chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

brief paper zeros of structured linear systems this paper provides a detailed study of transmission zeros in the context of linear structured systems for such systems the entries of the state space model matrices are supposed to be either fixed zeros or free parameters a directed graph can be associated to such systems and structural properties can be studied i e properties which are valid for almost all values of the parameters we show that there exists two kinds of transmission zeros the parameter dependent ones and the constant ones which can only occur in zero moreover the constant transmission zeros can be directly obtained on the associated graph in terms of particular paths and cycles

sbil nikov hopf bifurcation abstract we derive a 2 parameter poincare map modeling a 3 dimensional vector flow near the codimension 2 point defined by a sil nikov homoclinic orbit to a fixed point undergoing a hopf bifurcation in one parameter regime the map reproduces glendinning and sparrow s results for the sil nikov bifurcation while in another regime it provides new results for a small hopf cycle with a global reinjection mechanism the map is analyzed finding primary homoclinic tangencies to the small hopf cycle parameter dependence of periodic orbits associated with these tangencies is described global behavior of the tube like unstable manifold of the hopf cycle is studied in depth analytically and numerically locating its simplest subsidiary tangencies

the role of anomalous modes in taylor couette flow one of the classic examples of a fluid mechanical bifurcation is the appearance of taylor vortex flow in a viscous fluid contained between concentric rotating cylinders a natural consequence of the mathematical model due to d g schaeffer which has been most successful in describing this phenomenon in a finite system is the presence of anomalous modes in the steady solution set these are cellular flows that exist in contradiction to the commonly held belief that the radial flow close to the ends is necessarily directed towards the inner cylinder because of the presence of an ekman boundary layer to date anomalous modes have only been studied in a restricted parameter range but here we report the results of numerical and experimental studies that cover a wide range of parameter space it is shown that the lower stability limits for anomalous modes are removed to high reynolds numbers when the gap between the cylinders is reduced this perhaps explains why this class of flow remained undiscovered in over fifty years of study of the taylor couette problem until the work of t b benjamin we also consider the implications of these results for the organization of dynamical behaviour in the taylor couette system

spirals vortices in taylor couette flow with rotating endwalls a hopf bifurcation with translational invariance has been widely considered as an appropriate model for the appearance of spiral vortices in counter rotating taylor couette flow our experimental work demonstrates that flow conditions close to the axial boundaries are responsible for the type of bifurcation scenario i e either asymmetric pure traveling waves or more complex behavior such as defect states or symmetric mixed states appearing from a hopf bifurcation the measurements were performed in the first taylor couette experiment with independently rotating endwalls confining the system in axial direction the rotation rate of the synchronous endwalls is found to be an essential control parameter for the spatial amplitude distribution of the traveling waves and also reflects symmetry of the corresponding flow pattern appearing from the hopf bifurcation

heteroclinic cycles in dynamical systems with broken spherical symmetry in mathematics and physics the phrase symmetry breaking has two distinct meanings the first refers to the frequently observed phenomenon that a configuration of a physical system satisfying a law a set of equations which is invariant under a group of transformations may itself only be invariant under a subgroup of this group this is referred to as spontaneous symmetry breaking the second meaning refers to the process of explicitly adding symmetry breaking terms to the equations which describe the system this we call induced or forced symmetry breaking over the past decade spontaneous symmetry breaking has attracted a considerable amount of attention within the context of bifurcation and dynamical systems theory see for example 13 previous work on forced symmetry breaking has mainly concentrated on the persistance of equilibrium solutions under small symmetry breaking perturbations of symmetric systems 3 4 2 7 8 12 22 231 the main aim of this paper is to show that such perturbations may also give rise to more complex dynamical behavior in particular to heteroclinic cycles we first present a simple example of our constructions for the particular choices of groups g so 3 h so 2 and k t described below in order to illustrate these general ideas without too many technicalities we return to a rigorous discussion of these types of problems in section 2

