Some Global Aspects of Homoclinic Bifurcations of Vector Fields PDF Download
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Author: Ale Jan Homburg
Publisher:
ISBN:
Category :
Languages : en
Pages : 136
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Book Description
Author: Ale Jan Homburg
Publisher:
ISBN:
Category :
Languages : en
Pages : 136
Get Book Here
Book Description
Author: Ale Jan Homburg
Publisher: American Mathematical Soc.
ISBN: 0821804413
Category : Mathematics
Languages : en
Pages : 143
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Book Description
In this book, the author investigates a class of smooth one parameter families of vector fields on some $n$-dimensional manifold, exhibiting a homoclinic bifurcation. That is, he considers generic families $x_\mu$, where $x_0$ has a distinguished hyperbolic singularity $p$ and a homoclinic orbit; an orbit converging to $p$ both for positive and negative time. It is assumed that this homoclinic orbit is of saddle-saddle type, characterized by the existence of well-defined directions along which it converges to the singularity $p$. The study is not confined to a small neighborhood of the homoclinic orbit. Instead, the position of the stable and unstable set of the homoclinic orbit is incorporated and it is shown that homoclinic bifurcations can lead to complicated bifurcations and dynamics, including phenomena like intermittency and annihilation of suspended horseshoes.
Author: Pablo Aguirre
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
We consider certain kinds of homoclinic bifurcations in three-dimensional vector fields. These global bifurcations are characterized by the existence of a homo clinic orbit that converges to a saddle equilibrium in both forward and backward time. If the equilibrium has a complex pair of (stable) eigenvalues, it is a saddle-focus, and one speaks of a Shilnikov homoclinic orbit. In this case, the homoclinic orbit converges towards the equilibrium in a spiralling fashion. On the other hand, if the saddle equilibrium has two real (stable) eigenvalues, then the homoclinic orbit converges generically to the saddle along the direction given by the weak stable eigenvector. The possible unfoldings of a codimension-one homoclinic bifurcation depend on the sign of the saddle quantity: when it is negative, breaking the homoclinic orbit results in a single stable periodic orbit from a saddle-focus homoclinic orbit; one speaks of a simple Shilnikov bifurcation. However, when the saddle quantity is positive, then the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of a chaotic Shilnikov bifurcation. For a homoclinic orbit to a real saddle, on the other hand, always a single periodic orbit bifurcates, which is attracting when the saddle quantity is negative and of saddle type when it is positive. In this thesis we show how the global three-dimensional phase space is organized near certain homoclinic bifurcations by the two-dimensional global stable manifolds of equilibria and periodic orbits. To this end, we consider a model of a laser with optical injection that contains Shilnikov homoclinic orbits and a model by Sandstede that features different kinds of homoclinic bifurca- tions to a saddle. We find that, in the simple Shilnikov case, the stable manifold ofthe saddle-focus forms the basin boundary of the bifurcating stable periodic orbit. On the other hand, in the chaotic case, the stable manifold of the equilibrium is the accessible set of a chaotic saddle that contains countably many periodic orbits of saddle type. In the case of a homoclinic bifurcation to a saddle, the stable manifold of the saddle is either an orientable or nonorientable two-dimensional surface. A change of orientability occurs at two kinds of codimension-two homoclinic bifurcations, called inclination flip and orbit flip bifurcations. At either of these flip bifurcation points, the stable manifold is neither orientable nor nonorientable, but just at the transition between both states. We show how this transition occurs for the case of negative saddle quantity, and how the basin of attraction of the stable periodic orbit is organized in different ways by the stable manifold of the saddle depending on the (non)orientability of the bifurcation. Finally, we show how the stable manifold rearranges both itself and the overall dynamics in phase space near the codimension-two transition from a saddle to saddle-focus homoclinic bifurcation that occurs at a so-called Belyakov point.
Author: William F. Langford
Publisher: American Mathematical Soc.
