DOCSLIB.ORG

On a List of Ordinary Differential Equations Problems ∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

ON A LIST OF ORDINARY DIFFERENTIAL EQUATIONS PROBLEMS ∗ Jorge Sotomayor y September 3, 2018 arXiv:1808.10571v1 [math.HO] 31 Aug 2018 ∗This essay contains a very free translation with comments, updates, annotations, additions, correc- tions and abridgment of \Uma Lista de Problemas de E.D.O", [54]. yThe author is fellow of CNPq. Grant: PQ-SR- 307690/2016-4. 1 Abstract This evocative essay focuses on the mathematical activities witnessed by the au- thor along 1962-64 at IMPA. The list of research problems proposed in September 1962 by Mauricio Peixoto at the Seminar on the Qualitative Theory of Differential Equations is pointed out as a landmark for the genesis of the research interest in the Qualitative Theory of Differential Equations and Dynamical Systems in Brazil. Mathematical Subject Classification: 01A60, 01A67, 37C20. Just as every human undertaking pursues certain objectives, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon. D. Hilbert, ICM, Paris. 1900. 1 The diary of a journey. To find an old mathematics notebook, protected from total deterioration by a plastic cover, was like an encounter with the diary of a distant journey. The contact with its yellowish pages triggered an avalanche of recollections of the years 1962 − 64; when I was a doctoral student in Brazil. Its sequential structure prompted the reconstruction of the chronology of my initiation into mathematical research. The essential landmarks { the stations { of a mathematical peregrination, starting from a bookish approach, heading toward an attempt to tackle research problems, pulsed latent in the rough writing and scribbled drawings. Some pages were missing, a few of them had faded away. 1 2 An afternoon in September 1962. In a seminar room of the Institute of Pure and Applied Mathematics (IMPA), in the Botafogo quarter of the city of Rio de Janeiro, gathered a group of around ten people: mathematicians, research fellows of disparate backgrounds, candidates to become math- ematicians, and one or two voyeurs. Figure 1: Present view of the facade of the two store house, located at the corner of the streets S~aoClemente and Sorocaba, hosting IMPA in 1962 − 64: The occasion was the Seminar on the Qualitative Theory of Differential Equations (QTDE), directed by Prof. Mauricio M. Peixoto (hereafter Prof. Peixoto, or simply Peixoto) who had announced the following title for his lecture: \Open Problems on The Qualitative Theory of Differential Equations". The seminar activity had been interrupted for a few sessions. It was restarting after the return of Prof. Peixoto who had travelled to attend the International Congress of Mathematicians at Stockholm, August 15 - 22. At that time Prof. Peixoto was the only Research Director resident at IMPA. Prof. Leopoldo Nachbin (1922 - 1993), also a research Director, was on leave of absence. Peixoto had very explicit views concerning Mathematics learning in graduate level. They where well known to most participants: \This science is assimilated through solv- ing problems and thinking", he had clearly stated, contesting insinuations of those who favored the preponderance of more intensive lecture courses and bookish learning. This 2 contrasted with the naive and easy going vision I had partially acquired in my undergrad- uate predominantly reading contact with mathematics. Excitement and a tense expectation could be noticed in the audience. For the most experienced participants of the Seminar, the time to face true research problems had arrived. As a newcomer into advanced studies, and the youngest of all, I could be implicitly regarded outside such group. 3 A flash of mathematics at IMPA in 1962 - 63. Among the experienced researchers, besides Prof. Peixoto who stayed at IMPA the whole period from 1962 to 1964, were Elon Lima (1930 - 2017), Djairo de Figueiredo (1934 - ) and Otto Endler (1929 - 1988) who sojourned for shorter periods. Elon stayed the whole first semester and Djairo only a couple of months. Otto stayed along 1962 and part of 1963. Few lecturers visited the Institute. Among them I mention below those which captured my interest. Charles Pugh (1940 - ). Subject: Closing Lemma, crucial in the work of Peixoto. Gilberto Loibel (1932 - 2013). Subject: Stratified Sets, introduced by Ren´eThom (1923 - 2004), founder, together with Hassler Whitney, of the Theory of Singularities of Mappings. Wilhelm Klingenberg (1924 - 2010). Subject: Closed Geodesics, located in the inter- section of Differential Geometry, Differential Equations and Calculus of Variations. At that time, IMPA did not have a minimal program of courses to be offered along the year. This activity depended on the research fellows present, in a quickly changing regime. However, there was a permanent basis providing scientific stability to the Institute along those years, around Prof. Peixoto research project. Furthermore, due to an agree- ment with the University of Brazil, later denominated UFRJ (Federal University of Rio de Janeiro), a doctoral program started at IMPA in 1963. This pioneer research project, under the leadership of Peixoto, is the land- mark of the systematic interest in Dynamical Systems (DS) in Brazil. This was the first explicit effort to stimulate the initiation into research in this area of mathematics in Brazil. 