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

perturbation of a hopf bifurcation by an external time periodic forcing contenrs i introduction ii the mathematical model iii dynamics of the poincark map in the non resonant and weakly resonant cases iv dynamics of the poincark map in the p l resonant case v dynamics of the poincart map in the p 2 resonant case vi dynamics of the poincark map in the p 3 resonant case

pattern formation and bistability in flow between counterrotating cylinders abstract in the taykor coutte experiment on fluid flow counterrotating cylinders there is a bicritical point where the onset of instabilities to taylor vortex flow a steady state bifurcation and spiral vortex flow a hopf bifurcation meet the nonlinear mode interactions near this bicritical point are analyzed exploiting the role of symmetry in the bifurcation theory and with emphasis of the relevance to experiments for a range of raduis ratios 0 43 0 98 the mechanism of the pattern formation is elucidated and several new flow patterns and transitions are predicted including wavy vortices bistability hysteresis and up to 7 quasiperiodic flows

global bifurcations in rayleigh be nard convection experiments empirical maps and numerical bifurcation analysis abstract we use nonlinear signal processing techniques based on artificial neural networks to construct an empirical mapping from experimental rayleigh benard convection data in the quasiperiodic regime the data in the form of a one parameter sequence of poincare sections in the interior of a mode locked region resonance horn are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied rayleigh number the dynamic phenomena apparent in the data include period doublings complex intermittent behavior secondary hopf bifurcations and chaotic dynamics we use the fitted map to reconstruct the experimental dynamics and to explore the associated local and global bifurcation structures in phase space using numerical bifurcation techniques we locate the stable and unstable periodic solutions calculate eigenvalues approximate invariant manifolds of saddle type solutions and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings

onset of wavy vortices in taylor couette flow with imperfect reflection symmetry we reveal experimentally a mechanism that alters the critical behavior of a hopf bifurcation substantially due to the presence of imperfections of the reflection symmetry in a hydrodynamic system the onset of rotating waves in taylor vortex flow which is widely considered as a classical example for a hopf bifurcation in hydrodynamics is investigated primarily by transient response experiments while wavy vortex flow is not influenced by such unavoidable experimental imperfections the critical behavior of the axially subharmonic rotating wave with wavy outflow boundaries also called the small jet mode is qualitatively altered experimental evidence is provided that the modified critical behavior at the hopf bifurcation is associated with imperfections of the reflection symmetry of the taylor couette setup the experimental results on hopf bifurcation are discussed in the context of a cusp hopf bifurcation model recently proposed by harlim and langford int j bifurcation chaos appl sci eng 17 2547 2007 and compared to experimental results on imperfect pitchfork bifurcation in small aspect ratio taylor couette flow

chaos from hopf bifurcation in a fluid flow experiment results of an experimental study of a hopf bifurcation with broken translation symmetry that organizes chaotic homoclinic dynamics from a t2 torus in a fluid flow as a direct consequence of physical boundaries are presented it is shown that the central features of the theory of hopf bifurcation in o 2 symmetric systems where the translation symmetry is broken are robust and are appropriate to describe the appearance of modulated waves homoclinic bifurcation takens bogdanov point and chaotic dynamics in a fluid flow experiment

symmetries and dynamics for 2 d navier stokes flow abstract simulations of forced 2 d navier stokes equations are analyzed the forcing is spatially periodic and temporally steady a karhunen loeve analysis is used to identify the structures in phase space that generate the pde behavior their relationship to the invariant subspaces generated by the symmetry group is discussed it is shown that certain modes that are in the stable eigenspace of the kolmogorov flow solution play an essential role for the dynamics of the attractor for the 2 d navier stokes equations below a reynolds number of about 30 in this regime all stable solutions are identified and thier relation to the symmetry structure is elucidated a new type of gluing bifurcation generated by the symmetry is found and analyzed a mechanism for the generation of bursting behavior is suggested

instability and localisation of patterns due to a conserved quantity abstract we describe the influence of a conserved quantity on the stability properties of roll square and hexagonal patterns near stationary onset the appropriate systems of amplitude equations are analysed they comprise equations for the amplitudes of the pattern modes together with an additional coupled equation governing the evolution of a large scale mode associated with the conserved quantity we find that the presence of this large scale mode may result in the destabilisation of all regular roll square or hexagonal patterns leading to amplitude modulation and strong localisation of the pattern our analytical results are complemented throughout by numerical simulations of a model partial differential equation to illustrate the modulational instabilities and their nonlinear development

bifurcations with imperfect so 2 symmetry and pinning of rotating waves rotating waves are periodic solutions in so 2 equivariant dynamical systems their precession frequency changes with parameters and it may change sign passing through zero when this happens the

