2.2.01 CO
2.02 Professor
2.03 Director, Mathematics Research Centre
2.04 M
2.05 1947
2.06 David
2.07
2.09 Arrowsmith
2.09 UNI
2.10 Queen Mary and Westfield College, London University
2.11 School of Mathematical Sciences
2.12 Mathematics Research Centre
2.13 Mile End Road
2.14
2.15 E1 4NS
2.16 London
2.17 UK
2.18 0044 (0)171 975 5464
2.19 0044 (0)181 981 9587
2.20 d.k.arrowsmith@qmw.ac.uk
2.21 3
2.22 0
2.23 0
2.24 5000

TEAM 2

2.01 CR
2.02 Professor
2.03 Leading senior researcher
2.04 M
2.05 1950
2.06 Rifkat
2.07 Ibragimovich
2.09 Bogdanov
2.09 UNT
2.10 Moscow State University by M.V. Lomonosov
2.11 Nuclear Physics Institute by D.V. Skobelcin
2.12
2.13 Lomonosov Prosp.
2.14
2.15 119899
2.16 Moscow
2.17 RV
2.18 007 (095) 939-56-41
2.19 007 (095) 939-68--96
2.20 bogdanov@bogdan.npi.msu.su
2.21 4
2.22 1
2.23 2
2.24 24000
 
 

TEAM 3
 
2.01 CR
2.02 Research Scientist
2.03 Investigator
2.04 M
2.05 1967
2.06 Julyan
2.07
2.08 Cartwright
2.09 PUB
2.10 Higher Council for Scientific Research(Consejo Superior de Investigaciones Cientificas)
2.11 Andalusian Institute for Earth Sciences(Instituto Andaluz de Ciencias de la Tierra)
2.12 Laboratory for Crystallographic Studies (Laboratorio de Estudios Cristalograficos)
2.13 Facultad de Ciencias, Campus Fuentenueva
2.14
2.15 E-18071
2.16 Granada
2.17 ES
2.18 34 958 243360
2.19 34 958 243384
2.20 julyan@galiota.uib.es
2.21 1
2.22 1
2.23
2.24 4000

TEAM 4. UMIST

2.01 CR
2.02 Professor
2.03 Senior researcher
2.04 M
2.05 1959
2.06 Paul
2.07 Alexander
2.08 Glendinning
2.09 UNI
2.10 University of Manchester Institute of Science and Technology (UMIST)
2.11 Department of Mathematics
2.12 Applied Mathematics
2.13 P.O. Box 88
2.14
2.15 M60 1QD
2.16 Manchester
2.17 UK
2.18 0044 (0)161 3680
2.19
2.20 p.a.glendinning@umist.ac.uk
2.21 3
2.22 1
2.23 0
2.24 3000
 
 

TEAM 5. MOSCOW ENERGY INSTITUTE
2.01 CR
2.02 Docent
2.03
2.04 M
2.05 1943
2.06 Aleksandr
2.07 Ivanovich
2.08 Plis
2.09 UNI
2.10 Moscow Energy Institute (Technical University)
2.11 Department “Mathematics”
2.12
2.13 Krasnokazarmennaya
2.14
2.15 119899
2.16 Moscow
2.17 RU
2.18 007 (095) 939-56-41
2.19 007 (095) 939-68--96
2.20 bogdanov@bogdan.npi.msu.su
2.21 3
2.22
2.23
2.24 23000

3. Work Programme

3.1 Research projects

3.1.1 Title

The Bogdanov map and weakly-dissipative theory of Kolmogorov-Arnold-Moser

3.1.2. Objectives

The key objective is to use the weakly-dissipative Kolmogorov-Arnold-Moser (KAM) theory to:

(a) calculate discrete invariants (with integer values) such as: the distribution of periods of asymptotically (un)stable periodic orbits in the set of all positive integer numbers; consider the decomposition of periods into prime factors and obtain corresponding distribution of degrees of prime factors;

(b) calculate adiabatic invariants ( over the real numbers) in weakly-dissipative KAM systems for example - total energy, kinetic energy, dissipation, Lyapunov exponent, length of free motion; random values for periodic asymptotic (un)stable periodic orbits determined by the measure of their basin of (repulsion) attraction(corresponding estimation of above measure);

(c) investigate the dependence of above invariants upon the periods of periodic orbits. The invariants will be found from the statisical and numerical studies once the appropriate algorithms have been finalised.

(d) interpret the BM as a weakly-dissipative perturbation of a classical system, the so-called "anharmonic oscillator". This property allows ideas and investigations from mathematical physics to be used to understand the maps properties.

Other intentions are to:

(e) investigate the approximating diffeomorphisms given by the flow of versal equivariant vector fields with higher symmetries and develop appropriate discrete models for the application of the weakly-dissipative KAM theory;

(f) carry out initial numerical tests on the new maps for the appropriate parameter regimes and liaise on the appropriateness of the new models;

(g) adapt the basic Bogdanov models to higher dimensions by considering product structures and coupled map lattice models;

(h) investigate the relevance of the periodic orbit map data to

1. (i) round-off error analysis for discretizations of vector fields;

3.1.3. Background

The aim is to study nonlinear phenomena of discrete dynamical systems by using approximating continuous-time systems. The term "strange attractor" was introduced in the 1970's to describe minimal non-trivial invariant sets which asymptotically attracted neighbouring. Typically the manifolds are braided with a fractal dimension reflecting the complex geometry of the folded unstable manifolds of the embedded saddle orbits of the attractor. Many computer examples of strange attractors have since been introduced. However, even the simplest examples, such as the dissipative Henon attractor, are not amenable to elementary interpretation. In [BC] and [MV], the existence of strange attractors is proven within certain families but not for any given map with a specific parameter value.