3 I had the unique chance and privilege to be part of the first group of doctoral candidates, under the supervision of Prof. Peixoto. After the influential works of the American mathematician Stephen Smale (sixties and seventies) the denomination DS nominally assimilated a significant part of the QTDE from which the separating border is not well defined. See Smale's 1967 landmark article [40]. DS is also the name of a famous book of George Birkhoff, printed in 1927. Ordered by the preponderance they had in my initiation into research, below I list some of the courses and seminars held at IMPA in 19621: 1. Seminar on the Qualitative Theory of Differential Equations; 2. Seminar on Differentiable Manifolds following the inspiring Porto Alegre Elon Lima book [22]; 3. Seminar based on the reading of the, now classic, book of J. Dieudonn´e:Foundations of Modern Analysis [7]; 4. Course on Algebraic Topology, taught by Peixoto, based on the books of Cairns and Hocking - Young; 5. Course on Multilinear Algebra and Exterior Differential Calculus, based on Bour- baki and Flanders, taught by Elon Lima; 6. Seminar on the reading of the book \Lectures on Ordinary Differential Equations" by W. Hurewicz [17]. I list bellow, with no ordering, some of the research fellows and habitu´ees of IMPA, related to Prof. Peixoto Project, in 1962 - 63: Ivan Kupka (1938 - ), Maria L´uciaAlvarenga (1937 - ), Jacob Palis (1940 - ), Lindolpho C. Dias (1930 - ), Alcil´eaAugusto, (1937 - ), Aristides C. Barreto, (1935 - 2000). Much of the contents of the activities 1 to 5 listed above was new to me. I devoted to them special attention, with library search and extensive complementary readings. In this endeavor the friendly interaction with Ivan Kupka, by far the most knowledgeable in the group, was auspicious. 1This abridged version concerns activities on the QTDE. A wider description can be found in [54]. 4 (a) (b) Figure 2: At the entrance of IMPA, August 1962. (a) Top row, left to right: Lindolpho Dias, M. Peixoto, Ivan Kupka, E. Isla, Aristides Barreto, Jacob Palis, Eliana Rocha, Alcil´eaAugusto; descending the right margin and continuing clock- wise on the bottom margin: C. Marquez, M. L. Alvarenga, M. H. Cerqueira, J. Sotomayor, Lia Velloso and Adarcy P. Costa. Three gentlemen sitting at the center, descending counter clockwise: L. Nachbin, H. Machado and J. A. Barroso. (b) Top row from left to right: M. Peixoto, E. Isla, Celina Marquez, Lindolpho Dias and Ja- cob Palis. L. Nachbin at center, with necktie. Bottom right corner: Alcil´eaAugusto, M. L. Alvarenga, Adarcy P. Costa. 4 Peixoto's Seminar on QTDE, 1962. Along my sojourn at IMPA, 1962-64, the most remarkable of all the activities in which I engaged was the Seminar on the Qualitative Theory of Ordinary Differential Equations. Following the epistolary reading directions sent to me by Peixoto in 1961, recounted in the evocative essay \Mathematical Encounters" [55], with the inspiring master presen- tations of Coddington and Levinson [6] and Hurewicz [17], I had already been initiated into the first steps of the Qualitative Theory. The phase portrait, limit cycles and singu- lar points local structures: saddles, nodes, foci and centers, were fiound there. A bright synthesis of these elements in the Poincar´e- Bendixson Theorem complemented the in- troductory contact. While the subjects presented and discussed in the Seminar did not require a heavy 5 background knowledge, their in depth appreciation depended on a level of maturity and on an inquisitive disposition, beyond the initiation outlined above, to which I had not yet been exposed. 4.1 Non singular ODEs on the Klein Bottle and the Torus. The first lecture at the Seminar TQEDO, was delivered by Peixoto at the beginning of April. The subject was The Theorem of Kneser, whose conclusion is that \Every vector field with no singular points in the Klein Bottle has a periodic orbit". See [19]. At the two next meetings of the seminar, Peixoto presented the basic theory of the rotation number following the last chapter of Coddington and Levinson. The second presentation included also a geometric construction of the example of Denoy of a C1 non singular vector field for which all its orbits cluster in a closed invariant set, which transversally is a Cantor set, with no proper subset sharing these properties. Such a set is denominated \minimal non-trivial". The \trivial minimal" sets are the singular points {equlilibria{ and periodic orbits. 4.2 Invariant Manifolds. Elon Lima continued the Seminar.
Recommended publications
  • Chapter 2 Structural Stability