survey generic properties and control of linear structured systems a survey in this survey paper we consider linear structured systems in state space form where a linear system is structured when each entry of its matrices like a b c and d is either a fixed zero or a free parameter the location of the fixed zeros in these matrices constitutes the structure of the system indeed a lot of man made physical systems which admit a linear model are structured a structured system is representative of a class of linear systems in the usual sense it is of interest to investigate properties of structured systems which are true for almost any value of the free parameters therefore also called generic properties interestingly a lot of classical properties of linear systems can be studied in terms of genericity moreover these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way we review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms by properties we mean here system specific properties like controllability the finite and infinite zero structure and so on as well as solvability issues of certain classical control problems like disturbance rejection input output decoupling and so on in this paper we do not try to be exhaustive but instead by a selection of results we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool we emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well known polynomially bounded combinatorial techniques for larger systems

heteroclinic dynamics in a model of faraday waves in a square container we study periodic orbits associated with heteroclinic bifurcations in a model of the faraday system for containers with square cross section and single frequency forcing these periodic orbits correspond to quasiperiodic surface waves in the physical system the heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable hamiltonian limit some of which persist in the presence of small damping the dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate poincare maps with predictions that agree with numerical computations the results suggest a great richness of possible dynamics of faraday waves even in simple geometries and with single frequency forcing

mean flow suppression by endwalls in oscillatory binary fluid convection abstract competition between traveling and standing waves in binary fluid mixtures with a negative separation ratio s is studied for realistic boundary conditions for typical experimental values of the prandtl and lewis numbers the suppression of net mean flow in traveling waves due to the presence of endwalls is found to have negligible effect 1 part in 10 4 on the coefficients of the nonlinear terms in the amplitude equations the dominant effect of the endwalls enters therefore at linear order the results are used to predict the range of s in which the initial instability is to stable chevron states

the generic number of invariant zeros of a structured linear system in this paper we consider a structured linear system represented by means of a directed graph we present a graph theoretic method to compute the generic number of invariant zeros of the corresponding rosenbrock matrix the method is based on a fundamental decomposition of the directed graph representing the structured system the generic number of invariant zeros is important in generic versions of problems involving controllability and pole placement

on the structure at infinity of a structured system abstract we develop a graph theoretic characterization of the generic structure at infinity of the transfer matrix of a structured system we show that the generic structure at infinity can be determined by means of algorithms from combinatorial optimization based on the max flow min cut theorem and on results concerning minimal cost flows as an application of the obtained characterization we propose a structural version of two well known disturbance decoupling problems and we derive graph theoretic necessary and sufficient conditions for the solvability of each of the two problems

hopf bifurcation and symmetry travelling and standing waves on the circle in this paper we consider hopf bifurcation in the presence of o 2 symmetry the system of reaction diffusion equations u t d u xx f u provided with periodic boundary conditions may serve as a model problem we prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability with asymptotic phase of these periodic solutions giving explicit formulae finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation

spatio temporal structures in phase turbulent convection abstract the problem of phase turbulence for convection in a horizontal layer heated from below with stress free boundaries is considered in the case of low prandtl numbers p when steady rolls become unstable to the monotone or oscillatory skewed varicose instabilities numerical computations are reported for a square horizontal periodicity interval with finite aspect ratio transitions to chaotic solutions occur when exceeds a value of the order 8 various three dimensional steady and time periodic solutions can be described by a reduced set of differential equations in time in particular the evolution of convection flows induced by the monotone skewed varicose instability is followed in detail

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

duality and yang mills fields on quaternionic kahler manifolds the concept of a self dual connection on a four dimensional riemannian manifold is generalized to the 4n dimensional case of any quaternionic kahler manifold the generalized self dual connections are minima of a modified yang mills functional it is shown that our definitions give a correct framework for a mapping theory of quaternionic kahler manifolds the mapping theory is closely related to the construction of yang mills fields on such manifolds some monopole like equations are discussed

lie symmetry analysis and exact solutions of the quasigeostrophic two layer problem the quasigeostrophic two layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics the complete set of point symmetries of the two layer equations is determined an optimal set of one and two dimensional inequivalent subalgebras of the maximal lie invariance algebra is constructed on the basis of these subalgebras we exhaustively carry out group invariant reduction and compute various classes of exact solutions wherever possible reference to the physical meaning of the exact solutions is given in particular the well known baroclinic rossby wave solutions in the two layer model are rediscovered

pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem taylor vortices in axial flow a unique pattern selection in the absolutely unstable regime of a driven nonlinear open flow system is analyzed the spatiotemporal structures of rotationally symmetric vortices that propagate downstream in the annulus of the rotating taylor couette system due to an externally imposed axial through flow are investigated for two different axial boundary conditions at the inlet and outlet detailed quantitative results for the oscillation frequency the axial profile of the wave number and the temporal fourier amplitudes of the propagating vortex patterns obtained by numerical simulations of the navier stokes equations are compared with results of the appropriate ginzburg landau amplitude equation approximation and also with experiments unlike the stationary patterns in systems without through flow the spatiotemporal structures of propagating vortices are independent of parameter history initial conditions and system length they do however depend on the axial boundary conditions in addition to the driving rate of the inner cylinder and the through flow rate our analysis of the amplitude equation shows that the pattern selection can be described by a nonlinear eigenvalue problem with the frequency being the eigenvalue the complex amplitude being the corresponding eigenfunction describes the axial structure of intensity and wave number small but characteristic differences in the structural dynamics between the navier stokes equations and the amplitude equation are mainly due to the different dispersion relations approaching the border between absolute and convective instability the eigenvalue problem becomes effectively linear and the selection mechanism approaches that of linear front propagation

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

complex dynamics in the hopf bifurcation with broken translation symmetry abstract the dynamics of the normal form for the o 2 equivariant hopf bifurcation perturbed by linear translation symmetry breaking terms are investigated a variety of stable small amplitude complex behavior is uncovered even for parameters for which the unperturbed system has no stable small amplitude solutions in some cases this behavior can be attributed to the proximity to global bifurcations of sil nikov and sil nikov hopf type in other cases cascades of gluing bifurcations involving various two frequency states are present since the equations studied can be obtained by center manifold reduction of the coupled complex ginzburg landau ccgl equations in a finite container this behavior must also be present in the ccgl equations the theory may explain the origin of erratic reversals in traveling wave convection and in spiral vortex flow

a hopf steady state mode interaction in rotating convection bursts and heteroclinic cycles in a square periodic domain abstract we investigate the transition between oscillatory and steady convection at onset in low prandtl number rotating convection in a plane layer this transition is dominated by mode interactions which at one point in parameter space can be posed on a square lattice this allows a rigorous reduction to a finite dimensional bifurcation problem we construct the normal form and compute the normal form coefficients for this codimension 2 bifurcation directly from the pdes for boussinesq rotating convection with stress free upper and lower boundaries the dynamics near the codimension 2 point are investigated fully they explain behaviour found in numerical simulations of the pdes at parameter values near the transition boundary in particular the normal form exhibits bursting dynamics created by a heteroclinic cycle containing points at infinity

wrinkled tori and bursts due to resonant temporal forcing the effect of resonant temporal forcing on a system undergoing a hopf bifurcation with d4 square symmetry is studied the forcing breaks the continuous normal form symmetry of the governing amplitude equations but the d4 symmetry is preserved for this system it is shown that bursts with very large dynamic range may occur the bursts are associated with visits near solutions at infinity and are related to those found previously for the hopf bifurcation with broken d4 symmetry moehlis and knobloch physica d 135 2000 263 the regime in which an attracting quasiperiodic solution that exists in the absence of forcing wrinkles into chaos as the amplitude of the forcing increases is investigated in detail the overall behavior is governed by the approach of the resulting attractor to solutions at infinity windows with stable periodic solutions are found and associated with the traversal in parameter space through arnol d tongues other aspects of the dynamics are related to the presence of a new type of gluing bifurcation which we call a supergluing bifurcation 2001 elsevier science b v all rights reserved

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

an efficient spectral projection method for the navier stokes equations in cylindrical geometries an efficient and accurate numerical scheme is presented for the three dimensional navier stokes equations in primitive variables in a cylinder the scheme is based on a spectral galerkin approximation for the space variables and a second order projection scheme for time the new spectral projection scheme is implemented to simulate unsteady incompressible flows in a cylinder

bifurcations in a model of magnetoconvection abstract bifurcations from oscillatory solutions are studied in a truncated model of two dimensional boussinesq magnetoconvection the fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period doubling cascade after the accumulation point there is a semiperiodic cascade leading to chaotic behaviour then the semiperiodic cascade is repeated in reverse followed by a period halving cascade and a bifurcation back to symmetry finally the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle foci the interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions each with its own cascade this pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle foci with eigenvalues that satisfy shil nikov s inequality