The proposed work leading up to the INTAS programme is centred around the study of the behaviour of generic planar diffeomorphisms for which there are approximating vector fields. The full orbital behaviour of such a diffeomorphisms is impossible to describe and so a fundamental and important way in which its behaviour is characterized is by the use of invariants. An "invariant" is a quantity which is shared by all diffeomorphisms which are either topologically or differentiably conjugate to each other. Simple examples of historically important invariants are:

(a) the asymptotic stability of a fixed point in the topologically conjugate case; or

(b) the set of eigenvalues at a fixed point in the differentiable case. Thus pursuit of invariants is a fundamental research project which had it origins with the Poincare-Dulac [A] theorem and remain ongoing.

The diffeomorphisms with approximating vector fields are developed as follows. If the linearization of the diffeomorphism is a rotation through the angle 2pi p/q where p and q are integers, then the q-th iterate of the map is approximated up to any order of its Taylor expansion by the time-one map of an appropriate vector field. This important result is a key to understanding the behaviour of planar diffeomorphisms. The collection of approximating vector fields, which have versal unfolding properties, was developed during the 1970's by Arnold [A] and Bogdanov [B1]. A contemporary development by Takens [T1] realised these results in the context of Poincare maps of forced oscillators. The hamiltonian case was treated in [T2]. Complementary important numerical work on the behaviour of planar diffeomorphisms is discussed in [ACLP] and [AP2].

The "Bogdanov map" (BM), which is at the heart of the numerical study, is a three-parameter discrete map developed in [AP1] from the "Bogdanov-Takens" (BT) vector field [B1] for the case(q=1) of no rotational symmetry. Similar maps are considered in [MGBC]. The BT vector field exhibits bifurcations involving Hopf and saddle connections and arises from a double zero eigenvalue at a fixed point. The vector field has further importance in that it provides a versal deformation [A], [B1] of a generic singularity with double zero eigenvalues. A sequence of such "equivariant versal families" of vector fields was developed in [A] for the various cases of rational (p/q) rotational symmetry for q=1,2,3,4 and a unified case q>4.

The BT bifurcation is a key method for the prediction of a global saddle-connection which emanates from a degenerate point. The Bogdanov map has several important features: (i) it models the growth of an attractor from a Hopf bifurcation, (ii) it develops a homoclinic saddle-connection, (iii) the attractor grows to disappear in the homoclinic tangle. This is, of course, much more complicated than the corresponding vector field behaviour where a limit cycle from a Hopf bifurcation grows to coincide with a homoclinic saddle connection before vanishing. A critical property of the Bogdanov map is that it is also unfolds the Henon area-preserving map (the anharmonic oscillator). The rich island-chain structure present in the Henon map is preserved at certain levels for small parameters in the Bogdanov family and thus the KAM type behaviour present in the Henon map, modified in the presence of weak dissipation, also occurs.

The detailed work on the Bogdanov vector field by Cartwright, Arrowsmith et al [ACLP] and the q=4 work on Birkhoff Attractors [AP2] encouraged Bogdanov and his co-workers to obtain more explicit information on the periodic structures of the BM via weakly-dissipative KAM theory, eg [B2], [B4], [B5], [B6], [B7], [B12] as an aid to giving insight into the structure of strange attractors.

The asymptotic (un)stable orbits for the BM are isolated from other periodic orbits and thus their numerical analysis is usually more straightforward than when a map has a dense sets of hyperbolic periodic points. Moreover, the integer number given by the period of an isolated periodic orbit is well defined for a suitable neighborhood of asymptotically (un)stable periodic orbits. Weakly-dissipative KAM theory allows research into structure within the neighborhood of strange hyperbolic attractors for appropriate maps.

Hence the Bogdanov map is a prototypical model for studying the way in which the systems undergo a continuous breakdown of KAM phenomena. The development of the various strands of work by several members of this proposal's teams over the last two decades has given three components to the research programme:

(a) the general results on vector field approximations;

(b) the introduction of discrete maps which mimic the key vector fields, the preliminary investigation of the bifurcational behaviour of their associated strange attractors;

and

(c) the detailed numerical follow-up work on the periodic structures neighbouring a strange attractor by Bogdanov and his co-workers.
 
 

This current situation together with the addition to the teams of further strong research expertise in dynamical systems, numerical analysis, round-off error and noisy systems leaves the proposed INTAS group in a powerful position to move forward on many new fronts during the programme.

References

[A] V.I. Arnold. Geometrical Methods in the theory of Ordinary Differential Equations. Berlin and New York : Springer-Verlag, 1983.

[AP1], [ACLP], [AP2] see 5.1.3.2(team1).

[B1], [B2], [B4], [B5], [B6], [B7], [B12] see 5.1.3.2(team2).