    Chapter 2 Structural Stability

    Chapter 2 Structural stability 2.1 Denitions and one-dimensional examples A very important notion, both from a theoretical point of view and for applications, is that of stability: the qualitative behavior should not change under small perturbations. Denition 2.1.1: A Cr map f is Cm structurally stable (with 1 m r ∞) if there exists a neighbour- hood U of f in the Cm topology such that every g ∈ U is topologically conjugated to f. Remark 2.1.1. The reason that for structural stability we just ask the existence of a topological conju- gacy with close maps is because we are interested only in the qualitative properties of the dynamics. For 1 1 R instance, the maps f(x)= 2xand g(x)= 3xhave the same qualitative dynamics over (and indeed are topologically conjugated; see below) but they cannot be C1-conjugated. Indeed, assume there is a C1- dieomorphism h: R → R such that h g = f h. Then we must have h(0) = 0 (because the origin is the unique xed point of both f and g) and 1 1 h0(0) = h0 g(0) g0(0)=(hg)0(0)=(fh)0(0) = f 0 h(0) h0(0) = h0(0); 3 2 but this implies h0(0) = 0, which is impossible. Let us begin with examples of non-structurally stable maps. 2 Example 2.1.1. For ε ∈ R let Fε: R → R given by Fε(x)=xx +ε. We have kFε F0kr = |ε| for r all r 0, and hence Fε → F0 in the C topology.
  • Writing the History of Dynamical Systems and Chaos

    Writing the History of Dynamical Systems and Chaos

    Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as
  • Structural Stability of Piecewise Smooth Systems

