perm filename FOO1[1,JMC]1 blob
sn#697393 filedate 1983-01-21 generic text, type C, neo UTF8
COMMENT ā VALID 00022 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00002 00002 1/5/1
C00004 00003 1/5/2
C00009 00004 1/5/3
C00012 00005 1/5/4
C00015 00006 1/5/5
C00017 00007 1/5/6
C00020 00008 1/5/7
C00022 00009 1/5/8
C00025 00010 1/5/9
C00028 00011 1/5/10
C00032 00012 1/5/11
C00035 00013 1/5/12
C00040 00014 1/5/13
C00044 00015 1/5/14
C00048 00016 1/5/15
C00051 00017 1/5/16
C00054 00018 1/5/17
C00058 00019 1/5/18
C00061 00020 1/5/19
C00063 00021 1/5/21
C00064 00022 20jan83 15:53:48 User29356
C00066 ENDMK
Cā;
1/5/1
828495 82j#34038
Entrainment domains.
Proceedings of the VIIIth International Conference on Nonlinear
Oscillations, Vol. I (Prague, 1978)
Meyer, K. R.; Schmidt, D. S.
Publ: Academia, Prague
1979, pp. 479 - 482,
Document Type: Collection
Authors' summary: ''We develop effective algorithms for computing the
domains in a parameter space where a differential equation which admits an
invariant torus has a periodic solution with fixed rotation number. The
algorithms are based on the method of Lie transforms and have been
implemented on a computer. The algorithms are applied to two variations of
van der Pol's equation.''
(For the entire collection see MR 82k:70002a.)
Reviewer: Authors' summary
Descriptors: *ORDINARY DIFFERENTIAL EQUATIONS -Qualitative theory
--Periodic solutions (34C25)
1/5/2
827401 82j#00001
Turtle geometry.
Abelson, Harold; diSessa, Andrea A.
Publ: MIT Press, Cambridge, Mass.-London
1981, xx+477 pp. 0-262-01063-1
Price: $20.00.
Document Type: Book
The computer as a medium for exploring mathematics. MIT Press Series in
Artificial Intelligence.
This book contains a computational introduction to geometry and advanced
mathematics at the undergraduate level. Its conception is based on the idea
that approaching mathematics by computation -- especially by the activity
of programming -- and by doing it is more effective than only learning
about it. Because of the emergence of powerful inexpensive personal
computers the student today gets the chance to experience mathematics in
terms of constructive process-oriented formulations instead of axiomatic-
deductive formalisms.
The first turtles designed at the Massachusetts Institute of Technology
where this material was created during the past ten years were computer-
controlled robots that moved around the floor in response to the commands,
''FORWARD'' and ''RIGHT''. The turtle that brings ''Turtle geometry'' to
life is a glowing tracer moving on the display screen of a small computer
system. Chapter 1 shows how to regard plane geometric figures not as static
entities but as tracings of this turtle controlled by suitable computer
programs leading to such concepts as rotation number of closed paths in a
natural way. Chapter 2 demonstrates procedures for random and directed
motion and for simulating growth processes in mathematical biology. Chapter
3 compares turtle methods and coordinate methods for representation of two-
or three- dimensional geometric objects on the screen. Chapter 4 discusses
the topology of curves in the plane. In Chapter 5 the turtle escapes the
plane and begins to move on nonflat surfaces (Chapters 6, 7, 8). In Chapter
9 this computational exploration of curved surfaces provides a conceptional
framework for studying Einstein's general theory of relativity.
Although most parts of the book are accessible with only pencil and
paper, the contents will be most useful to those who have access to
interactive computer graphics. Many of the sections contain extended
descriptions of computer projects to implement and investigate.
An appendix includes a typical implementation of turtle commands in
BASIC, which can be used for most computer graphic systems. In addition,
there are listed a few commercially available computer systems that have
turtle graphics built in.
Reviewer: KAHMANN, J.