[BC] M. Benedicks and L. Carleson, The dynamics of the Henon Map, Ann. Math., v.133, 73-169, (1991).

[GM] J. Gumowski and C. Mira, Dynamique Chaotique, Cepadues Editions, (1980).

[MV] L. Mora & M. Viani, Abundance of strange attractors, Acta Math., v171, 1-71 (1993).

[MGBC] C. Mira, L. Gardini, A. Barugola and J.C. Cathala, Chaotic Dynamics in Two-dimensional Noninvertible Maps, (1996) World Scientific, Singapore.

[T1] F.Takens, Forced oscillations and bifurcations, Commun. Math. Inst. Rijksuniversiteit, Utrecht, 3, (1974) 1-59.

[T2] F.Takens, Introduction to global analysis, Commun. Math. Inst. Rijksuniversiteit, Utrecht, 2, (1973) 1-111.

3.1.4 Scientific/Technical description

3.1.4.1 Research programme

The key feature of the research programme is to develop the techniques required for the calculation of new invariants of planar diffeomorphisms and expand the base of maps for the techniques applicability. The numerical techniques involve locating period orbits, their periodicities and approximate basins of attraction in the neighbourhoods of strange attractors as a necessary step towards promoting a new approach to finding invariants of planar diffeomorphisms. Large periods have to be detectable. Moreover, it needs to be shown that the techniques currently being applied to the Bogdanov map can be extended to other maps of dynamical significance. The algorithmic and numerical studies have to be intensive to obtain sufficient data and, hence, it is important that the techniques being developed are made known to a wide group of researchers to allow multi-centre calculations to develop. The research will be published in ways which maximise its circulation within the academic community.

The Russian groups are already heavily involved in the numerical use of the weakly-dissipative KAM theory for the Bogdanov map. The 2-dimensional nature of the Bogdanov map allows the invariant codimension-one foliations (i.e. orbits) of the vector fields to be used more effectively than for higher dimensional phase spaces. Specifically, the low dimension allows numerical calculation of periodic orbits up to extremely high periods of 100,000,000, (see [B6],[B7]). This will enable the project to:

(a) obtain data on approximately 1000 stable and separately 1000 unstable periodic orbits for analysis of their statistical properties (this is needed to obtain suitable values of parameters for the BM; the idea is to present the above data for 4-6 points in the parameter space of the BM);

(b) calculate adiabatic invariants for the above data (it allows a comparison of the properties of models of continuous mechanics with the discrete particle motion models of statistical mechanics);

(c) calculate estimates for positive measure of the basins of (repulsion)attraction of asymptotic (un)stable periodic orbits;

(d) use the invariant foliation of the approximating vector fields to determine regions of phase space where stochastic (Arnold) diffusion occurs (ordinarily, the regions have infinite numbers of hyperbolic periodic orbits), cf. [AP2].

Furthermore, the new results of qualitative theory of dynamical systems on the plane relevant to weakly-dissipative KAM enables a more detailed study of the following:

(e1) properties of asymptotic (un)stable orbits randomized by positive measure of its basin of attraction;

(e2) regular properties of stochastic dynamics due to the existence of smooth asymptotics for adiabatic invariants near generic periodic orbits;

(e3) dependence of the above characteristics on the period of the orbit (parameters for the BM can be chosen for which small period periodic orbits are not present. The statistical properties of the dynamical system is clearer for high values of periods).

Finally, the periodic orbit data files will enable:

(f1) a check on the efficiency and accuracy of new software;(f2) the ability to improve precision of calculations;

(f3) the calculation of new invariants valid for other maps.

Initially, the UK and Spanish groups (QMW, Granada, UMIST) will concentrate on two areas, both fundamental to bifurcation theory: symmetric systems and imperfection theory. The effect of symmetry constraints on bifurcations has always been important in applications, and the BT bifurcation with Z(2) symmetry can be found in [GH], is discussed in detail in [K] whilst Dangelmayr and Knobloch [DK] consider the effect of O(2) symmetry. To this end the UK and Spanish groups will:

(g) investigate the various equivariant vector fields which arise from the imposition of symmetry and consider the behaviour of the corresponding maps from the viewpoint of near area-preserving criteria;

(h) collaborate with the Moscow group as to use of symmetry and symmetry breaking in the context weakly-dissipative KAM theory;

(i) understand the dynamics of planar map with higher order symmetries which have related versal properties: for example: x'= x+y'; y'=y+mx+ny-kx²(y+ax) with k>0 and a=+/-1, which is the Bogdanov map with Z(2) symmetry, and carry out initial numerical work on appropriate parameter regimes.

Imperfection theory describes the way in which small symmetry breaking terms alter the symmetric picture. This is particularly important in applications because the symmetries of a system are often idealized (parallel plates are never truely parallel in practice). Although the effect of such terms is well understood for codimension one local bifurcations [Sh] there have been few attempts at producing a coherent analysis for codimension two bifurcations. Glendinning and Mullin [GM] have considered the effect of small symmetry breaking terms on global bifurcations, and shown that these considerations are important in applications. Thus the UK and Spanish groups will

(j) extend these ideas to codimension-two bifurcations for both maps and flows.