    Structural Stability of Piecewise Smooth Systems

    STRUCTURAL STABILITY OF PIECEWISE SMOOTH SYSTEMS MIREILLE E. BROUCKE, CHARLES C. PUGH, AND SLOBODAN N. SIMIC´ † Abstract A.F. Filippov has developed a theory of dynamical systems that are governed by piecewise smooth vector fields [2]. It is mainly a local theory. In this article we concentrate on some of its global and generic aspects. We establish a generic structural stability theorem for Filip- pov systems on surfaces, which is a natural generalization of Mauricio Peixoto’s classic result [12]. We show that the generic Filippov system can be obtained from a smooth system by a process called pinching. Lastly, we give examples. Our work has precursors in an announcement by V.S. Kozlova [6] about structural stability for the case of planar Fil- ippov systems, and also the papers of Jorge Sotomayor and Jaume Llibre [8] and Marco Antonio Teixiera [17], [18]. Partially supported by NASA grant NAG-2-1039 and EPRI grant EPRI-35352- 6089.† STRUCTURAL STABILITY OF PIECEWISE SMOOTH SYSTEMS 1 1. Introduction Imagine two independently defined smooth vector fields on the 2- sphere, say X+ and X−. While a point p is in the Northern hemisphere let it move under the influence of X+, and while it is in the Southern hemisphere, let it move under the influence of X−. At the equator, make some intelligent decision about the motion of p. See Figure 1. This will give an orbit portrait on the sphere. What can it look like? Figure 1. A piecewise smooth vector field on the 2-sphere. How do perturbations affect it? How does it differ from the standard vector field case in which X+ = X−? These topics will be put in proper context and addressed in Sections 2-7.
  • Stability and Explanatory Significance of Some Simple Evolutionary Models

    Stability and Explanatory Significance of Some Simple Evolutionary Models

    Stability and Explanatory Significance of Some Simple Evolutionary Models Brian Skyrms University of California, Irvine 1. Introduction. The explanatory value of equilibrium depends on the underlying dynamics. First there are questions of dynamical stability of the equilibrium that are internal to the dynamical system in question. Is the equilibrium locally stable, so that states near to it stay near to it, or better, asymptotically stable, so that states near to it are carried to it by the dynamics? If not, we should not expect to see this equilibrium. But even if an equilibrium is asymptotically stable, that is no guarantee that the system will reach that equilibrium unless we know that the system's initial state is sufficiently close to the equilibrium. Global stability of an equilibrium, when we have it, gives the equilibrium a much more powerful explanatory role. An equilibrium is globally asymptotically stable if the dynamics carries every possible initial state in the interior of the state space to that equilibrium. If an equilibrium is globally stable, it can have explanatory value even when we are completely uncertain about the initial state of the system. Once questions of dynamical stability are answered with respect to the dynamical system in question, there is the further question of structural stability of that system itself. That is to say, are dynamical systems close to the one in question (in a sense to be made 1 precise) topologically equivalent to that system? If not, a slight mispecification of the model may make predictions that are drastically wrong. Structural stability is defined in terms of small changes in the model.
  • Stability and Bifurcation of Dynamical Systems

    Stability and Bifurcation of Dynamical Systems

    STABILITY AND BIFURCATION OF DYNAMICAL SYSTEMS Scope: • To remind basic notions of Dynamical Systems and Stability Theory; • To introduce fundaments of Bifurcation Theory, and establish a link with Stability Theory; • To give an outline of the Center Manifold Method and Normal Form theory. 1 Outline: 1. General definitions 2. Fundaments of Stability Theory 3. Fundaments of Bifurcation Theory 4. Multiple bifurcations from a known path 5. The Center Manifold Method (CMM) 6. The Normal Form Theory (NFT) 2 1. GENERAL DEFINITIONS We give general definitions for a N-dimensional autonomous systems. ••• Equations of motion: x(t )= Fx ( ( t )), x ∈ »N where x are state-variables , { x} the state-space , and F the vector field . ••• Orbits: Let xS (t ) be the solution to equations which satisfies prescribed initial conditions: x S(t )= F ( x S ( t )) 0 xS (0) = x The set of all the values assumed by xS (t ) for t > 0is called an orbit of the dynamical system. Geometrically, an orbit is a curve in the phase-space, originating from x0. The set of all orbits is the phase-portrait or phase-flow . 3 ••• Classifications of orbits: Orbits are classified according to their time-behavior. Equilibrium (or fixed -) point : it is an orbit xS (t ) =: xE independent of time (represented by a point in the phase-space); Periodic orbit : it is an orbit xS (t ) =:xP (t ) such that xP(t+ T ) = x P () t , with T the period (it is a closed curve, called cycle ); Quasi-periodic orbit : it is an orbit xS (t ) =:xQ (t ) such that, given an arbitrary small ε > 0 , there exists a time τ for which xQ(t+τ ) − x Q () t ≤ ε holds for any t; (it is a curve that densely fills a ‘tubular’ space); Non-periodic orbit : orbit xS (t ) with no special properties.
  • Dynamical System: an Overview Srikanth Nuthanapati* Department of Aeronautics, IIT Madras, Chennai, India

