The Creation of Strange Non-Chaotic Attractors in Non-Smooth Saddle-Node Bifurcations
The Creation of Strange Non-Chaotic Attractors in Non-Smooth Saddle-Node Bifurcations

Author: Tobias H. JŠger

Publisher: American Mathematical Soc.

Published: 2009-08-07

Total Pages: 120

ISBN-13: 082184427X

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The author proposes a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism, and its implementation in different models, is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which the author calls `exponential evolution of peaks'.

Strange Nonchaotic Attractors
Strange Nonchaotic Attractors

Author: Ulrike Feudel

Publisher: World Scientific

Published: 2006

Total Pages: 226

ISBN-13: 9812774408

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This book is the first monograph devoted exclusively to strange nonchaotic attractors (SNA), recently discovered objects with a special kind of dynamical behavior between order and chaos in dissipative nonlinear systems under quasiperiodic driving. A historical review of the discovery and study of SNA, mathematical and physically-motivated examples, and a review of known experimental studies of SNA are presented. The main focus is on the theoretical analysis of strange nonchaotic behavior by means of different tools of nonlinear dynamics and statistical physics (bifurcation analysis, Lyapunov exponents, correlations and spectra, renormalization group). The relations of the subject to other fields of physics such as quantum chaos and solid state physics are also discussed. Sample Chapter(s). Chapter 1: Introduction (122 KB). Contents: Models; Rational Approximations; Stability and Instability; Fractal and Statistical Properties; Bifurcations in Quasiperiodically Forced Systems and Transitions to SNA; Renormalization Group Approach to the Onset of SNA in Maps with the Golden-Mean Quasiperiodic Driving. Readership: Graduate students and researchers in nonlinear science.

The Parameterization Method for Invariant Manifolds
The Parameterization Method for Invariant Manifolds

Author: Àlex Haro

Publisher: Springer

Published: 2016-04-18

Total Pages: 280

ISBN-13: 3319296620

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This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.

Strange Nonchaotic Attractors
Strange Nonchaotic Attractors

Author: Ulrike Feudel

Publisher: World Scientific

Published: 2006

Total Pages: 226

ISBN-13: 9812566333

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This book is the first monograph devoted exclusively to strange nonchaotic attractors (SNA), recently discovered objects with a special kind of dynamical behavior between order and chaos in dissipative nonlinear systems under quasiperiodic driving. A historical review of the discovery and study of SNA, mathematical and physically-motivated examples, and a review of known experimental studies of SNA are presented. The main focus is on the theoretical analysis of strange nonchaotic behavior by means of different tools of nonlinear dynamics and statistical physics (bifurcation analysis, Lyapunov exponents, correlations and spectra, renormalization group). The relations of the subject to other fields of physics such as quantum chaos and solid state physics are also discussed. Key Features Topics are suitable for various disciplines dealing with nonlinear dynamics (mechanics, physics, nonlinear optics, hydrodynamics, chemical kinetics, etc.) A variety of theoretical tools is supplied to reveal different characteristics of strange nonchaotic behavior Readership: Graduate students and researchers in nonlinear science.

Nonautonomous Bifurcation Theory
Nonautonomous Bifurcation Theory

Author: Vasso Anagnostopoulou

Publisher: Springer Nature

Published: 2023-05-31

Total Pages: 159

ISBN-13: 303129842X

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Bifurcation theory is a major topic in dynamical systems theory with profound applications. However, in contrast to autonomous dynamical systems, it is not clear what a bifurcation of a nonautonomous dynamical system actually is, and so far, various different approaches to describe qualitative changes have been suggested in the literature. The aim of this book is to provide a concise survey of the area and equip the reader with suitable tools to tackle nonautonomous problems. A review, discussion and comparison of several concepts of bifurcation is provided, and these are formulated in a unified notation and illustrated by means of comprehensible examples. Additionally, certain relevant tools needed in a corresponding analysis are presented.

Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models
Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models

Author: Pierre Magal

Publisher: American Mathematical Soc.

Published: 2009

Total Pages: 84

ISBN-13: 0821846531

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Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, the authors study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, they use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.

Towards Non-Abelian P-adic Hodge Theory in the Good Reduction Case
Towards Non-Abelian P-adic Hodge Theory in the Good Reduction Case

Author: Martin C. Olsson

Publisher: American Mathematical Soc.

Published: 2011-02-07

Total Pages: 170

ISBN-13: 082185240X

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The author develops a non-abelian version of $p$-adic Hodge Theory for varieties (possibly open with ``nice compactification'') with good reduction. This theory yields in particular a comparison between smooth $p$-adic sheaves and $F$-isocrystals on the level of certain Tannakian categories, $p$-adic Hodge theory for relative Malcev completions of fundamental groups and their Lie algebras, and gives information about the action of Galois on fundamental groups.

Classification of Radial Solutions Arising in the Study of Thermal Structures with Thermal Equilibrium or No Flux at the Boundary
Classification of Radial Solutions Arising in the Study of Thermal Structures with Thermal Equilibrium or No Flux at the Boundary

Author: Alfonso Castro

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 87

ISBN-13: 0821847260

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The authors provide a complete classification of the radial solutions to a class of reaction diffusion equations arising in the study of thermal structures such as plasmas with thermal equilibrium or no flux at the boundary. In particular, their study includes rapidly growing nonlinearities, that is, those where an exponent exceeds the critical exponent. They describe the corresponding bifurcation diagrams and determine existence and uniqueness of ground states, which play a central role in characterizing those diagrams. They also provide information on the stability-unstability of the radial steady states.

Affine Insertion and Pieri Rules for the Affine Grassmannian
Affine Insertion and Pieri Rules for the Affine Grassmannian

Author: Thomas Lam

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 103

ISBN-13: 0821846582

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The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${\rm Gr}$ associated with $SL(n,\mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({\rm Gr})$ and $H_*({\rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({\rm Gr})$. A combinatorial interpretation of the pairing $H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z$ induced by the cap product.