In recent years there has been a great deal of effort into understanding spatial structure via coupled lattice models, [BS]. Models in which all lattice sites are coupled to each other (global coupling [G, 1999]) are also relevant to the understanding of control and synchronization [PC]. Related to this is the issue of blowout bifurcations [ABS] . None of the recent work considers blowout and synchronization from weakly dissipative states, and the intention is to:

(k) look at issues of synchronization and blowout bifurcations [G3] in globally coupled Bogdanov maps.

This approach is often related to issues of synchronization [PC], [BG],[G3] (where all systems behave the same way), and this leads in to issues in control theory, although in the first case there are also interesting issues about travelling waves through the lattice.

Much of the work here uses logistic maps as the basic system, or piecewise linear systems. The Bogdanov map is therefore an interesting system to use as a base map in that it is two-dimensional (which already introduces an extra element of complexity), and has Hopf bifurcations which have not been studied in this context. As a first step

(l) a four dimensional system of coupled Bogdanov maps will be considered.

Having understood some of the basic features of this model synchronization will be considered in larger systems of coupled maps of this type. As well as understanding how the stability boundary of synchronized states changes as a function of dynamic behaviour it would also be possible to investigate whether there are other ways of enhancing synchronization (cf. [ALM])

Finally, the algorithms developed during the proposal can be used to move research forward in the direction of round-off errors after the project is formally concluded. Vivaldi (team 1) has considerable expertise in round-off error in numerical processes, for example [LHV], [VB], [LV] and has given a full analysis of a the discretized rotation map. The numerical data on the Bogdanov map will give further insight on round-off in discretized systems (cf. [VB]) and its relevance will be assessed for progression of the theory. Furthermore, noise induced perturbations of maps, (studied by Team 4) [MBHH], [SBDH] can destroy periodic orbits and the relevance of the periodic orbit data to this approach will also be investigated.

References

[ABS] P. Ashwin, J. Buescu & I. Stewart, (1996) From attractor to chaotic saddle: a tale of
                    transverse instability, Nonlinearity v.9, (1996) 703-738.

[ACLP] see section 3.1.5.2(team1)

[ALM] D.K.Arrowsmith, A.N. Lansbury and R.J. Mondragon, Control of the Arnold Circle Map, Int Jnl of Bifurcation and Chaos, 1996, v.6 (iii), 437-453.

[BS] C. Beck & F. Schogl, The thermodynamics of chaotic systems, CUP (1993), pp284

[B1], [B2], [B3], [B4], [B5], [B6], [B7], [B8], [B9], [B10] (see section 3.1.5.2(team2) )

[B11] R.I. Bogdanov, Symplectic orbital equivalence of vector fields on plane (Elementary Singular Points). In comp. of. sc. works "Mathematics and Modeling", Pushino, 1990, 32-45. In Russian: the English Translation is Mathematics and Modeling", Nova Science Publishers, Inc., 1993, p. 41-61.

[B12] R.I. Bogdanov, Applications of weakly-dissipative theory by Kolmogorov-Arnold-Moser. M.: Preprint of Nuclear Physics Institute MSU. 96-22/489, 138p.

[B13] R.I. Bogdanov, Singular relative integral invariants and adiabatic process in thermodynamics. - Itogi Naiki i Tekhniki: Sovremennye Problemy Mat.:Dinamicheskie systemy - 7, VINITI, t.47 (to appear in Russian. English trans. To appear in J. of Math. Sc.).

[B14] R.I. Bogdanov, Solution Hilbert's problem about finite number of auto-oscillation for nonconservative dynamical systems on the plane. M.: Preprint of Nuclear Physics Institute MSU 98-46/547, pp64.

[BG] see section 3.1.5.2(team1)

[CAV] see section 3.1.5.2(team1)

[DK] G. Dangelmayr & E. Knobloch (1987) The Takens-Bogdanov bifurcationwith O(2) symmetry, Phil. Trans. Roy. Soc. (London) A v.233 243-279.

[GH] J. Guckenheimer & P. Holmes (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer.

[G1], [G2],[G4] see section 3.1.5.2(team4)

[G3] P.A. Glendinning The stability boundary of synchronized states in globally coupled dynamical systems, to appear, Phys. Lett. A (1999).

[GM] P. Glendinning & T. Mullin (1999) Imperfection theory for global bifurcations, in preparation.

[K] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer Verlag, 1994.

[LHV] see section 3.1.5.2(team1)

[MBHH] see section 3.1.5.2(team4)

[SBDH] see section 3.1.5.2(team4)

[Sh] D. Shaeffer (1981) General introduction to steady state bifurcation, (in Dynamical Systems and Turbulence), Warwick (1980} eds. D.A. Rand and L-S. Young, Springer LNM 898.

[VB] F. Vivaldi & D. Bosio, Round-off errors and p-adic numbers, preprint.

[PC] L.M. Pecora & T.L. Carrol (1990) Synchronization in chaotic systems, Phys. Rev. Lett. v. 64, 821-824.

[VB] F.Vivaldi & D. Bosio, Round-off errors and p-adic numbers, preprint.

[LV] J.H.  Lowenstien & F. Vivaldi, Anomalous Transport in a model of Hamiltonian round-off, Nonlinearity, 1998.

3.1.4.2 Deliverables, exploitation and dissemination of results

Year 1:

Algorithms will be described and desseminated in publications.

