Global analysis of dynamical systems on low-dimensional manifolds.

The interaction of topology and dynamics has attracted a great deal of attention from numerous mathematicians. This thesis is devoted to the study of dynamical systems on low-dimensional manifolds. In the order of dimensions, we first look at the case of two-manifolds (surfaces) and derive explicit differential equations for dynamical systems defined on generic surfaces by applying elliptic and automorphic function theory to uniformise the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series, we will determine general meromorphic systems on a fundamental domain in the upper half plane, the solution trajectories of which 'roll up' onto an appropriate surface of any given genus. Meanwhile, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by analyzing the topology of inversely unstable solutions contained within each invariant set. Then the thesis concerns a study of three-dimensional systems. We give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tubular neighbourhood of the braid. The second one, generating chaotic systems, is realized by modelling the Smale horseshoe. Moreover, we shall consider the analytical and topological structure of systems on 2- and 3- manifolds. By considering surgery operations, such as Dehn surgery, Heegaard splittings and connected sums, we shall show that it is possible to obtain systems with 'arbitrarily strange' behaviour, Le., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We will also consider diffeomorphisms which are defined on closed 3-manifolds and contain generalized Smale solenoids as the non-wandering sets. Motivated by the result in [Jiang, Ni and Wang, 2004], we will investigate the possibility of generating dynamical systems containing an arbitrary number of solenoids on any closed, orientable 3-manifold. This shall also include the study of branched coverings and Reeb foliations. Based on the intense development from four-manifold theory recently, we shall consider four-dimensional dynamical systems at the end. However, this part of the thesis will be mainly speculative

On dynamical systems

Part A. We prove existence of smooth invariant circles for area preserving twist maps close enough to integrable using renormalisation. The smoothness depends upon that of the map and the Liouville exponent of the rotation number. Part B. Ruelle and Capocaccia gave a new definition of Gibbs states on Smale spaces. Equilibrium states of suitable function there on are known to be Gibbs states. The converse in discussed in this paper, where the problem is reduced to shift spaces and there solved by constructing suitable conjugating homeomorphisms in order to verify the conditions for Gibbs states which Bowen gave for shift spaces, where the equivalence to equilibrium states is known. Part C. On subshifts which are derived from Markov partitions exists an equivalence relation which idendifies points that lie on the boundary set of the partition. In this paper we restrict to symbolic dynamics. We express the quotient space in terms of a non-transitive subshift of finite type, give a necessary and sufficient condition for the existence of a local product structure and evaluate the Zeta function of the quotient space. Finally we give an example where the quotient space is again a subshift of finite type

The Hopf Bifurcation and Its Applications

The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations. Historically, the subject had its origins in the works of Poincaré [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term "Poincaré-Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle-Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper. These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and Kirchgässner and Kielhoffer [1] (contributed by G. Childs and O. Ruiz) are given. The contributions of S. Smale, J. Guckenheimer and G. Oster indicate applications to the biological sciences and that of D. Schmidt to Hamiltonian systems. For other applications and related topics, we refer to the monographs of Andronov and Chaiken [1], Minorsky [1] and Thom [1]. The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value. In Hopf's original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation. We have given explicit algorithms for this calculation which are easy to apply in examples. (See Section 4, and Section 5A for comparison with Hopf's formulae). The method of averaging, exposed here by S. Chow and J. Mallet-Paret in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly. In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for t > O. This enables the invariant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]). To aid in determining smoothness in examples we have presented parts of the results of Dorroh-Marsden. [1]. Similar ideas for utilizing smoothness have been introduced independently by other authors, such as D. Henry [1]. Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here. We have selected results of Chafee [1] and Ruelle [3] (the latter is exposed here by S. Schecter) to indicate some generalizations that are possible. The subject is by no means closed. Applications to instabilities in biology (see, e.g. Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf. Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field (cf. Durand [1]) are appearing at a rapid rate. Also, the qualitative theory proposed by Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9). In this direction, the papers of Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important. Stable oscillations in nonlinear waves may be another fruitful area for application; cf. Whitham [1]. We hope these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods. After we completed our stability calculations we were happy to learn that others had found similar difficulty in applying Hopf's result as it had existed in the literature to concrete examples in dimension ≥ 3. They have developed similar formulae to deal with the problem; cf. Hsü and Kazarinoff [1, 2] and Poore [1]. The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type. The new proof, relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations. These notes originated in a seminar given at Berkeley in 1973-4. We wish to thank those who contributed to this volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive. Many other references are contained in the lengthy bibliography in Cesari [1]. We also thank those who have taken an interest in the notes and have contributed valuable comments. These include R. Abraham, D. Aronson, A. Chorin, M. Crandall., R. Cushman, C. Desoer, A. Fischer, L. Glass, J. M. Greenberg, O. Gurel, J. Hale, B. Hassard, S. Hastings, M. Hirsch, E. Hopf, N. D. Kazarinoff, J. P. LaSalle, A. Mees, C. Pugh, D. Ruelle, F. Takens, Y. Wan and A. Weinstein. Special thanks go to J. A. Yorke for informing us of the material in Section 3C and to both he and D. Ruelle for pointing out the example of the Lorentz equations (See Example 4B.8). Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript. Jerrold Marsden Marjorie McCracke