Descriptors: *GENERAL --General mathematics (00A05); GEOMETRY --Explicit
machine computation and programs (51-04); COMPUTER SCIENCE (including
AUTOMATA) -Simulation --None of the above, but in this section (68J99);
RELATIVITY --Explicit machine computation and programs (83-04); BIOLOGY
AND BEHAVIORAL SCIENCES --Explicit machine computation and programs
(92-04)
1/5/3
825895 82i#58061
Rotation sets are closed.
Ito, Ryuichi
Math. Proc. Cambridge Philos. Soc., 1981, 89, no. 1, 107 - 111.
Document Type: Journal
The rotation number of an orientation-preserving homeomorphism of the
circle S=R/Z measures the average rotation of the orbit of each point under
iterations of the homeomorphisms. S. Newhouse, J. Palis and F. Takens
(''Bifurcations and stability of families of diffeomorphisms'', Inst.
Hautes Etudes Sci. Publ. Math., to appear) generalized this concept to
include maps of S (continuous endomorphisms of S which are not necessarily
injective) of degree one. In particular, given such a map f: S{arrr}S and a
lift F of f to the covering space R, the rotation number is defined for
each x{in}R to be rho (F, x)=lim sup{sub}(n{arrr}{infin}) n{sup}( -
1)(F{sup}n(x) - x). When f is a homeomorphism, rho (F, x) is independent of
x. Hence, the appropriate generalization of the rotation number for maps is
the rotation set rho (F)= (rho (F, x): x{in}R). Newhouse, Palis and Takens
proved that the closure of rho (F) is either a point or a closed interval.
In the present article this result is strengthened; the author proves that
rho (F) is closed.
Reviewer: CHICONE, CARMEN (Columbia, Mo.)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --None of the
above, but in this section (58F99); MEASURE AND INTEGRATION
-Measure-theoretic ergodc theory --None of the above, but in this section
(28D99)
1/5/4
817982 82g#05053
On rotation numbers for complete bipartite graphs.
Cockayne, E. J.; Lorimer, P. J.
Congr. Numer., 1980, 28, , 281 - 291.
Document Type: Journal
Proceedings of the Eleventh Southeastern Conference on Combinatorics,
Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla.,
1980), Vol. I.
The rotation number h(G, x) of a graph G having one vertex x designated
as its root is the minimum integer m for which there exists a graph F
having the same number of vertices as G and having m edges, such that, for
any vertex y of F, there exists an embedding f of G into F for which
f(x)=y. F is an extremal (G, x) graph if its number of edges realizes this
minimum. Section 2 is concerned with rotation numbers for complete
bipartite graphs.
For each integer 2{sup}k the authors define a rooted tree (T{sub}k, u).
''Broadcasting'' in time k is possible from any source y in a graph G on 2
{sup}k vertices if and only if there exists an embedding f of T{sub}k into
G such that f(x)=y. (''By broadcasting, we mean information dissemination
in communications networks whereby a single message known initially to one
member (the source) becomes known to all members. The dissemination is
accomplished by a series of phone calls over the edges of the network,
subject to the following constraints: (i) Each phone call requires one unit
of time; (ii) a vertex may participate in only one call per unit time; and
(iii) a vertex may only call an adjacent vertex.'')
Reviewer: BROWN, W. G. (Montreal, Que.)
Descriptors: *COMBINATORICS -Graph theory --Extremal problems (05C35);
COMBINATORICS -Graph theory --Trees (05C05)
1/5/5
815323 82f#34051
Systems of differential equations with switching that are periodic with
respect to two arguments.
Ivanov, A. F.
Visnik Kiiv. UnivMat. Meh., 1978, , No. 20, 28 - 32, 164.
Languages: Ukrainian Summary Languages: English and Russian summaries
Document Type: Journal
The author considers the system of differential equations d theta /d phi
=f{sub}i(phi , theta ) (i=1, 2, ... ,k) with switchings on the line q theta
=p phi , where the functions f{sub}i are 2 pi -periodic with respect to the
scalar variables phi , theta , and p, q are integers. He proves that in the
absence of a sliding regime every solution has a rotation number, and that
a solution is periodic precisely when its rotation number is rational.