Data sets with characteristics of periodic orbits will be created.

Listings of new maps with symmetry for analysis by weakly-dissipative KAM theory and parameter regimes.

The generic behaviour of some of these maps under symmetry breaking.

Web information on progress of project.

Initial papers on the techniques and background theory

INTAS annual report will be assembled and produced
 
 

Year 2:

Further analysis of the Bogdanov map data and preliminary investigation of using the algorithms completed in year 1.

Delivery of dynamical invariants for certain parameter points of the Bogdanov map.

Investigation of the relevance of the data to other studies on periodic orbit structure of maps.

Extensions to toroidal attractors.

Relevance of the data to round-off errors in by discretizing vector fields.

Comparison of small noise-induced perturbations with the weak-dissipation for the Bogdanov map.

Involvement of other groups in research project.

INTAS annual report and final report will be assembled and produced.

Detailed academic papers and surveys of the results obtained will be produced. It is very important that the techniques are fully explained to co-workers. One of the secondary aims of the proposal is to involve more research teams in the calculation of invariants of planar diffeomorphisms.

3.1.5 Description of the consortium

3.1.5.1 Research teams

In 3.1.3 the contributions of some of the teams showed the development of the work to the present day. The various teams will make distinct contributions to the overall project.

Team 1. Queen Mary and Westfield College, University of London, UK

Personnel

(CO) (DKA) Prof. D.K. Arrowsmith

(CR) (PAG) Prof. P.A. Glendinning (until October 2000, then moves to team 4)

(CR) (FV) Dr. F.Vivaldi

Primary Tasks

DKA - monitor progress and achievement of objectives and organise the managerial aspects of the project with RIB(team2) and JHC(team 3) and PAG(team 4) at a primary level and team 5 at a secondary level ;

PAG - lead the co-investigate the development of new maps from equivariant versal unfolding;

DKA, PAG - produce overview paper of results for Bogdanov map and discretizations;

FV - consider the dynamical implications of round-off error on the statistics of the periodic behaviour detected;

DKA PAG - oversee dissemination of results in appropriate format and journals throughout programme;

Team 2. Nuclear Physics Institute, Moscow State University, RU.

Personnel

(CR) (RIB) R.I. Bogdanov

(CR) (YuIT) Yu.I. Tarasov

(CR) (IVG) I.V.Gayduchenko

(CR) (VVS) V,V. Sucharevskiy

Primary Tasks

RIB - liaise throughout the project with DKA(CO) and JHC regularly throughout project on managerial matters;

RIB – organize the work of Russian co-investigators, separate the problem and integrate corresponding results and coordinate co-investigations, background for algorithms, calculations and analysis of data;

YuIT – calculate and organize database of files of initial data given program for PC, support standard software packages like Windows98, TEX and another suitable software program;

IVG – prepare procedures for analysis data from files of initial data (base language Pascal), prepared output tables, diagrams, illustration, etc.;

VVS –prepare procedures for analysis of data from initial data (base language C⁺⁺).

Team 3. University of Granada, SPAIN

Personnel

(CR) (JHC) Dr J.H. Cartwright

Primary tasks

JHC - liaise with DKA(CO) and RIB(team 2/5) regularly throughout project on managerial matters;

JHC - interpret essentials of algorithms/numerical work and coordinator for their publication;

JHC - support development of new maps and with parameter investigation and initial suggestions for detailed exploration of orbit space using algorithms;

JHC -produce overview paper of results for Bogdanov map and discretizations;

JHC - liaise with teams 2 and 5 on a programme for adapting algorithms for new maps.

Team 4. University of Manchester, Institute of Science & Technology, UK

Personnel

(CR) (PAG) Prof P. Glendinning, (from October,2000)

(CR) (DB) Prof. D. Broomhead

(CR) (MM) Dr. M.Muldoon

Primary Tasks

PAG - provide liaison to DKA(CO) with UMIST;

PAG - consider problems of coupling/synchronization of Bogdanov maps;

MM,DB - liaise to compare and contrast the loss of periodic structure in the statistical data for the weakly-dissipative case with that arising from noise-induced perturbation.
 
 

Team 5. Moscow Energy Institute(Technical University), RUSSIA

Personnel

(CR) (AIP) A.I. Plis

(CR) (VAR)V.A. Rastorguev

(CR) (TAS)T.A. Salnikova

Primary Tasks

AIP – organize communications with co-investigators with RIB(team 2);

AIP and TAS – prepare procedures for statistical analysis of data;

VAR – prepare and instigate downloaded software for support of needed database (including graph structure of data, graph structure of windows in interactive working on PC); system support of suggested algorithms(e.g. format of data, storing, organizing archive and so on);integrating procedures in software packages;

VAR and TAS – checking and verifying integrability of programs system.

3.1.5.2 Scientific references

Team 1 (Mathematics Research Centre, Queen Mary and Westfield College, UK)

[AP1] D.K.Arrowsmith & C.M.Place, An Introduction to dynamical systems, Cambridge

University Press, 1990, pp423.

[AP2] D.K. Arrowsmith & C.M. Place, Examples of Birkhoff Attractors, Acta Appl. Mathematicae, v.21 (1990), 315-329.