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

oscillatory bifurcation with broken translation symmetry the effect of distant endwalls on the bifurcation to traveling waves is considered previous approaches have treated the problem by assuming that it is a weak perturbation of the translation invariant problem when the problem is formulated instead in a finite box of length l and the limit l ensuremath rightarrow ensuremath infty is taken one obtains amplitude equations that differ from the usual ginzburg landau description by the presence of an additional nonlinear term this formulation leads to a description in terms of the amplitudes of the primary box modes which are odd and even parity standing waves for large l the equations that result take the form of a hopf bifurcation with approximate d 4 symmetry these equations are able to describe qualitatively not only traveling and blinking states but also asymmetrical blinking states and repeated transients all of which have been observed in binary fluid convection experiments

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

disordered fluid motion in a small closed system abstract we present the results of an experimental study of the development of disordered motion in a small scale fluid mechanical system as the reynolds number is increased the experimental arrangement is a variant of the taylor coutte problem with two inner rotating cylinders and thus the flow is triply connected two different dynamical attractors are investigated and it is shown that while one exhibits features of low dimensional chaos the other appears to lack any obvious structure this finding appears to contradict the currently held view that small aspect ratio fluid mechanical systems will always exhibit low dimensional routes to chaos

pattern selection in three dimensional magnetoconvection abstract weakly nonlinear boussinesq convection in an imposed vertical magnetic field is studied stress free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical both steady and oscillatory convection are considered among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters these include in particular a pattern called alternating rolls the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found in certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence the analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field

holonomy groups and w symmetries irreducible sigma models i e those for which the partition function does not factorise are defined on riemannian spaces with irreducible holonomy groups these special geometries are characterised by the existence of covariantly constant forms which in turn give rise to symmetries of the supersymmetric sigma model actions the poisson bracket algebra of the corresponding currents is aw algebra extended supersymmetries arise as special cases

turbulent transport and the random occurrence of coherent events abstract we suggest that the transport properties and dissipation rates of a wide class of turbulent flows are determined by the random occurrence of coherent events which correspond to certain orbits which we call homoclinic excursions in the high dimensional strange attractor homoclinic excursions are trajectories in the noncompact phase space that are attracted to special orbits which connect saddle points in the finite region of phase space to infinity and represent organized structures in the flow field this picture also suggests that one can compute fluxes using a relatively low dimensional description of the flow a method for extracting the organized structures from a time series is given and provides a local analogue of the notion of lyapunov exponents

arnold tongues in a billiard problem in nonlinear and nonequilibrium systems abstract we study a billiard problem in nonlinear and nonequilibrium systems this is motivated by the motions of a traveling spot in a reaction diffusion system rds in a rectangular domain we consider a four dimensional dynamical system defined by ordinary differential equations this was first derived by s i ei et al 2006 based on a reduced system on the center manifold in a neighborhood of a pitchfork bifurcation of a stationary spot for the rds in contrast to the classical billiard problem this defines a dynamical system that is dissipative rather than conservative and has an attractor according to previous numerical studies the attractor of the system changes depending on parameters such as the aspect ratio of the domain it may be periodic quasi periodic or chaotic in this paper we elucidate that it results from parameters crossing arnold tongues and that the organizing center is a hopf hopf bifurcation of the trivial equilibrium

classification of robust heteroclinic cycles for vector fields in with symmetry we consider a classification of robust heteroclinic cycles in the positive octant of under the action of the symmetry group we introduce a coding system to represent different classes up to a topological equivalence and produce a characterization of all types of robust heteroclinic cycle that can arise in this situation these cycles may or may not contain the origin within the cycle we proceed to find a connection between our problem and meandric numbers we find a direct correlation between the number of classes of robust heteroclinic cycle that do not include the origin and the mercedes benz sequence of integers characterizing meanders through a y shaped configuration we investigate upper and lower bounds for the number of classes possible for robust cycles between n equilibria one of which may be the origin

a geometric approach for structured systems application to disturbance decoupling structured systems described by state space models are considered for these systems the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters for such systems one can study structural properties i e properties that are valid for almost any value of the parameters we revisit the classical geometric theory in the context of structured systems in particular the well known notions of a b invariant and c a invariant subspaces are redefined and analysed in this context we characterize fixed subspaces included or containing these invariant subspaces and show that they play a key role in solving control problems as an application of this geometric approach for structured systems we propose new solutions to the disturbance decoupling problems by state feedback and by output measurement feedback the solvability conditions are easily checked on the associated graph of the system