    Dynamical System: an Overview Srikanth Nuthanapati* Department of Aeronautics, IIT Madras, Chennai, India

    OPEN ACCESS Freely available online s & Aero tic sp au a n c o e r e E n A g f i n Journal of o e l a e n r i r n u g o J Aeronautics & Aerospace Engineering ISSN: 2168-9792 Perspective Dynamical System: An Overview Srikanth Nuthanapati* Department of Aeronautics, IIT Madras, Chennai, India PERSPECTIVE that involved a large number of people. Knowing the trajectory is often sufficient for simple dynamical systems, but most dynamical The idea of a dynamical system emerges from Newtonian mechanics. systems are too complicated to be understood in terms of individual There, as in other natural sciences and engineering disciplines, the trajectories. The problems arise because: evolution rule of dynamical systems is an implicit relation that gives the system’s state only for a short time in the future. Finding an (a) The studied systems may only be known approximately—the orbit required sophisticated mathematical techniques and could system’s parameters may not be known precisely, or terms only be accomplished for a small class of dynamical systems prior may be missing from the equations. The approximations to the advent of computers. The task of determining the orbits of a used call the validity or relevance of numerical solutions dynamical system has been simplified thanks to numerical methods into question. To address these issues, several notions of implemented on electronic computing machines. A dynamical stability, such as Lyapunov stability or structural stability, system is a mathematical system in which a function describes the have been introduced into the study of dynamical systems.
  • Structural STABILITY MORPHOGENESIS

    Structural STABILITY MORPHOGENESIS

    1 • IMtlM" Hco_ E=~t:~'!. .. "'...-.",·Yv_ /sTRUCTURAL STABILITY and MORPHOGENESIS An Outline of a General Theory of Models Tranllated from the French edition, as updated by the author, by D. H. FOWlER U....... ".04 w ....... + Witt. 0 ForOJWOfd by C. H. WADDINGTON 197.5 (!) W. A. BENJAMIN, INC. ,,""once;;! &ook Program Reading, Mauochvsetfs I: 'I) _,...... O""MiIo .O,~c;;.,· 5,..,. lolyo J r If ! "• f• Q f iffi Enlll'" £d,,;on. 1m Orilon,l", pubh5l1ed in 19'12. • 5tdN1i1. Itruclut. t1 r'IMIfphas~oe b~ d'uroe Ihlone ghw!:,,1e des modiO'le!o bv W. A. IIMlalnln, 11><:., ',,;h.-naQ Book Pros.,m, Lib ...., or Cell lr"' Cltiliollinl! In Publication Data Them , Ren ~. 1923 - Structural stablll ty and morphogensis . 1. Btology--Mathematlcal. lIIOdels . 2 . Morphogenesis-­ Mathematical. models. 3. Topology-. I . Title. QH323 .5.T4813 574 .4'01'514 74-8829 ISBN 0-8053 -~76 -9 ISBN 0-8053 - S~?77 - 7 (pbk.) American M.,hftT1.,IQI Society (MOS) Subje<l Oassifoc"ion SdI~....., (1970); 'J2o\101Si'A2!l Copy.lllht elW5 by W. A. 1IM~IT"n . Inc. Published 1,"",1",,_sly'" CMid• • AllIt_l "JlHsned res,1t! , .... ed. mN... , I • , permIssion~::'~~:t.:~ oft""" ';:;~~~~~~~pYbllsl!e. , ~~~~~ BooIt Pros •• m, RNdinl. ~nKhuseltJ ()'867. U.S.A. END PAPERS; The hyperbolic: umbillc In hydrody... mIa : Waves . t Plum ItJllnd, M8Uachvselts. f't1oto courtesy 01 J. GLH;klnheirner. CONTENTS , FOREWORD by C. 1-1 . Waddington • • . • • • • • • • FOREWORD to the angmal French edItIO-n. by C. II . Waddington . • • • • • • • PREFACE(tran,lated from the French edition) · . xx iii TRANSLATOR'S NOTE . ... C HA PTER I.
  • A PROOF of the C1 STABILITY CONJECTURE by RICARDO MAN£