[ACLP] D.K.Arrowsmith, J.H.E. Cartwright, A.N. Lansbury & C.M.Place, The Bogdanov map: bifurcation, mode-locking and chaos in a dissipative system, Int Jnl Bif. and Chaos, 1993, v.3 (iv), 803-842.

[ALM] D.K.Arrowsmith, A.N. Lansbury & R.J. Mondragon, Control of the Arnold

Circle Map, Int Jnl of Bifurcation and Chaos, 1996, v.6 (iii), 437-453.

[GP] P.A. Glendinning & M.R.E. Proctor, Travelling waves with spatially resonant forcing: bifurcations of a modified Landau equation, Int. Jnl Bif. and Chaos v.3, 1993, 1447-1455.

[G1] P.A. Glendinning, Stability, Instability and Chaos: an introduction to the theory of nonlinear differential equations, Cambridge University Press, 1994, pp388.

[G2] P.A. Glendinning, Differential equations with bifocal homoclinic orbits, Intl.Jnl Bif. and Chaos 1997, v.7, 27-37.

[CAV] R. Carretero, D.K. Arrowsmith & F. Vivaldi, Mode-Locking in Coupled Map Lattices, Physica D 103 (1997) 381-403.

[BG] M. Banaji & P. Glendinning (1999) Towards a quasi-periodic mean field theory for globally coupled oscillators, Phys. Lett. A, v.251 297-302.

[LHV] J. Lowenstein, S.Hatjispryros & F. Vivaldi, Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off, Chaos v. 7(1997) 49-66.

Team 2 (Nuclear Physics Institute, Moscow State University)

[B1] R.I. Bogdanov, Bifurcation of the limit cycle of a family of plane vector fields, Trudy Sem. Petrovsk, v.2 1976 23-35. (In Russian: The English translation is Sel.Math.Sov., v.1 (1981),389 - 421).

[B2] R.I. Bogdanov, Factorization of diffeomorphisms over phase portrait of vector fields on the plane, Func. Anal. and its Appl., 1996, t.31, v2, p. 67- 70.

[B3] R.I. Bogdanov, Groups of symmetries for thermodynamics with one degree of freedom, Abstracts of Internatinal Geometrical Colloqium, UNESCO, Moscow, May 10-14, 1993, 6 p.

[B4] R.I. Bogdanov, Integrals of semi-integrable k-tuples of germs of vector fields in the plane, Trudy Sem. Petrovsk., 16,70. In Russian: The English translation is Journal of Math.Sciences, 69, 2, 974 p. (1994).

[B5] R.I. Bogdanov, Local relative integral invariants determined by the phase portrait of a vector field on the plane, Trudy Sem. Petrovsk., 1997, v17, 249. In Russian: The English translation is Journal of Math. Sciences, v75 (3)1773 (1995).

[B6] R.I. Bogdanov, Introduction to weakly-dissipative theory by Kolmogorov-Arnold-Moser, Abstracts of ''Dynamic Systems and Pattern Formation, Leiden, Lorenz Center, 29 September - 1 October, 1997, p. 8-9.

[B7] R.I. Bogdanov, Weakly-dissipative Theory by Kolmogorov-Arnold-Moser, Abstacts of Int. Conf. On Contemp. Problems in Th. Of Dynamic Systems, Nizny Novgorod, 1- 6 July, 1996, p. 14.

[B8] R.I. Bogdanov, & V.A. Rastorguev, Fermi-Pasta-Ulam's problem in weakly dissipative theory by Kolmogorov-Arnold-Moser. Moscow: Preprint Moscow Energy Institute, 12-17, 1997.

[B9] R.I. Bogdanov & V.A. Rastorguev, Principle of indeterminacy in classical mechanics, Pr. 5-th mathematical session of MGSU, (January, 26-31, 1998). Moscow: MSSV ''Sojuz'', 1998 (Russian).

[B10] R.I. Bogdanov, I.V. Gayduchenko, V.A. Rastorguev, Yu. I. Tarasov. Spectro-analysis in weakly-dissipative theory by Kolmogorov-Arnold-Moser, Tr.Seminara ''Time,chaos and mathematical problems}, i.1. Moscow: Knignyidom ''Universitat'', 1999.

Team 3 University of Granada, SPAIN

[CGP1] J. H. E. Cartwright, D. L. Gonzalez, & O. Piro, Nonlinear dynamics of the perceived pitch of complex sounds, Phys. Rev. Lett., 82, 5389-5392, 1999.

[CCGPR] O. Calvo, J. H. E. Cartwright, D. L. Gonzalez, O. Piro, & O. Rosso, Three-frequency resonances in dynamical systems, Int. J. Bifurcation and Chaos, 9, 1999.

[C1] J. H. E. Cartwright, Newton maps: fractals from Newton's method for the circle map,
Computers and Graphics, 23, 607-612, 1999.

[CGP2] J. H. E. Cartwright, D. L. Gonzalez, & O. Piro, Universality in three-frequency resonances, Phys. Rev. E, 59, 2902-2906, 1999.

[CHP] J. H. E. Cartwright, E. Hernandez-Garcia, & O. Piro, Burridge-Knopoff models as elastic excitable media, Phys. Rev. Lett., 79, 527-30, 1997.