Reviewer: ZADOROZNII, V. G. (Voronezh)
Descriptors: *ORDINARY DIFFERENTIAL EQUATIONS -Qualitative theory
--Dynamical systems (34C35)
1/5/6
797483 81m#85003
Orbits near a 2/3 resonance.
Michaelidis, P.
Astronom. and Astrophys., 1980, 91, no. 1-2, 165 - 174.
Document Type: Journal
In this paper the author has studied the orbits in a galactic type
potential V={1/2}(Ax{sup}2+By{sup}2) - epsilon xy{sup}2 in the near
resonance case {root}(A/B){simeq}(2/3). A large number of periodic orbits
have been found empirically and by means of the third integral. The orbits
are determined by using the Lie transforms, or by a computer programme that
gives the ''third integral'' up to 14th degree. Resonant periodic orbits
appear for epsilon > 1.65.
Some nonperiodic orbits have also been calculated in order to find the
rotation number as a function of the distance from the central periodic
orbit.
The theoretical results so obtained have been compared with the empirical
results regarding (i) the position of periodic orbits, (ii) the shape of
different characteristics and (iii) the values of rotation number. It is
found that there is agreement between empirical and theoretical values for
0 <= epsilon <= 2.1. If epsilon > 2.1 the positions of periodic orbits do
not agree with those found empirically.
Reviewer: BHATNAGAR, K. B. (Delhi)
Descriptors: *ASTRONOMY AND ASTROPHYSICS --Galactic and stellar
dynamics (85A05); GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Periodic
solutions (58F22); MECHANICS OF PARTICLES AND SYSTEMS -Dynamics of a
system of particles, including celestial mechanics --Celestial mechanics
(70F15)
1/5/7
796331 81m#58049
Centralisateur d'un diffeomorphisme du cercle dont le nombre de rotation
est irrationnel.
Yoccoz, Jean-Christophe
C. R. Acad. Sci. Paris Ser. A-B, 1980, 291, no. 9, A523 - A526.
Languages: French Summary Languages: English summary
Document Type: Journal
Author's summary: ''Let f be a C{sup}{infin} orientation- preserving
diffeomorphism with irrational rotation number alpha , (p{sub}n/q
{sub}n){sub}(n{in}N) the convergents of alpha . Using a theorem which
estimates f{sup}(q{sub}n{sub}+{sub}1) from f{sup}(q{sub}n) when
q{sub}(n+1)/q {sub}n is big, we obtain properties of centralizers in
Diff{sub}+{sup} {infin}(T{sup}1); for example, we construct a
diffeomorphism with irrational rotation number and discrete centralizer in
Diff{sub}+{sup}{infin}(T{sup}1).''
Reviewer: Author's summary
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Ergodic theory;
invariant measures (58F11); MEASURE AND INTEGRATION -Measure-theoretic
ergodic theory --None of the above, but in this section (28D99)
1/5/8
789772 81j#58055
Chaotic behavior in piecewise continuous difference equations.
Keener, James P.
Trans. Amer. Math. Soc., 1980, 261, no. 2, 589 - 604.
Document Type: Journal
A class of difference equations: x{sub}(k+1)=F(x{sub}k), where F: (0,
1){arrr}(0, 1) is piecewise continuous, is investigated. The following are
assumed: (i) F is piecewise continuously differentiable with a single jump
discontinuity at some theta {in}(0, 1); (ii) for x{neq} theta , 0 < t <=
dF/dx <= s < {infin}; (iii) F(x){arrr}0 as x{arrr} theta {sup}+,
F(x){arrr}1 as x{arrr} theta {sup} - , and F(theta ) equals 0 or 1.
