms with some examples. For this, we
have checked and veri ed these algorithms by implementing on some noninvertible
dynamical systems including a nonlinear Leslie model. / Includes bibliography. / Dissertation (Ph.D.)Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

32 
Some new results on hyperbolic gauss curvature flows. / CUHK electronic theses & dissertations collectionJanuary 2011 (has links)
Wo, Weifeng. / Thesis (Ph.D.)Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 99102). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

33 
Estudo de difusão caótica em um modelo de poço de potencial dependente do tempo /Graciano, Flávio Heleno. January 2018 (has links)
Orientador: Edson Denis Leonel / Banca: Juliano Antônio de Oliveira / Banca: Renê Orlando Medrado Torricos / Resumo: Neste trabalho consideramos o modelo do poço de potencial dependente do tempo e construimos de forma detalhada o mapeamento discreto bidimensional nas variáveis energia e fase que descreve a dinâmica do sistema. Mostramos que o espaço de fases é do tipo misto, contendo mares de caos, curvas invariantes e ilhas de estabilidade. Encontramos a matriz Jacobiana para o mapeamento assim como seu determinante, confirmando a propriedade de preservação de área. Estudamos a evolução no tempo da energia quadrática média e discutimos leis de escala para o comportamento dessa evolução. Por fim demos início à resolução da equação da difusão a fim de encontrarmos uma equação analitíca para energia quadrática média / Abstract: In this work we consider the model of the timedependent potential well and we construct in detail the twodimensional discrete mapping in the energy and phase variables that describes the dynamics of the system. We show that the phase space is of the mixed type, containing chaotic seas, invariant curves and stability islands. We obtain the Jacobian matrix for the mapping as well as its determinant, confirming the area preservation property. We study the evolution in time of the average squared energy and discuss scaling laws for the behavior of this evolution. Finally we started the resolution of the diffusion equation in order to find an analytical equation for mean quadratic energy / Mestre

34 
Group invariant solutions for some curvature driven flows. / CUHK electronic theses & dissertations collectionJanuary 1999 (has links)
by Guanxin Li. / "January 1999." / Thesis (Ph.D.)Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 223225). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

35 
Lagrangian angles of foliation in R² under curve shortening flow.January 2011 (has links)
Ma, Man Shun. / Thesis (M.Phil.)Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 7576). / Abstracts in English and Chinese. / Chapter 1  Introduction  p.6 / Chapter 2  Basic notions in Riemannian geometry  p.10 / Chapter 2.1  Basic manifold theory  p.11 / Chapter 2.2  "Connection, curvature"  p.19 / Chapter 2.3  Submanifold theory  p.29 / Chapter 3  Basic facts in symplectic and complex geometry  p.33 / Chapter 3.1  "Symplectic manifolds, Lagrangian submanifolds"  p.34 / Chapter 3.2  Kahler and CalabiYau manifolds  p.39 / Chapter 3.3  Calibration  p.49 / Chapter 4  Mean curvature flow  p.52 / Chapter 4.1  Basic equations in Lagrangian immersions  p.53 / Chapter 4.2  Evolution equation for  p.57 / Chapter 4.3  Evolution equations for H and θ  p.62 / Chapter 5  Lagrangian angle of a foliation  p.67 / Chapter 5.1  "Proof of equation (5.1), (5.2)"  p.68 / Chapter 5.2  Main theorem  p.70 / Chapter 5.3  Examples of invariant solution  p.73 / Bibliography  p.75

36 
On manifolds of nonpositive curvature.January 1997 (has links)
by Yiu Chun Chit. / Thesis (M.Phil.)Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 8182). / Chapter 1  Introduction  p.7 / Chapter 1.1  Riemannian Manifolds  p.7 / Chapter 1.1.1  Completeness  p.8 / Chapter 1.1.2  Curvature tensor  p.9 / Chapter 1.1.3  Holonomy  p.11 / Chapter 1.2  Simplyconnected Manifold of Nonpositive Sectional Curvature  p.11 / Chapter 1.2.1  Topological structure  p.12 / Chapter 1.2.2  Basic geometric properties  p.13 / Chapter 1.2.3  Examples of nonpositively curved manifold  p.20 / Chapter 1.2.4  Convexity properties  p.23 / Chapter 1.2.5  Points at infinity for M  p.27 / Chapter 2  Symmetric Spaces  p.36 / Chapter 2.1  Symmetric Spaces of Noncompact Type  p.36 / Chapter 2.1.1  Symmetric diffeomorphisms  p.36 / Chapter 2.1.2  Transvections in I(M)  p.38 / Chapter 2.1.3  Symmetric spaces as coset manifolds G/K  p.39 / Chapter 2.1.4  Metric on TpM and the adjoint representation of Lie group  p.41 / Chapter 2.1.5  Curvature tensor of M  p.43 / Chapter 2.1.6  Killing form and classification of symmetric spaces  p.44 / Chapter 2.1.7  Holonomy of M at p  p.44 / Chapter 2.1.8  Rank of a symmetric space M  p.45 / Chapter 2.1.9  Regular and singular points at infinity  p.46 / Chapter 2.2  "The Symmetric Space Mn = SL(n,R)/SO(n,R)"  p.46 / Chapter 2.2.1  Metric on TIMn  p.47 / Chapter 2.2.2  Geodesic and symmetries of Mn  p.48 / Chapter 2.2.3  Curvature of Mn  p.48 / Chapter 2.2.4  Rank and flats in Mn  p.49 / Chapter 2.2.5  Holonomy of Mn at I  p.49 / Chapter 2.2.6  Eigenvalueflag pair for a point in Mn(∞ )  p.50 / Chapter 2.2.7  Action of I0(Mn) on Mn(∞ )  p.52 / Chapter 2.2.8  Flags in opposition  p.53 / Chapter 2.2.9  Joining points at infinity  p.53 / Chapter 3  Group Action  p.56 / Chapter 3.1  Action of Isometries on M(oo)  p.56 / Chapter 3.1.1  Fundamental group as a group of isometries  p.56 / Chapter 3.1.2  Lattices  p.58 / Chapter 3.1.3  Duality condition  p.59 / Chapter 3.1.4  Geodesic flows  p.61 / Chapter 3.2  Action of Geodesic Symmetries on M(oo)  p.62 / Chapter 3.3  Rank  p.66 / Chapter 3.3.1  Rank of a manifold of nonpositive curvature  p.66 / Chapter 3.3.2  Rank of the fundamental group  p.68 / Chapter 3.4  Rigidity Theorems of Locally Symmetric Spaces  p.69