ISBN: 9780821885871
Category : Mathematics
Languages : en
Pages : 316
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Book Description
This volume presents new research on normal forms, symmetry, homoclinic cycles, and chaos, from the Workshop on Normal Forms and Homoclinic Chaos held during The Fields Institute Program Year on Dynamical Systems and Bifurcation Theory in November 1992, in Waterloo, Canada. The workshop bridged the local and global analysis of dynamical systems with emphasis on normal forms and the recently discovered homoclinic cycles which may arise in normal forms. Specific topics covered in this volume include normal forms for dissipative, conservative, and reversible vector fields, and for symplectic maps; the effects of symmetry on normal forms; the persistence of homoclinic cycles; symmetry-breaking, both spontaneous and induced; mode interactions; resonances; intermittency; numerical computation of orbits in phase space; applications to flow-induced vibrations and to mechanical and structural systems; general methods for calculation of normal forms; and chaotic dynamics arising from normal forms. Of the 32 presentations given at this workshop, 14 of them are represented by papers in this volume.
Author: Shui-Nee Chow
Publisher: Cambridge University Press
ISBN: 0521372267
Category : Mathematics
Languages : en
Pages : 482
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Book Description
This book is concerned with the bifurcation theory, the study of the changes in the structures of the solution of ordinary differential equations as parameters of the model vary.
Author: Leon O Chua
Publisher: World Scientific
ISBN: 9814494291
Category : Science
Languages : en
Pages : 591
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Book Description
Bifurcation and chaos has dominated research in nonlinear dynamics for over two decades, and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book has been written to serve that unfulfilled need.Following the footsteps of Poincaré, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in the book have been developed only recently and have not yet appeared in textbook form.In keeping with the self-contained nature of the book, all the topics are developed with introductory background and complete mathematical rigor. Generously illustrated and written at a high level of exposition, this invaluable book will appeal to both the beginner and the advanced student of nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject.
Author: Gardini Laura
Publisher: World Scientific
ISBN: 9811204713
Category : Mathematics
Languages : en
Pages : 648
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Book Description
The investigation of dynamics of piecewise-smooth maps is both intriguing from the mathematical point of view and important for applications in various fields, ranging from mechanical and electrical engineering up to financial markets. In this book, we review the attracting and repelling invariant sets of continuous and discontinuous one-dimensional piecewise-smooth maps. We describe the bifurcations occurring in these maps (border collision and degenerate bifurcations, as well as homoclinic bifurcations and the related transformations of chaotic attractors) and survey the basic scenarios and structures involving these bifurcations. In particular, the bifurcation structures in the skew tent map and its application as a border collision normal form are discussed. We describe the period adding and incrementing bifurcation structures in the domain of regular dynamics of a discontinuous piecewise-linear map, and the related bandcount adding and incrementing structures in the domain of robust chaos. Also, we explain how these structures originate from particular codimension-two bifurcation points which act as organizing centers. In addition, we present the map replacement technique which provides a powerful tool for the description of bifurcation structures in piecewise-linear and other form of invariant maps to a much further extent than the other approaches.
Author: Yuri A. Kuznetsov
Publisher: Springer Nature
ISBN: 3031220072
Category : Mathematics
Languages : en
Pages : 722
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Book Description
Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
Author: Moshé Flato
Publisher: American Mathematical Soc.
ISBN: 0821806831
Category : Mathematics
Languages : en
Pages : 328
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Book Description
The purpose of this work is to present and give full proofs of new original research results concerning integration of and scattering for the classical Maxwell-Dirac equations.
Author: Jin Nakagawa
Publisher: American Mathematical Soc.
ISBN: 0821804723
Category : Mathematics
Languages : en
Pages : 90
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Book Description
In this book, the author studies the Dirichlet series whose coefficients are the number of orders of a quartic field with given indices. Nakagawa gives an explicit expression of the Dirichlet series. Using this expression, its analytic properties are deduced. He also presents an asymptotic formula for the number of orders in a quartic field with index less than a given positive number.