For x{in}(0, 1) the rotation number is defined (roughly) to be the
fraction of the iterates of x under F contained within (theta , 1). The
difference equation is considered to be chaotic if the range of the
rotation numbers rho (x) contains a nontrivial interval. Results are proven
for two cases: For F(0) > F(1): rho (x) is constant; if F depends smoothly
on some lambda , then rho (x, lambda ) is rational except for a Cantor set
in lambda ; when rho is rational all solutions are eventually periodic,
and when rho is irrational {hat}{sub}(k=0){sup}{infin} F{sup}((k))((0, 1))
is a Cantor set. For F(0) < F(1): rho (x) covers a ntrivial interval if
F(1) - F(0) is sufficiently large; if dF/dx >= t > 1 then rho (x) is
nowhere continuous; if t < 1 then rho (x) may be constant on an interval
having other values on at most a Cantor set.
Reviewer: MAROTTO,FREDERICK R. (New York)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Strange
attractrs; chaos and other pathologies (58F13); FINITE DIFFERENCES AND
FUNCTIONAL EQUATIONS -Finite differences --Difference equations (39A10)
1/5/9
786512 81i#58035
Nonconjugacy of a minimal distal diffeomorphism of the torus to a C
{sup}1 skew-product.
Rees, M. (Rees, Mary)
Israel J. Math., 1979, 32, no. 2-3, 193 - 200.
Document Type: Journal
Let T{sub}f(x, y)=(f(x), x+y) be a minimal distal homeomorphism of the
torus, where f is a minimal homeomorphism of the circle. Then T{sub}f is
topologically conjugate to the skew product S{sub}(alpha , phi )(x, y)= (x+
alpha , y+ phi (x)), where alpha is the rotation number of f and phi is a
homeomorphism conjugating f and rotation by alpha . A. Finzi (Ann. Sci.
Ecole Norm. Sup. (3) 67 (1950), 243 - 305; MR 12, 434) has shown that if f
is C{sup}3 and alpha lies outside some exceptional set, then the (in this
case unique) conjugating homeomorphism phi , and hence the skew product S
{sub}(alpha , phi ), must be C{sup}1.
The author shows by example that even if f is C{sup}{infin}, T{sub}f need
not be topologically conjugate to any C{sup}1 skew product S{sub}(alpha
,g), where g is a C{sup}1 map of the circle to itself. The construction is
similar to V. I. Arnol'd's construction (Izv. Akad. Nauk SSSR Ser. Mat. 25
(1961), 21 - 86; MR 25#4113; translated in Amer. Math. Soc. Transl. (2) 46
(1965), 213 - 284; see MR 32 #1081) of a minimal analytic homeomorphism of
the circle such that the unique homeomorphism conjugating it and a rotation
is not absolutely continuous.
Reviewer: COVEN, ETHAN M. (Middletown, Conn.)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --None of the
above, but in this section (58F99); MEASURE AND INTEGRATION
-Measure-theoretic ergodic theory --None of the above, but in this section
(28D99)
1/5/10
786506 81i#58029
Topological invariants of germs of analytic mappings and area preserving
mappings, and their application to analytic differential equations in
CP{sup}2.
Naishul', V. A.
Funkcional. Anal. i Prilozhen., 1980, 14, no. 1, 73 - 74.
Languages: Russian
Document Type: Journal
The author considers two classes of mappings (C, 0){arrr}(C, 0), the
class A of analytic mappings and the class G of mappings differentiable at
the point zero and preserving two-dimensional Lebesgue measure. A mapping
from A or G is called a nonlinear rotation by an angle phi {in}R (mod 2 pi
) if its linear part is a rotation by phi . A lifting of a nonlinear
rotation to the universal covering of a punctured neighborhood of the
origin is called a covering rotation; the number phi /2 pi is called the
rotation number of the covering rotation. The author proves the following
results for mappings from the classes A and G: (1) If two mappings are
topologically equivalent and one of them is a nonlinear rotation (about an
angle phi {neq}0, pi (mod 2 pi ) in the case of G), then the second one is
a nonlinear rotation. (2) The linear part of a nonlinear rotation is a
topological invariant. (3) The rotation number of a covering rotation is a
topological invariant. It is shown that the assertions (2) and (3) are not
true for infinitely differentiable and real analytic mappings (R{sup}2,
0){arrr}(R{sup}2, 0). An application to the classification of analytic
differential equations in CP {sup}2 is given.