37 
Stability theory and numerical analysis of nonautonomous dynamical systems.Stonier, D. J., mikewood@deakin.edu.au January 2003 (has links)
The development and use of cocycles for analysis of nonautonomous behaviour is a technique that has been known for several years. Initially developed as an extension to semigroup theory for studying rionautonornous behaviour, it was extensively used in analysing random dynamical systems [2, 9, 10, 12].
Many of the results regarding asymptotic behaviour developed for random dynamical systems, including the concept of cocycle attractors were successfully transferred and reinterpreted for deterministic nonautonomous systems primarily by P. Kloeden and B. Schmalfuss [20, 21, 28, 29]. The theory concerning cocycle attractors was later developed in various contexts specific to particular classes of dynamical systems [6, 7, 13], although a comprehensive understanding of cocycle attractors (redefined as pullback attractors within this thesis) and their role in the stability of nonautonomous dynamical systems was still at this stage incomplete.
It was this purpose that motivated Chapters 13 to define and formalise the concept of stability within nonautonomous dynamical systems. The approach taken incorporates the elements of classical asymptotic theory, and refines the notion of pullback attraction with further development towards a study of pullback stability arid pullback asymptotic stability. In a comprehensive manner, it clearly establishes both pullback and forward (classical) stability theory as fundamentally unique and essential components of nonautonomous stability. Many of the introductory theorems and examples highlight the key properties arid differences between pullback and forward stability. The theory also cohesively retains all the properties of classical asymptotic stability theory in an autonomous environment. These chapters are intended as a fundamental framework from which further research in the various fields of nonautonomous
dynamical systems may be extended.
A preliminary version of a Lyapunovlike theory that characterises pullback attraction is created as a tool for examining nonautonomous behaviour in Chapter 5. The nature of its usefulness however is at this stage restricted to the converse theorem of asymptotic stability.
Chapter 7 introduces the theory of Loci Dynamics. A transformation is made to an alternative dynamical system where forward asymptotic (classical asymptotic) behaviour characterises pullback attraction to a particular point in the original dynamical system. This has the advantage in that certain conventional techniques for a forward analysis may be applied.
The remainder of the thesis, Chapters 4, 6 and Section 7.3, investigates the effects of perturbations and discretisations on nonautonomous dynamical systems known to possess structures that exhibit some form of stability or attraction. Chapter 4 investigates autonomous systems with semigroup attractors, that have been nonautonomously perturbed, whilst Chapter 6 observes the effects of discretisation on nonautonomous dynamical systems that exhibit properties of forward asymptotic stability. Chapter 7 explores the same problem of discretisation, but for pullback asymptotically stable systems. The theory of Loci Dynamics is used to analyse the nature of the discretisation, but establishment of results directly analogous to those discovered in Chapter 6 is shown to be unachievable. Instead a case by case analysis is provided for specific classes of dynamical systems, for which the results generate a numerical approximation of the pullback attraction in the original continuous dynamical system.
The nature of the results regarding discretisation provide a nonautonomous extension to the work initiated by A. Stuart and J. Humphries [34, 35] for the numerical approximation of semigroup attractors within autonomous systems. . Of particular importance is the effect on the system's asymptotic behaviour over nonfinite intervals of discretisation.

38 
Semihyperbolic mappings in Banach spaces.AlNayef, Anwar Ali Bayer, mikewood@deakin.edu.au January 1997 (has links)
The definition of semihyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional.
Bishadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudotrajectories calculated by a computer with the true trajectories. In this thesis, the concept of bishadowing in a Banach space is defined and proved for semihyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bishadowing with respect to pseudotrajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations.
Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semihyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation.
It is shown that semihyperbolic mappings are locally φcontracting, where 0 is the Hausdorff measure of noncompactness, and that a linear operator is semihyperbolic if and only if it is φcontracting and has no spectral values on the unit circle. The definition of φbishadowing is given and it is shown that semihyperbolic mappings in Banach spaces are φbishadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations.
Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is chaotic.

39 
Singular perturbation, state aggregation and nonlinear filteringJanuary 1981 (has links)
Omar Hijab, Shankar Sastry. / Bibliography: leaf [4]. / Caption title. "September, 1981." / Supported in part by NASA Grant no. 2384 Office of Naval Research under the JSEP Contract N0001475C0648 DOE Grant no. ETA012295T050

40 
Methods of dynamical systems, harmonic analysis and wavelets applied to several physical systemsPetrov, Nikola Petrov 28 August 2008 (has links)
Not available / text

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