    A PROOF of the C1 STABILITY CONJECTURE by RICARDO MAN£

    PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. RICARDO MAÑÉ A proof of theC1 stability conjecture Publications mathématiques de l’I.H.É.S., tome 66 (1987), p. 161-210 <http://www.numdam.org/item?id=PMIHES_1987__66__161_0> © Publications mathématiques de l’I.H.É.S., 1987, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A PROOF OF THE C1 STABILITY CONJECTURE by RICARDO MAN£ INTRODUCTION Two continuous maps, f^:X^ and f^:X^ are topologically equivalent if there exists a homeomorphism h: X^ -> X^ such that h~lf^h ==f^. A G1' diffeomorphism/ of a closed manifold M is C*" structurally stable if it has a G*" neighborhood ^ such that every g e W is topologically equivalent to /. This concept was introduced in the thirties by Andronov and Pontrjagin [I], in the limited (when compared with its present range) framework of flows on the two dimensional disk. The turning point of its development that connected it with much richer possibilities, came in the early sixties, through the work of Smale who, as a consequence of his improved version of a classical result of Birkhoff about homoclinic points, showed that structural stability can coexist with highly developed forms of recurrence [24].
  • Structurally Stable Systems Are Not Dense

    Structurally Stable Systems Are Not Dense

    STRUCTURALLY STABLE SYSTEMS ARE NOT DENSE. By S. SMALE. In this note, we give a negative answer to "the problem of structural stability"; are the structurally stable differential equations dense in the C' topology in all (first order, ordinary, autonomous) differential equations? MAIN THEOREM. There exists a compact 4 dimensional manifold M, an open set U in the space of Cr vector fields, Cr topology, r > 0, on MI such that no X C U is structurally stable. This problem has been stated explicitly in the above form by the author on several occasions, see e. g. [10] and [11]. However the problem is older, considered for example by Soviet mathematicians, and by Peixoto who gave an affirmative answer [7] for the 2-disk and [8] for compact 2-manifolds. Evidence for a positive answer in higher dimensions was given by the author [12], [13] where it was shown that certain examples of differential equations having an infinite number of periodic solutions were structurally stable, and Anosov [2] gave further examples with the same properties. The notion of structural stability was introduced by Andronov and Pontriagin in 1937 [1] (see also Lefschetz [5]). We recall the definitions now. A differential equation (lst order, ordinary) X, will be a Cr tangent vector field on a C? manifold M, which we will assume compact for purposes of this paper. Of course X, through its solution curves, generates a 1 para- meter group of diffeomorphisms ct of M. An equivalence between differential equations X and X' on M is a homeomorphism h: 11M-->M which sends a sensed solution curve of X onto one of X'.
  • Determining the D-Dimensional Symbolic Dynamics