(English translation: Functional Anal. Appl. 14 (1980), no. 1, 59 - 60.)
Reviewer: GLIKLIH, JU. E. (Voronezh)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Stability theory
(58F10); REAL FUNCTIONS -Miscellaneous topics --Real-analytic functions
(26E05); ORDINARY DIFFERENTIAL EQUATIONS -General theory --Differential
equations i the complex domain (34A20)
1/5/11
785552 81i#28023
The action of diffeomorphism of the circle on the Lebesgue measure.
Katznelson, Y.
J. Analyse Math., 1979, 36, , 156 - 166 (1980).
Document Type: Journal
Any diffeomorphism of the circle is a nonsingular isomorphism relative to
Lebesgue measure mu . Suppose that f is an orientation-preserving
diffeomorphism of the circle whose rotation number rho (f) has unbounded
continued fraction coefficients. It is shown that if f is C{sup}2 then the
nonsingular system (f, mu ) is orbit equivalent to an odometer (adding
machine) of product type. (Every ergodic nonsingular isomorphism of a
Lebesgue space is orbit equivalent to some odometer, but not necessarily to
one of product type.) The following converse is also proved. For a given
odometer theta of product type, the set of orientation-preserving C{sup}
{infin} diffeomorphisms of the circle which are orbit equivalent to theta
is C{sup}{infin} dense in the set of all C{sup}{infin}
orientation-preserving diffeomorphisms with irrational rotation numbers.
Reviewer: WALTERS, PETER (Coventry)
Descriptors: *MEASURE AND INTEGRATION -Measure-theoretic ergodic theory
--None of the above, but in this section (28D99); GLOBAL ANALYSIS, ANALYSIS
ON MANIFOLDS -Ordinary differential equations on manifolds; dynamical
systems --Ergodic theory; invariant measures (58F11)
1/5/12
783273 81h#58039
Sur la conjugaison differentiable des diffeomorphismes du cercle a des
rotations.
Herman, Michael-Robert
Inst. Hautes Etudes Sci. Publ. Math., 1979, , No. 49, 5 - 233.
Languages: French
Document Type: Journal
If f is a diffeomorphism of the circle, isotopic to the identity, with a
lift f to the real line, the rotation number rho (f) is defined to be alpha
= rho (f)=lim{sub}(n{arrr}{infin}) f{sup}n(x)/n (which is independent of
x). V. I. Arnol'd proved in 1961 (Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961),
21 - 86; MR 25#4113; translated in Amer. Math. Soc. Transl. (2) 46 (1965),
213 - 284; MR 32#1081) that if f is a C{sup} omega -diffeomorphism and rho
(f)= alpha satisfies a certain Diophantine condition, then f is C {sup}
omega -conjugate to a rotation. He further conjectured that there exists a
set A{lhk}(0, 1){sbs}Q of measure 1 such that if f is a C{sup} omega
-diffeomorphism and alpha {in}A then f is C{sup} omega -conjugate to R{sub}
alpha , the rotation through an angle 2 pi alpha .
The principal result of this paper is a proof of a stronger version of
the conjecture. If f is a C{sup}r-diffeomorphism, 3 <= r <= omega and rho
(f)= alpha satisfies a certain condition (which determines a set of full
Lebesgue measure) then f is C{sup}(r - 1- beta )-conjugate to R{sub} alpha
for all beta > 0, beta {in}R (beta is a Holder condition on the (r - 1)st
derivative of f).
The paper is, however, much more than just a proof of this result; it is
a complete, detailed study of diffeomorphisms of the circle. It contains
many new results, now well known due to expository articles by H. Rosenberg
and P. Deligne (for a more complete review, the reader is referred to the
review of the last-mentioned papers (Rosenberg, Seminaire Bourbaki, Vol.
1975/76, 28eme annee, Exp. No. 476, pp. 81 - 98, Lecture Notes in Math.,
Vol. 567, Springer, Berlin, 1977; MR 56 #13291a; Deligne, ibid., Exp. No.