anomalous modes in the taylor experiment an experimental investigation and a theoretical appraisal are presented which enlarge upon previous observations of multiple steady flows in a taylor apparatus where the fluid filled annulus is comparatively short the aim of the experiments was a systematic exploration of all anomalous modes in this apparatus namely those stable flows that exist only at sufficiently high reynolds number r and are always distinct from the primary flow developed by gradually increasing r from small values having been established at high r an anomalous mode will always collapse eventually if r is then gradually reduced the theoretical material bearing on the interpretation of the experiments is reviewed in s 2 where mathematical details are largely deferred but several new conclusions are demonstrated in particular some curious facts are revealed about the limits of stability for anomalous modes comprising even numbers of taylor cells and about the properties of flows comprising a single cell only following a description of the apparatus in s 3 which includes an account of several improvements that have been introduced since the earlier experiments the new experimental results are reported in s 4 they include stability curves critical r against aspect ratio of annulus for anomalous modes consisting of from two to seven taylor cells s 4 1 observations on collapse routes for all these modes s 4 3 and measurements of various phenomena exhibited by the single cell mode s 4 5 these and other phenomena recorded here for the first time are seen to accord with the theoretical prospectus

homoclinic chaos in chemical systems abstract we first focus on complex dynamical phenomena generated by chemical kinetics in homogeneous media we comment on the alternating sequences of periodic and chaotic states observed in some experiments on the belousov zhabotinsky reaction when conducted in a continuously stirred tank reactor we present numerical results obtained with a seven variable oregonator model which reproduce most of the features of the experimental sequences we discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity we construct poincare map models which reduce to either multi humped or multi branched 1d maps in the strong area contraction limit we report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of spiral type strange attractors satistifies the symbolic dynamics predicted by sil nikov theory of homoclinic chaos we then investigate spatio temporal pattern forming phenomena in a one dimensional reaction diffusion system with equal diffusion coefficients when imposing a concentration gradient through the system this model mimics the sustained stationary and periodically oscillating front structures observed in recent experiments conducted in an open couette flow reactor we emphasize the possibility that the oscillations of the spatial structure become chaotic we report on numerical simulations of a diffusion induced intermittent bursting phenomenon that is likely to be observed in bench experiments we elaborate on the interpretation of this intermittent occurrence of spatially localized structures in an extended system in terms of sil nikov s homoclinic chaos

imperfect hopf bifurcation in spiral poiseuille flow we present the results of an experimental study on the transition to spiral vortices in flow between concentric counter rotating cylinders in the presence of an axial through flow i e in spiral poiseuille flow the experiments were performed in an apparatus having an aspect ratio ensuremath gamma l d 22 8 l axial length d gap width between cylinders and end plates enabling an in and outflow of mass as a result of an applied axial through flow the classical hopf bifurcation to spiral vortices spi splits up and a primary and secondary branch of down and upstream propagating spi respectively as well as a transient quasiperiodic flow appear downstream propagating spi resulting from the primary supercritical hopf bifurcation are either convectively or absolutely unstable the bifurcation structure observed in this open flow experiment is in qualitative agreement with predictions from theory of hopf bifurcation with broken reflection symmetry j d crawford and e knobloch nonlinearity 1 617 1988 and also in quantitative agreement with results from recent numerical calculations a pinter m l ucke and c hoffmann phys rev e 67 026318 2003 c hoffmann m l ucke and a pinter phys rev e 69 056309 2004

geometry of hyper kahler connections with torsion the internal space of a n 4 supersymmetric model with wess zumino term has a connection with totally skew symmetric torsion and holonomy in sp n we study the mathematical background of this type of connection in particular we relate it to classical hermitian geometry construct homogeneous as well as inhomogeneous examples characterize it in terms of holomorphic data develop its potential theory and reduction theory