Analysis and Computation of Equilibria and Regions of Stability, With Applications in Chemistry, Climatology, Ecology, Economics

This record has been put together in a limited time for prompt distribution. It is not a proceedings volume. Rather it is a collection of all memoranda, diagrams, and literature references that were circulated before the workshop, used to support presentations during the workshop, or written down to preserve some ideas and some outcomes of computations that arose from the workshop. The only organizing principle is the temporal sequence in which the materials were presented or prepared

Complexity, Language, and Life: Mathematical Approaches

In May 1984 the Swedish Council for Scientific Research convened a small group of investigators at the scientific research station at Abisko, Sweden, for the purpose of examining various conceptual and mathematical views of the evolution of complex systems. The stated theme of the meeting was deliberately kept vague, with only the purpose of discussing alternative mathematically based approaches to the modeling of evolving processes being given as a guideline to the participants. In order to limit the scope to some degree, it was decided to emphasize living rather than nonliving processes and to invite participants from a range of disciplinary specialities spanning the spectrum from pure and applied mathematics to geography and analytic philosophy. The results of the meeting were quite extraordinary; while there was no intent to focus the papers and discussion into predefined channels, an immediate self-organizing effect took place and the deliberations quickly oriented themselves into three main streams: conceptual and formal structures for characterizing system complexity; evolutionary processes in biology and ecology; the emergence of complexity through evolution in natural languages. The chapters presented in this volume are not the proceedings of the meeting. Following the meeting, the organizers felt that the ideas and spirit of the gathering should be preserved in some written form, so the participants were each requested to produce a chapter, explicating the views they presented at Abisko, written specifically for this volume. The results of this exercise are contained in this book

Foundations of Mechanics, Second Edition

Preface to the Second Edition. Since the first edition of this book appeared in 1967, there has been a great deal of activity in the field of symplectic geometry and Hamiltonian systems. In addition to the recent textbooks of Arnold, Arnold-Avez, Godbillon, Guillemin-Sternberg, Siegel-Moser, and Souriau, there have been many research articles published. Two good collections are "Symposia Mathematica," vol. XIV, and "Géométrie Symplectique el Physique Mathématique," CNRS, Colloque Internationaux, no. 237. There are also important survey articles, such as Weinstein [1977b]. The text and bibliography contain many of the important new references we are aware of. We have continued to find the classic works, especially Whittaker [1959], invaluable. The basic audience for the book remains the same: mathematicians, physicists, and engineers interested in geometrical methods in mechanics, assuming a background in calculus, linear algebra, some classical analysis, and point set topology. We include most of the basic results in manifold theory, as well as some key facts from point set topology and Lie group theory. Other things used without proof are clearly noted. We have updated the material on symmetry groups and qualitative theory, added new sections on the rigid body, topology and mechanics, and quantization, and other topics, and have made numerous corrections and additions. In fact, some of the results in this edition are new. We have made two major changes in notation: we now use f^* for pull-back (the first edition used f[sub]*), in accordance with standard usage, and have adopted the "Bourbaki" convention for wedge product. The latter eliminates many annoying factors of 2. A. N. Kolmogorov's address at the 1954 International Congress of Mathematicians marked an important historical point in the development of the theory, and is reproduced as an appendix. The work of Kolmogorov, Arnold, and Moser and its application to Laplace's question of stability of the solar system remains one of the goals of the exposition. For complete details of all tbe theorems needed in this direction, outside references will have to be consulted, such as Siegel-Moser [1971] and Moser [1973a]. We are pleased to acknowledge valuable assistance from Paul Chernoff, Wlodek Tulczyjew, Morris Hirsh, Alan Weinstein, and our invaluable assistant authors, Richard Cushman and Tudor Ratiu, who all contributed some of their original material for incorporation into the text. Also, we are grateful to Ethan Akin, Kentaro Mikami, Judy Arms, Harold Naparst, Michael Buchner, Ed Nelson, Robert Cahn, Sheldon Newhouse, Emil Chorosoff, George Oster, André Deprit, Jean-Paul Penot, Bob Devaney, Joel Robbin, Hans Duistermaat, Clark Robinson, John Guckenheimer, David Rod, Martin Gutzwiller, William Satzer, Richard Hansen, Dieter Schmidt, Morris Kirsch, Mike Shub, Michael Hoffman, Steve Smale, Andrei Iacob, Rich Spencer, Robert Jantzen, Mike Spivak, Therese Langer, Dan Sunday, Ken Meyer, Floris Takens, [and] Randy Wohl for contributions, remarks, and corrections which we have included in this edition. Further, we express our gratitude to Chris Shaw, who made exceptional efforts to transfom our sketches into the graphics which illustrate the text, to Peter Coha for his assistance in organizing the Museum and Bibliography, and to Ruthie Cephas, Jody Hilbun, Marnie McElhiney, Ruth (Bionic Fingers) Suzuki, and Ikuko Workman for their superb typing job. Theoretical mechanics is an ever-expanding subject. We will appreciate comments from readers regarding new results and shortcomings in this edition. RALPH ABRAHAM, JERROLD E. MARSDEN</p