477, pp. 99 - 121; MR 56 #13291b)). The present paper also contains a
complete discussion of the classical results concerning diffeomorphisms of
the circle together with new proofs, examples and insightful analyses of
the consequences which result from the author's unique point of view. The
last chapter gives examples and generalizations on T{sup}n and an
independent appendix discusses and gives a new proof for Arnol'd's theorem
on small divisors.
Reviewer: HARTZMAN, C. S. (Halifax, N.S.)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Ergodic theory;
invariant measures (58F11); MEASURE AND INTEGRATION -Measure-theoretic
ergodic theory --None of the above, but in this section (28D99)
1/5/13
780029 81g#58030b
Periodic and conditionally periodic trajectories in conservative systems
under minimal conditions of smoothness. II.
Antonov, V. A.
Leningrad. Gos. Univ. Uchen. Zap., 1979, , No. 400 Ser. Mat. Nauk
Vyp. 57 Trudy Astronom. Obser. 35, 127 - 142, 220.
Languages: Russian Summary Languages: English summary
Document Type: Journal
The author considers periodic trajectories (cycles) of an area-
preserving homeomorphism P of a plane circular annulus. Various dynamical
problems (orbits of stars in stellar accumulations and others) lead to such
mappings (nonsmooth or mappings of small order of smoothness). The mapping
P shifts all radii into one side and satisfies the ''radii distortion''
condition (roughly speaking, a point with a larger radius is shifted by a
larger angle). Let q be the number of points in a cycle, p the number of
complete revolutions of a cycle, nu =p/q the rotation number. The author
studies the so-called regular cycles: nu =p/q is an irreducible fraction,
and the azimuths theta {sub}i of points in a cycle satisfy the condition
((theta {sub}j - theta {sub}i)/2 pi )=((j - i) nu ). The main result of the
first paper is the following: For every rational nu lying between the
rotation numbers of the interior and the exterior circles there exists a
regular cycle with the rotation number nu . For q=1 this result was proved
by Poincare (a fixed point is a regular cycle). In the second paper the
author studies ''minimal'' (in the sense of the variational definition)
regular cycles and ''alternating'' regular cycles lying between ''minimal''
cycles (all cycles for a fixed nu ). He shows that there is always an
alternating cyclelying between two minimal cycles. In the known examples
the minimal and alternating cycles correspond to stable and unstable closed
trajectories. The articles are notable for their detailed exposition.
Reviewer: GLIKLIH, JU. E. (Voronezh)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Periodic
solutions (58F22); MECHANICS OF PARTICLES AND SYSTEMS -Linear vibration
theory --Conservative systems (70J15); ASTRONOMY AND ASTROPHYSICS
--Galactic and stellar dynamics (85A05)
1/5/14
764883 81b#34028
Computer studies of nonlinear oscillators.
Nonlinear oscillations in biology (Proc. Tenth Summer Sem. Appl. Math.,
Univ. Utah, Salt Lake City, Utah, 1978)
Hoppensteadt, Frank C.
Publ: Amer. Math. Soc., Providence, R.I.
1979, pp. 131 - 139, Lectures in Appl. Math., 17,
Document Type: Collection
A sample of nonlinear oscillation problems is presented to illustrate
several numerical methods which have been successfully applied to the study
of oscillanomena. Results are summarized in this survey article.
First, studies of Mathieu's equation suggest computer experiments which can
be performed to study the response of a pendulum to oscillation of its
support point. A pendulum having a vertically oscillating support point can
have three (at least) stable responses coexisting for the same parameter
values of forcing and tuning; the straight up and straight down positions
as well as a continual rotation are all stable solutions of this problem.
Computer experiments can be used in these cases to determine the domains of
attraction of the various modes of response. This example also demonstrates
the interesting fact that otherwise unstable static states can be
stabilized by external oscillatory forcing.
Next, stable oscillatory responses of the van der Pol equation to
periodic forcing are described. The concept of rotation number can be used
to summarize numerical experiments on subharmonic solutions. Then it is
described how computation of the power spectrum can be used to study the
presence of higher harmonics.