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

spatial and temporal resonances in a periodically forced hydrodynamic system abstract the occurrence of a naimark sacker bifurcation in a periodically forced spatially extended system is analyzed with floquet theory the basic periodic state of the flow in an annular region consists of the superposition of circular couette flow driven by the constant rotation of the inner cylinder and annular stokes flow driven by the harmonic oscillation of the inner cylinder in the axial direction windows in parameter space where onset of instability is via naimark sacker correspond to forcing frequencies close to the natural frequencies of the unforced system when the forcing amplitude is sufficiently large in these windows not only have strong temporal resonances been identified but also competition and resonances between various spatial modes that are simultaneously excited take place an excitation diagram mapping out these windows and loci of resonance points is produced providing a guide for future explorations into the nonlinear regime by either experimental or computational techniques

imperfect homoclinic bifurcations experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations these results are discussed in the light of a theory of imperfect global bifurcations it is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system

bursts in oscillatory systems with broken d 4 symmetry a new mechanism responsible for generating regular and irregular bursts of large dynamic range near onset of an oscillatory instability is identified the bursts are present in systems with nearly square symmetry and are the result of heteroclinic cycles involving infinite amplitude states created when the square symmetry is broken all possible cycles of this type are identified and the resulting bursts described global connections involving finite amplitude states are also present the intricate sequence of bifurcations that result is described in several cases 2000 elsevier science b v all rights reserved

comparison of algorithms calculating optimal embedding parameters for delay time coordinates the success of noise reduction processes and forecasting methods as well as accurate calculations of fractal dimensions and lyapunov exponents from scalar time series depends strongly on the quality of the reconstructed phase space we present a comparison of four methods which simultaneously estimate the two embedding parameters delay time and sufficiently large embedding dimension dime for takens delay time coordinates recently we introduced a global geometrical measure the fill factor and a local dynamical measure the averaged integral local deformation we will discuss these methods briefly and compare them to the topological wavering product in addition a simple algorithm is presented which observes the spreading of trajectories at the transition from dime to dime 1 and gives the opportunity to estimate the correlation entropy k2 we applied these algorithms to experimental data series measured from a rotational taylor couette experiment and verified the quality of the obtained embedding parameters by the estimation of correlation dimensions for different reconstructed attractors

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

heteroclinic cycles involving periodic solutions in mode interactions with o 2 symmetry in this paper we show that in o 2 symmetric systems structurally stable asymptoticallystable heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions these cycles are found in the third order truncated normal forms of specific codimension two steady state hopf and hopf hopf mode interactions we find these cycles using group theoretic techniques in particular we look for certainpatterns in the lattice of isotropy subgroups once the pattern has been identified the heteroclinic cycle can be constructed by decomposing the vector field on fixed point subspaces into phase amplitude equations it is here that we use the assumption of normal form the final proof of existence and stability relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied

a periodically forced flow displaying symmetry breaking via a three tori tori gluing birurcation and two tori resonances the dynamics due to a periodic forcing harmonic axial oscillations in a taylor couette apparatus of finite length is examined numerically in an axisymmetric subspace the forcing delays the onset of centrifugal instability and introduces a z2 symmetry that involves both space and time this paper examines the influence of this symmetry on the subsequent bifurcations and route to chaos in a one dimensional path through parameter space as the centrifugal instability is enhanced we have observed a well known route to chaos via frequency locking and torus break up on a two tori branch once the z2 symmetry has been broken however this branch is not connected in a simple manner to the z2 invariant primary branch an intermediate branch of three tori solutions exhibiting heteroclinic and homoclinic bifurcations provides the connection on this three tori branch a new gluing bifurcation of three tori is seen to play a central role in the symmetry breaking process 2001 elsevier science b v all rights reserved

one dimensional spatial structure near a hopf bifurcation at finite wavenumber abstract the theory of the spatial structure in a finite geometry in the weakly non linear regime of an instability to a traveling wave state with x x symmetry is described very close to the threshold a direct expansion in the unstable modes yields a quantitative description of the structure in terms of the reflection coefficient of the ends the calculation of the reflection coefficient and its dependence on fluid and end wall properties is illustrated using the model of binary fluid convection between free slip boundaries further away from threshold the phase diagram of the spatial structure involving four distinct phases is calculated using the amplitude equation approach the validity of this method is judged by comparison with the free slip convection model the transition between a state where the disturbance fills the system and one where only a portion is filled filling to confined is analyzed in detail our theory is compared with experiments on the oscillatory instability in pure fluid convection

