UK Nonlinear News, November 2003


Review article Rotation numbers and instability sets

ABSTRACT. Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows.

Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G.D. Birkhoff was led to the concept of ``regions of instability'' for twist maps. We discuss the closely related notion of instability sets for a generic area preserving surface diffeomorphism and develop their properties.

J. Franks. 2003. Rotation numbers and instability sets. Bulletin (New Series) of the American Mathematical Society, Volume 40, Number 3, pages 263-279.

Journal: Dynamical Systems

Invitation to Contribute

The journal aims to publish in all areas of applications of dynamical systems to the sciences and engineering including bifurcation theory, differential equations, Hamiltonian and Lagrangian dynamics, ergodic theory and topological dynamics. High quality papers in all of these areas are welcome. The journal was given an impact factor of 1.300 in the ISI citation index for 2002.

We would especially encourage authors to consider submitting their papers to the journal in the following areas:


Papers for consideration should be sent to either of the following editors:
Matthew Nicol, Peter Ashwin
Editorial Office, Editorial Office
Department of Mathematics and Statistics, Department of Mathematical Sciences
University of Surrey, University of Exeter
Guildford, GU2 7XH, Exeter, EX4 4QE

Instructions for Authors:

Further information and details on submitting papers can be found at the instructions for authors webpage

Source: Matthew J. Nicol (

Journal: Journal of Nonlinear Mathematical Physics

The Journal of Nonlinear Mathematical Physics will have two supplementary issues in 2003.

Supplement 1, Volume 10, September 2003:
"The Oeresund Symposium on Partial Differential Equations" held in Lund (Sweden), May 23-25, 2002.
Editors: A. Constantin, G. Grubb, P.-A. Ivert, A. Melin and J.P. Solovej.

Currently in Press: Supplement 2, Volume 10, December 2003:
Symmetries and Integrability of Difference Equations, Proceedings of the EuroConference on Discrete Painlevť Equations and the Solvability of Difference Equations
Editors: Yu. Suris et al

For orders or more information contact

Professor Norbert Euler
Managing Editor
Journal of Nonlinear Mathematical Physics
Department of Mathematics
Lulea University of Technology
SE-971 87 Lulea, Sweden
Fax +46 920-491073

Source: Professor Norbert Euler

Newsletter: Society for Mathematical Biology

The September issue of the SMB Newsletter is on the web at

It contains a report on ICIAM2003 and other items of interest to your readers.

Center: Raymond Mejía

Table of Contents: Dynamical Systems Volume 18(1), March 2003

Ciprian D. Coman
pp. 1 - 22On a class of non-smooth oscillators
D. Turzík, M. Dubcová, A. Klí), P. Pokorny
pp. 23 - 33 Application of Bernoulli shift to investigation of stability of spatially periodic solutions in lattice dynamical systems
V√≠tor Ara√ļjo
pp. 35 - 55Random perturbations of codimension one homoclinic tangencies in dimension 3
J. A. Langa
pp. 57 - 68 Finite-dimensional limiting dynamics of random dynamical systems
Patrick Bonckaert, Vincent Naudot, Jiazhong Yang
pp. 69 - 88 Linearization of hyperbolic resonant germs
Ricardo Riaza, Pedro J. Zufiria
pp. 89 - 105 Differential-algebraic equations and singular perturbation methods in recurrent neural learning

Source: Matthew J. Nicol (

Table of Contents: Dynamical Systems Volume 18(2), June 2003

Z. Bishnani, R. S. MacKay
pp. 107 - 129 Safety criteria for aperiodically forced systems
Peter Walters
pp. 131 - 138 A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function
Peter Ashwin, Gunter Ochs
pp. 139 - 158 Convergence to local random attractors
Florian Wagener
pp. 159 - 163 A note on Gevrey regular KAM theory and the inverse approximation lemma
Rafael R. Souza
pp. 165 - 179 Sub-actions for weakly hyperbolic one-dimensional systems
C. A. Morales
pp. 181 - 189 Singular-hyperbolic sets and topological dimension

Source: Matthew J. Nicol (

Table of Contents: Dynamical Systems Volume 18(3), in press

P. McSharry and P. Ruffino
Asymptotic angular stability in non-linear systems: rotation numbers and winding numbers
A. Shcherbina
Gevrey regularity of the global attractor for the dissipative Zakharov system
A. Halanay
On the stability of some equilibrium points in a plankton population model
X. Liao and K. Wong
Convergence analysis of hybrid Bi-directional associative memory neural networks with discrete delays
N. Sri Namachchivaya and H. van Rossel
A centre manifold analysis of variable speed machining
P. Walters

Source: Matthew J. Nicol (

Table of Contents: Nonlinearity Volume 16(5), September 2003

All articles are free for 30 days after publication on the web.