[CFP1] J. H. E. Cartwright, M. Feingold, & O. Piro, Chaotic advection in three-dimensional unsteady incompressible laminar flow, J. Fluid Mech., 316, 259-84, 1996.

[CFP2] J. H. E. Cartwright, M. Feingold, & O. Piro, Global diffusion in a realistic three-dimensional time-dependent non-turbulent fluid flow, Phys. Rev. Lett., 75, 3669-72, 1995.

[CFP3] J. H. E. Cartwright, M. Feingold, & O. Piro, Passive scalars and three-dimensional Liouvillian maps, Physica D, 76, 22-33, 1994.

[ACLP] D. K. Arrowsmith, J. H. E. Cartwright, A. N. Lansbury & C. M. Place, The Bogdanov map: bifurcations, mode locking, and chaos in a dissipative system, Int. J. Bifurcation and Chaos, 3, 803-842, 1993.

[CP] J. H. E. Cartwright & O. Piro, The dynamics of Runge-Kutta methods, Int. J. Bifurcation and Chaos, 2, 427-449, 1992.

Team 4 (Department of Mathematics, UMIST, UK)

[MBHH] M.R. Muldoon, D.S. Broomhead, J.P. Huke & R. Hegger, Delay embedding in the presence of dynamical noise, Dyn. Stab. Syst. v.13, (1998) 175-186.

[SBDH] J Stark, D S Broomhead, M E Davies and J P Huke, Takens Embedding Theorems for Forced and Stochastic Systems, Nonl. Anal, Theory, Meth. and Appl. v.30, (1997) 5303-5314.

[GP] P.A. Glendinning & M.R.E. Proctor, Travelling waves with spatially resonant forcing: bifurcations of a modified Landau equation, Int. Jnl Bif. and Chaos v.3, 1993, 1447-1455.

[G1] P.A. Glendinning, Stability, Instability and Chaos: an introduction to the theory of nonlinear differential equations, Cambridge University Press, 1994, pp388.

[G2] P.A. Glendinning, Differential equations with bifocal homoclinic orbits, Intl.Jnl Bif. and Chaos v.7, 27-37.

[G3] P.A. Glendinning The stability boundary of synchronized states in globally coupled dynamical systems, to appear, Phys. Lett. A (1999).

[G4] P.A. Glendinning Transitivity and blowout bifurcations in a class of globally coupled maps, submitted to Phys. Lett. A, (1999).

[BG] M. Banaji and P. Glendinning (1999) Towards a quasi-periodic mean field theory for globally coupled oscillators, Phys. Lett. A, v.251 297-302.

Team 5 (Moscow Energy Institute, Russia)

[PS1] A.I. Plis & N.A. Slivina, "Mathcad: mathematics textbook for economists and engineers. Educational
Textbook.-M.:Finances and statistics,1999,656 p. (in Russian).

[PS2] A.I. Plis and N.A. Slivina, Laboratory practical work in mathematical analysis.-M.: "Vysshaya
skola", 1994 (in Russian).

3.1.6 Management planning

The primary international coordination will arise through close liaison between Bogdanov in Moscow, Cartwright in Spain and Arrowsmith in London. The liaison in Moscow will be led by Bogdanov and by Arrowsmith and Glendinning in the UK. To keep close control of the programme, the main coordination will arise at the following meetings evenly spaced throughout the period of support;

-visit of RIB and JHC to DKA and PAG (as early as financial support is received) – March-May, 2000 (4 days) London.

-visit of JHC to Moscow September 2000 (5 days)

-visit of DKA to Moscow – November 2000 (5 days) to MSU.

- visit of JHC and PAG to Moscow – April 2001 ( 5 days).

-visit of DKA to Moscow - October, 2001 (5 days)

-visit of RIB to London - December, 2001 (5 days)

-visit of JHC to London - December 2001 (5 days)

There will in addition be monthly e-mail statement on progress on meeting of targets and and a assessment of meeting targets/modifications every three months. This will be electronic mail unless there are difficulties in which extra meetings of key personnel will be involved. There will be other liaisons/travel on research as and when appropriate.

The CO is currently Director of the Mathematics Research Centre at QMW, University of London with an annual budget of 70,000 ecus (for non-perosnnel) expenditure of travel/equipment and research support with responsibility for 25 staff.

3.1.6.1 Planning and tasks allocation

T.1. Prepare foundations of necessary algorithms for

(i) calculations of periodic orbits and their invariants (Months 1-5)

(ii) analysis of output data (Months 12-17).

T.2. Write and test program for calculation of periodic orbits (Months 7-10).

T.3. Calculation of files of data for periodic orbits (10-24 months).

T.4. Write and test program for analysis of invariants of periodic orbits

(13-24 months).

T.5. Formulate hypotheses and mathematical theorems for understanding

numerical data and results (whole period)

T.6. Investigate new diffeomorphisms via discretizations and find useful parameter regimes for applications of algorithms (Months 5-10)

T.7 Liaise with Moscow groups on calculations for new maps (Months 13-18)

T.8. Perform independent check of algorithms and data (Months 1-18)

T.9 Collect information from the various teams and produce an outline web page of the project (1-6) months and a survey article of the project past and future (Months 7-12)

T.10. Assess the nature of the periodic orbit data (Months 7-12). Explore comparison of noise-induced perturbations with statistics of weak-dissipation periodic orbits (Months 13-24).