    Determining the D-Dimensional Symbolic Dynamics

    spatiotemporal cats or, try herding 6 cats siminos/spatiotemp, rev. 6798: last edit by Predrag Cvitanovi´c,03/08/2019 Predrag Cvitanovi´c,Boris Gutkin, Li Han, Rana Jafari, Han Liang, and Adrien K. Saremi April 1, 2019 Contents 1 Cat map7 1.1 Adler-Weiss partition of the Thom-Arnol’d cat map.......7 1.2 Adler-Weiss partition of the Percival-Vivaldi cat map......9 1.2.1 Adler-Weiss linear code partition of the phase space... 16 1.3 Perron-Frobenius operators and periodic orbits theory of cat maps 18 1.3.1 Cat map topological zeta function............. 19 1.3.2 Adler / Adler98........................ 20 1.3.3 Percival and Vivaldi / PerViv................ 21 1.3.4 Isola / Isola90......................... 22 1.3.5 Creagh / Creagh94...................... 23 1.3.6 Keating / Keating91..................... 24 1.4 Green’s function for 1-dimensional lattice............. 25 1.5 Green’s blog.............................. 28 1.6 Z2 = D1 factorization......................... 31 1.7 Any piecewise linear map has “linear code”............ 32 1.8 Cat map blog............................. 33 References.................................. 47 1.9 Examples................................ 53 exercises 61 2 Statistical mechanics applications 67 2.1 Cat map................................ 67 2.2 New example: Arnol’d cat map................... 67 2.3 Diffusion in Hamiltonian sawtooth and cat maps......... 70 References.................................. 75 3 Spatiotemporal cat 77 3.1 Elastodynamic equilibria of 2D solids............... 79 3.2 Cats’ GHJSC16blog.......................... 80 References.................................. 94 4 Ising model in 2D 97 4.1 Ihara zeta functions.......................... 101 4.1.1 Clair / Clair14......................... 101 2 CONTENTS 4.1.2 Maillard............................ 105 4.2 Zeta functions in d=2........................
  • Structural Stability and Hyperbolic Attractors Artur Oscar Lopes

    Structural Stability and Hyperbolic Attractors Artur Oscar Lopes

    Structural Stability and Hyperbolic Attractors Artur Oscar Lopes Transactions of the American Mathematical Society, Vol. 252. (Aug., 1979), pp. 205-219. Stable URL: http://links.jstor.org/sici?sici=0002-9947%28197908%29252%3C205%3ASSAHA%3E2.0.CO%3B2-I Transactions of the American Mathematical Society is currently published by American Mathematical Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ams.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Wed Mar 26 16:36:54 2008 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 252, August 1979 STRUCTURAL STABILITY AND HYPERBOLIC A'ITRACTORS BY ARTLJR OSCAR LOPES ABSTRACT.A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors.
  • Structural Stability and Bifurcations

    Structural Stability and Bifurcations

    Chapter 8 Structural Stability and Bifurcations 8.1 Smooth and Sudden Changes So far, we discussed the behavior of solutions of a dynamical system x_ = G(x); x(0) = x0; (8.1) under changes of the initial point x0. Since practical problems modelled by (8.1) can be expected to involve uncertainties in the mapping G, we should address also the question of the sensitivity of the solutions of (8.1) to perturbations of G. Then, however, we may see a very different behavior, than in the case of changes of x0. By ODE-theory, the solution of (8.1) depends continuously on the initial point, as is obvious for the solution x(t) = exp(tA)x0 of the linear problemx _ = Ax, x(0) = x0. On the other hand, small changes of the mapping G may well produce abrupt changes in the solution behavior. We saw this already in the case of the discrete logistics problem k+1 x = g(xk); k = 0; 1; 2; : : : ; g(x) := µx(1 − x); µ > 0: (8.2) Here, the fixed points are x∗ = 0 and x∗∗ = (µ − 1)/µ, where x∗ = 0 is asymptotically stable for 0 < µ < 1 and unstable for µ > 1, while x∗∗ is unstable for 0 < µ < 1 and asymptotically stable for 1 ≤ µ < 3. But, at µ = 3 the solution behavior changes significantly: In fact, for any µ > 3 the iterates tend to a 2-cycle consisting of the points 1 z± := (µ + 1) ± p(µ + 1)(µ − 3); 2µ 130 that is, the attractor consisting of the single point fx∗∗g has become an attractor-set fz+; z−g of cardinality two.