1565 M. Yampolsky
On the eigenvalues of a renormalization operator
1573 H.E. Lomel\'{\i} and J.D. Meiss
Heteroclinic intersections between invariant circles of volume-preserving maps
1597 A. Yu Pogromsky, G. Santoboni and H. Nijmeijer
An ultimate bound on the trajectories of the Lorenz system and its applications
1607 A. Novikov
Modulational stability of cellular flows
1641 C.G.T. de Araujo Moreira and E.M. Mu\~noz Morales
Sums of Cantor sets whose sum of dimensions is close to 1
1649 M. Rams
Absolute continuity of the SBR measure for non-linear fat baker maps
1657 L. Barreira
Dimension estimates in nonconformal hyperbolic dynamics
1673 C.M. Field and F.W. Nijhoff
A note on modified Hamiltonians for numerical integrations admitting an exact invariant
1685 S. Nonnenmacher
Spectral properties of noisy classical and quantum propagators
1715 P.B. Mucha
On the inviscid limit of the Navier--Stokes equations for flows with large flux
1733 B. Solomyak and H. Xu
On the `Mandelbrot set' for a pair of linear maps and complex Bernoulli convolutions
1751 H. Broer, H. Han{\ss}mann, \`A. Jorba, J. Villanueva and F. Wagener
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
1793 D.V. Zenkov and A.M. Bloch
Invariant measures of nonholonomic flows with internal degrees of freedom
1809 G. Fibich, B. Ilan and S. Schochet
Critical exponents and collapse of nonlinear Schr\"odinger equations with anisotropic fourth-order dispersion
1823 J.D. Gibbon, D.R. Moore and J.T. Stuart
Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations
1833 G. Arioli and P. Zgliczy\'nski
The H\'enon--Heiles Hamiltonian near the critical energy level---some rigorous results
1853 X-S Yang and S. Zhang
On the possibility of creating new asymptotically stable periodic orbits in continuous time dynamical systems by small feedback control
1861 D.H. Kim and B.K. Seo
The waiting time for irrational rotations
1869 E. Herrera and R. Herrera
A note on the relation between period and energy of periodic orbits near equilibrium points

Source: Elizabeth Martin (

Table of Contents: Nonlinearity Volume 16(6), November 2003

All articles are free for 30 days after publication on the web.

R27 P.J. Forrester
Growth models, random matrices and Painlev\'e transcendents
1875 A. Chertock, A. Kurganov and P. Rosenau
Formation of discontinuities in flux-saturated degenerate parabolic equations
1899 T. Kapela and P. Zgliczy\'nski
The existence of simple choreographies for the N-body problem - a computer-assisted proof
1919 P.J. Forrester and N.S. Witte
Discrete Painlev\'e equations and random matrix averages
1945 R. Dascaliuc
On backward-time behaviour of Burgers' original model for turbulence
1967 H. Bellout, S. Benachour and E.S. Titi
Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations
1991 B. Saussol and J. Wu
Recurrence spectrum in smooth dynamical systems
2003 K. Yagasaki
Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle centres
2013 G. Verzini
Bounded solutions to superlinear ODEs: a variational approach
2029 T. Ozawa and Y. Yamazaki
Life-span of smooth solutions to the complex Ginzburg--Landau type equation on a torus
2035 E. Bogomolny and C. Schmit
Asymptotic behaviour of multiple scattering on an infinite number of parallel half-planes
2061 Y. Yang
On Pokrovskii's anisotropic gap equations in superconductivity theory
2075 P.V. Gordon
An upper bound of the bulk burning rate in porous media combustion
2083 P. Plech\'a\v c and V. \v Sver\'ak
Singular and regular solutions of a nonlinear parabolic system
2099 T.K. Callahan
Hopf bifurcations on cubic lattices
2123 M. Mourragui, E. Orlandi and E. Saada
Macroscopic evolution of particle systems with random field Kac interactions
2149 K.N. Webster and J.N. Elgin
Asymptotic analysis of the Michelson system
2163 D. Turaev and S. Zelik
Homoclinic bifurcations and dimension of attractors for damped nonlinear hyperbolic equations
2199 A. Fruchard and R. Schäfke
Bifurcation delay and difference equations
AUTHOR INDEX (with titles), volume 16

Source: Elizabeth Martin (

Table of Contents: Nonlinearity Volume 17(1), January 2004

All articles are free for 30 days after publication on the web.

C1 B. Krauskopf and H.M. Osinga
The Lorenz manifold as a collection of geodesic level sets
1 N. Simanyi
Proving the ergodic hypothesis for billiards with disjoint cylindric scatterers
23 R. Coutinho and B. Fernandez
Fronts in extended systems of bistable maps coupled via convolutions
49 J.S. C\'anovas
Topological sequence entropy of interval maps
57 J. Benoit, E. von Hauff and A. Saxena
Self-dual bending theory for vesicles
67 P.C. Moan
On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth
85 J. Sieber and B. Krauskopf
Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity
105 V. Afraimovich, A. Morante and E. Ugalde
On the density of directional entropy in lattice dynamical systems
117 R. Giamb\`o, F. Giannoni and P. Piccione
Gravitational lensing in general relativity via bifurcation theory
133 J. Su, J. Rubin and D. Terman
Effects of noise on elliptic bursters
159 H. Fuk\'s
Probabilistic cellular automata with conserved quantities
175 F. Abdenur, C. Bonatti and L.J. D\' \i az
Non-wandering sets with non-empty interiors
193 M. Kamenskii, O. Makarenkov and P. Nistri
Small parameter perturbations of nonlinear periodic systems
207 G. Friesecke and R.L. Pego
Solitary waves on Fermi--Pasta--Ulam lattices: III. Howland-type Floquet theory
229 G. Friesecke and R.L. Pego
Solitary waves on Fermi--Pasta--Ulam lattices: IV. Proof of stability at low energy
253 R.E. Beardmore, R. Laister and A. Peplow
Trajectories of a DAE near a pseudo-equilibrium
281 U. Kirchgraber and D. Stoffer
Possible chaotic motion of comets in the Sun--Jupiter system---a computer-assisted approach based on shadowing
301 S.C. Venkataramani
Lower bounds for the energy in a crumpled elastic sheet---a minimal ridge
313 J. Billingham
Dynamics of a strongly nonlocal reaction--diffusion population model
347 B. Rink
A Cantor set of tori with monodromy near a focus--focus singularity
357 J. Campos and M. Tarallo
Large minimal period orbits of periodic autonomous systems

Source: Elizabeth Martin (

<< Move to UK Nonlinear News Issue 34 Index Page.
Page Created: 8 November 2003.
Last Updated: 1 December 2003.