Finally, computer studies of certain chaotic dynamical systems are
discussed. Numerical simulations are used to describe density functions
characterizing random behavior of deterministic oscillators.
(For the entire collection see MR 81a:92002.)
Reviewer: AMES, W. F. (Atlanta, Ga.)
Descriptors: *ORDINARY DIFFERENTIAL EQUATIONS -Qualitative theory
--Nonlinear oscillations (34C15); GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS
-Ordinary differential equations on manifolds; dynamical systems --Strange
attractors; chaos and other pathologies (58F13)
1/5/15
757303 80k#58080
Sur la structure des bifurcations des diffeomorphismes du cercle.
Mira, Christian
C. R. Acad. Sci. Paris Ser. A-B, 1978, 287, no. 13, A883 - A886.
Languages: French Summary Languages: English summary
Document Type: Journal
Piecewise linear mappings of the form x 3a+ lambda {sub}1x for x < 0, and
x 3 - b+ lambda {sub}2x for x > 0 with a, 3b > 0, x{in}R have been studied
by many authors, including N. N. Leonov (Izv. Vyssh. Uchebn. Zaved.
Radiofizika 2 (1959), no. 6, 942 - 956). In the present paper, the author
adapts Leonov's results to mappings of the circle having the form theta 3
theta + lambda +g(theta ); lambda {in}( - 2 pi , 30), theta {in}S{sup}1,
where g(theta )= g(theta +2 pi ) and g(0){neq}0, g'(0)=0, {vert}g'(theta )
{vert} < 1. It is shown that period-doubling bifurcations occur in infinite
sequences associatd with which one has a ''box within box'' structure. The
behavior of the rotation number mu = mu (lambda ) as the parameter lambda
varies is also discussed, it being shown that there is a Cantor set of
lambda values for which mu is irrational.
Reviewer: HOLMES, P. J. (Ithaca, N.Y.)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Bifurcation
theory and singularities (58F14)
1/5/16
722330 58#31279
Conjugaison C{sup}{infin} des diffeomorphismes du cercle pour presque
tout nombre de rotation.
Herman, Michael-Robert
C. R. Acad. Sci. Paris. Ser. A-B, 1976, 283, no. 8, Aii, A579 -
A582.
Languages: French Summary Languages: English summary
Document Type: Journal
Author's summary: ''We announce that Theorem 1 of our preceding note
(same C. R. Ser. A-B 282 (1976), no. 10, A503 - A506; MR 55#6448) is true
for a set A of numbers of T{sup}1 of Haar measure equal to 1 (A is also
meager in T{sup}1): If f is a diffeomorphism of S{sup}1 of class C{sup}
{infin}, of rotation number alpha {in}A, then f is C{sup}{infin}-conjugate
to the rotation R{sub} alpha of rotation number alpha . This settles a
conjecture of V. I. Arnol'd. ''
A detailed version of the author's work has recently appeared (Inst.
Hautes Etudes Sci. Publ., No. 49, (1979), 5 - 235).
Reviewer: Editors
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --None of the
above, but in this section (58F99); MEASURE AND INTEGRATION -Classical
measure theory --Measure-preserving transformations, flows (dynamical
systems), measure-theoretic ergodic theory (28A65)
1/5/17
722257 58#31210
Differentiable dynamics. An introduction to the orbit structure of
diffeomorphisms.
Nitecki, Zbigniew
Publ: The M.I.T. Press, Cambridge, Mass.-London
1971, xv+282 pp.
Price: $6.50.
Document Type: Book
The book is a revised version of lecture notes for a course given by the
author at Yale in 1969. It is based on ideas presented by S. Smale in his
1967 survey article (Bull. Amer. Math. Soc. 73 (1967), 747 - 817; MR
37#3598; erratum, MR 39, p. 1593) and subsequently developed by many
authors. Crudely speaking, the Smale program has two objectives: (1) to
find generic properties of dynamical systems, i.e., properties which hold
for ''almost all'' systems; and (2) to prove that certain systems are
''stable'', i.e., have qualitative behavior which is unchanged by small
perturbations of the system. The author explains the program, gives
detailed examples and proves many of the basic theorems.