temporal forcing of small amplitude waves in anisotropic systems we investigate the effect of resonant temporal forcing on an anisotropic system that exhibits a hopf bifurcation to obliquely traveling waves in the absence of this forcing we find that the forcing can excite various phase locked standing wave structures rolls rectangles and cross rolls at onset at most one of the two char22 rolls or rectangles char22 is stable the cross rolls can arise in a secondary bifurcation and can be stable experimentally they would appear as a periodic switching between a structure in which the zig component dominates and one with a dominating zag structure since there are two symmetry related states of this kind one may expect disordered structures to arise due to the breakup of the pattern into domains the results are consistent with recent experiments on electroconvection in nematic liquid crystals by de la torre ju arez and rehberg phys rev a 42 2096 1990 we also apply the general analysis to a model of the behavior near a lifshitz point where the angle of obliqueness vanishes this analysis indicates that phase locked standing rectangles are always unstable in this parameter regime

lie symmetries and exact solutions of the barotropic vorticity equation lie group methods are used for the study of various issues related to symmetries and exact solutions of the barotropic vorticity equation the lie symmetries of the barotropic vorticity equations on the f and planes as well as on the sphere in rotating and rest reference frames are determined a symmetry background for reducing the rotating reference frame to the rest frame is presented the one and two dimensional inequivalent subalgebras of the lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations this provides large classes of exact solutions which include both rossby and rossby haurwitz waves as special cases we also discuss the possibility of partial invariance for the plane equation thereby further extending the family of its exact solutions this is done in a more systematic and complete way than previously available in literature

reduction of hkt structures kt geometry is the geometry of a hermitian connection whose torsion is a 3 form hkt geometry is the geometry of a hyper hermitian connection whose torsion is a 3 form we identify nontrivial conditions for a reduction theory for these types of geometry

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

standing waves in flow between finite counterrotating cylinders experimental evidence for standing waves resulting from a supercritical hopf bifurcation that appears as the first pattern forming instability in counterrotating taylor couette flow is presented depending on the aspect ratio two different types of standing waves denoted as sw0 and swp could be observed both modes have an azimuthal wave number m51 but differ in symmetry while for swp a spatiotemporal glide reflection symmetry could be found sw0 is purely spatial reflection symmetric the transition between the two modes is found to be organized in a cusp bifurcation unfolded by variations of the aspect ratio the classical spiral vortex flow appears in this control parameter regime only as a result of a secondary steady bifurcation from sw0 this transition is found to be either subcritical or supercritical the experimentally observed bifurcation structure has been predicted by theory of hopf bifurcation to spiral vortex flow in finite counterrotating taylor couette systems

disturbance rejection for structured systems the disturbance rejection problem for structured systems is solved a graph is associated with such a structured system disturbance rejection is shown to be solvable by state feedback whenever the control inputs reach the outputs more quickly than the disturbances in the associated graph the infinite structure of a structured system is characterized in terms of input output paths in the associated graph

symmetry preserving parameterization schemes methods for the design of physical parameterization schemes that possess certain invariance properties are discussed these methods are based on different techniques of group classification and provide means to determine expressions for unclosed terms arising in the course of averaging of nonlinear differential equations the demand that the averaged equation is invariant with respect to a subalgebra of the maximal lie invariance algebra of the unaveraged equation leads to a problem of inverse group classification which is solved by the description of differential invariants of the selected subalgebra given no prescribed symmetry group the direct group classification problem is relevant within this framework the algebraic method or direct integration of determining equations for lie symmetries can be applied for cumbersome parameterizations a preliminary group classification can be carried out the methods presented are exemplified by parameterizing the eddy vorticity flux in the averaged vorticity

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a perturbed hopf bifurcation with reflection symmetry we study the effects of a small symmetry breaking perturbation on a system of differential equations at a coupled hopf bifurcation with o 2 symmetry where the perturbation breaks the continuous rotation symmetry but retains a reflection z 2 symmetry it is shown that for a large range of parameter values the invariant manifolds of the unperturbed bifurcation persist and that for some values of normal form coefficients there are secondary bifurcations of nonsymmetric periodic standing wave solutions

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number

an efficient spectral projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases abstract an efficient and accurate numerical scheme is presented for the axisymmetric navier stokes equations in primitive variables in a cylinder the scheme is based on a new spectral galerkin approximation for the space variables and a second order projection scheme for the time variable the new spectral projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition a sensible comparison is made with a standard second order in time and space finite difference scheme based on a stream function vorticity formulation and with available experimental data the numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition the spectral projection scheme is still more accurate in terms of a fixed number of unknowns and more efficient in terms of cpu time required for resolving the flow at a fixed reynolds number to within a prescribed accuracy than the finite difference scheme more importantly the spectral projection scheme can be readily extended to three dimensional nonaxisymmetric cases