T.11 Explore results on coupled systems involving the Bogdanov map for higher dimensional attractors using the basic "Bogdanov" algorithms.

T.12. Also consider round-off error/noise implications from data (Months 12-24)

Tasktable
 

Title/Particpants

Month

1/2

3/4

5/6

7/8

9/10

11/2

13/4

15/6

17/8

19/20

21/2

23/4

T.1. (i) RIB,VAR

$

$

$

T.1.(ii) RIB,IVG

$

$

$

$

T.2. RIB,VAR

$

$

T.3. YuTT,RIB

$

$

$

$

$

$

$

$

T.4. RIB,IVC

$

$

$

$

$

$

T.5. DKA,RIB,JHG

$

$

$

$

$

$

$

$

$

$

$

$

T.6. DKA,JHC,PAG

$

$

$

T.7. DKA,PAG,RIB,IVC

$

$

$

T.8. JHC,VAR

$

$

$

$

$

$

$

$

$

$

$

$

T.9. DKA,JHC,RIB

$

$

$

$

$

$

T.10. DB,MM

$

$

$

T.11. PAG,DKA,RIB

$

$

$

$

$

$

T.12. FV,VAR,DB,MM

$

$

$

$

$

$

 

3.1.6.2 Cost table
 

MAIN COST TABLE
INTAS MEMBER TEAMS
Team Name	Status	COST CATEGORIES						TOTAL
		Labour Cost	Over-heads	Travel and Subsistence	Consum-ables	Equip- ment	Other costs	Total Euro
Team1 DKA	CO	1000		5000				6000
Team3 JHC	CR	0		3000				3000
Team4 DB	CR	0		3000				3000
Sub Total	Euro	1000		11000				12000

NIS TEAMS
Team Name	Status	COST CATEGORIES						TOTAL
		Labour Cost	Over-heads	Travel and Subsistence	Consum-ables	Equip-ment	Other costs	Total Euro
Team2 RIB	CR	13440	500	8060		3000		25000
Team 5 AIP	CR	14400	1000	4600	1000	2000		23000
Sub Total	Euro	27840	1500	12660	1000	5000		48000

Total

Euro

39840

1500

12660

1000

5000

60000

NIS Labour Cost Summary Table
Team Name	Number of individual grants	Cost/month (Euro)	Number of Month	Total Cost (Euro)
Team2. RIB	4	560	24	13440
Team5. AIP	3	600	24	14400

3.1.7.Summary

Title: The Bogdanov map and weakly-dissipative theory of Kolmogorov-Arnold-Moser

The aim is to investigate the local behaviour of generic planar diffeomorphisms at a fixed point. The maps are close to area preservation and there exist approximating vector fields which help provide invariants for the orbit structure of the diffeomorphisms which in turn characterise their behaviour. The invariants being produced during the period of the proposal will be new and important.

The Bogdanov map was originally created by team1, and being currently investigated by teams 2 and 5 in this INTAS proposal, have the complexity of invariant curves, island chains and cantori. These are the remants of the anharmonic oscillator's orbital behaviour (henon-type area-preserving map) which persist for small perturbations. In these circumstances, weakly-dissipative KAM theory can then be used to investigate periodic structures in and close to the strange attractors. These attractors are the discretised version of the limit cycles of the approximating vector fields.

The intention is to;

(a) develop further data files for the approximating vector fields to the Bogdanov map, a planar diffeomorphism of some topological importance;

(b) produce tables, graphs and interpretation of statistical analysis of data obtained in (a);

(c) find numerical values of invariants for the Bogdanov map;

(d) develop new maps close to area preservation which exhibit further examples;

(e) make available on the Internet (i) algorithms for calculation of invariants and (ii) several examples of data files;

(f) extend the range of maps via symmetry considerations that can be considered by the algorithmic techniques developed;

(g) consider higher dimensional models via synchronisation techniques;

(h) publish on progress and results in papers, reports and conferences and electronic form.

All of the teams listed in this application will play a vital part in the success of this project:

Team 1 QMW,UK- overall coordination, bifurcation, development of new maps and round-off expertise;

Team 2 MSU,RU - Russian sub-coordination and development and application of theorems giving invariance under vector fields of generic planar diffeomorphisms;

Team 3 Granada, SP - strong expertise in numerical development of results for the maps to be considered and important independent check of numerical techniques;

Team 4 UMIST,UK - strength in bifurcation of maps and ordinary differential equations, noise induced perturbation

Team 5 MEI,RU - high technical involvement and computational expertise which is crucial to the success of the project.

An important part of the project is that the various groups come together to explain to a wider audience the relevance of the models and the calculation of the invariants. The work is labour-intensive and this is the only way that substantial investigations can take place in the longer term. A crucial aspect of extending interest in this investigation is the set of publications which will emanate from the group. There will be publications of various types (I) general informative papers on the precise mix of theoretical and algorithmic/numerical results; (II) development of algorithms and calculation of specific invariants for the Bogdanov map generated from numerical information on periodic orbit structure of high periods; (III) development of new maps for and their key behaviour in the near-area preserving case; (IV) relevance of the numerical work to the problem of round-off error in the discretization of continuous maps to noisy data.