Here is a list of the chapter titles together with an indication of the
contents: (1) The circle: rotation number, Denjoy's example, Peixoto's
generic theory; (2) Periodic points: local behavior (Hartman's theorem),
stable manifold theory, Kupka-Smale genericity theorem; (3) Anosov
diffeomorphisms: examples and stability theorem; (4) The horseshoe: a
detailed construction and its relation to Bernoulli shifts; (5) Hyperbolic
sets: stable manifold theory, local product structure, proof that for an
Axiom A system every point is asymptotic to a nonwandering point; (6) The
Omega -stability theorem: a detailed proof; (7) Survey of recent (at the
time of writing) work.
Reviewer: J. W. Robbin (Madison, Wis.)
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems (58Fxx); ORDINARY
DIFFERENTIAL EQUATIONS-Qualitative theory --Dynamical systems (34C35)
1/5/18
707397 58#17324
Frequency entrainment of a forced van der Pol oscillator.
Flaherty, J. E.; Hoppensteadt, F. C.
Studies in Appl. Math., 1978, 58, no. 1, 5 - 15.
Document Type: Journal
This is an interesting study of a van der Pol relaxation oscillator
subject to external sinusoidal forcing:
(d{sup}2u/dt{sup}2)+k(u{sup}2 - 1)(du/dt)+u= mu kB cos(mu t+ alpha ).
The model is studied for the parameters satisfying alpha =0, mu =1, 0 < B
< 0.8, 0 < 1/k < 0.2. The numerical computing of the rotation number (rho )
suggests that it defines a continuous but piecewise constant surface except
in overlap regions where it is double-valued, having what looks like folds.
The parameter ranges where rho is single-valued illustrate the phenomenon
of locking phase, the successive bifurcation of stable subharmonic and
almost periodic oscillations. The paper gives an indication of the nature
of stable responses of this system, which poses difficult analytic
problems.
Reviewer: G. V. Hmelevskaja-Plotnikov (Jambes)
Descriptors: *ORDINARY DIFFERENTIAL EQUATIONS -Qualitative theory
--Nonlinear oscillations (34C15); MECHANICS OF PARTICLES AND SYSTEMS
--Ordinary differential equations (70.34)
1/5/19
611858 53#6012
The stability of the rotation number of a doubly periodic differential
equation.
Ershov, E. K.
Differencial'nye Uravnenija, 1975, 11, no. 11, 1949 - 1955, 2107.
Languages: Russian
Document Type: JOURNAL
Reviewer: Ju. G. Borisovich
Descriptors: *ORDINARY DIFFERENTIAL EQUATIONS -Qualitative theory
--Periodic solutions (34C25)
1/5/20
597333 52#8566
Dependence on a parameter of the rotation number of a differential
equation defined on the torus.
Ershov, E. K.
Differencial'nye Uravnenija, 1975, 11, no. 10, 1774 - 1779, 1907.
Languages: Russian
Document Type: JOURNAL
Reviewer: L. Reizinsh
Descriptors: *ORDINARY DIFFERENTIAL EQUATIONS -Qualitative theory
--Periodic solutions (34c25); GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS
-Ordinary differential equations on manifolds; dynamical systems
--Periodic points and zeta functions (58f20)
1/5/21
556329 49#11567
Generic properties of the rotation number of one-parameter
diffeomorphisms of the circle.
Brunovsky, Pavol
Czechoslovak Math. J., 1974, 24(99), , 74 - 90.
Document Type: JOURNAL
Reviewer: L. Reizinsh
Descriptors: *GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS -Ordinary
differential equations on manifolds; dynamical systems --Periodic points
and zeta functions (58f20)
20jan83 15:53:48 User29356
$15.79 0.287 Hrs File239 4 Descriptors
LOGOFF 15:53:50