differential embedding of the lorenz attractor ideally an embedding of an n dimensional dynamical system is n dimensional ideally an embedding of a dynamical system with symmetry is symmetric ideally the symmetry of the embedding is the same as the symmetry of the original system this ideal often cannot be achieved differential embeddings of the lorenz system which possesses a twofold rotation symmetry are not ideal while the differential embedding technique happens to yield an embedding of the lorenz attractor in three dimensions it does not yield an embedding of the entire flow an embedding of the flow requires at least four dimensions the four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in r4 this inversion symmetric three manifold cannot be projected into any three dimensional euclidean subspace without singularities

effects of the computational time step on numerical solutions of turbulent flow effects of large computational time steps on the computed turbulence were investigated using a fully implicit method in turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined turbulence fluctuations could not be sustained if the computational time step was near or larger than the kolmogorov time scale

a robust method for generating inflow conditions for direct simulations of spatially developing turbulent boundary layers a method for generating inflow conditions for direct numerical simulations dns of spatially developing turbulent boundary layers is presented the method is a modification of that of lund et al j comput phys 140 1998 233 the approach of lund et al is based on having an auxiliary simulation code a in a three dimensional domain similar to that of the main simulation code b the instantaneous velocity field on a selected plane in code a is used as the instantaneous inflow conditions for code b the inflow conditions for code a are generated through a sequence of operations in which the velocity field at a downstream station is rescaled and re introduced at the inlet plane our present method modifies the operations in code a by introducing a set of additional steps preceding the rescaling process this set involves imposing at the inlet plane an appropriate spectrum e k for the turbulence kinetic energy tke and a condition for insuring that the statistical correlation u 1 u 3 between the streamwise and vertical velocity fluctuations retain a non vanishing magnitude this modification is essential for sustaining the production rate of tke near the wall throughout the domain our dns results obtained with the new modification are in excellent agreement with the experimental data of degraaff and eaton j fluid mech 422 2000 319 for re 0 1430

three dimensional simulations of compressible mixing layers visualizations and statistical analysis the effects of compressibility on the planar mixing layer are investigated by means of visualization and statistical analysis of the flowfield quantities computed using three dimensional temporally evolving inviscid simulations the levels of compressibility studied range from relative mach numbers of mr 0 2 2 4 the objectives of this research are to identify large scale structures present in the mixing layer at different levels of compressibility as well as to examine the statistical description of the flowfield in order to gain understanding of the physical entrainment and mixing processes three dimensional simulation visualizations of both the passive scalar and pressure fields show the nature of the large scale structure present in the planar mixing layer to change from nearly two dimensional and spanwise at low compressibility to highly three dimensional and oblique at increased compressibility statistical analysis of the flowfield quantities shows that the shear layer width reynolds shear stress and transverse turbulence intensity decrease with increasing compressibility levels whereas the streamwise turbulence intensity remains nearly constant and the spanwise turbulence intensity increases these statistics support the increasingly three dimensional nature of the large scale motion of the mixing layer with increasing mr

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

hybrid reynolds averaged navier stokes large eddy simulations of supersonic turbulent mixing aturbulentmixinglayerformedfromtwosupersonicstreamsinitially separatedbyasplitterplateisinvestigated withahybridcomputationalmethodusingareynolds averagednavier stokes rans approachforwall bounded regions and a large eddy simulation les approach for the turbulent mixing region simulations of the mixing layer predict a vortex shedding originating from the splitter base and then a rapid transition to turbulence which is in agreement with experimental observations as a result the potential of the hybrid method is demonstrated for e ows in which a geometric feature such as the blunt splitter plate considered here provides the dominant unsteady feature leading to turbulence without imposing additional ine ow disturbances parametric studies of les subgrid modeling settings variations in the spanwise computational domain and wall temperature settings in the rans region were conducted the subgrid model cases using a baseline computational grid with small spanwise computational domain overpredicted the mixing layer turbulencelevels but only showed small variation in the mixing layer predictions with large changes in the model parameters the cases examining wider spanwise domains enabled more turbulent energy to be released in the spanwise direction which in turn reduced axial and vertical turbulence levels finally prescribing the wall temperatures in the rans regions instead of using the more traditional adiabatic wall boundary conditions further reduced turbulence levels and enabled reasonable agreement with experimental data

covering dynamical systems twofold covers we study the relation between a dynamical system which is unchanged equivariant under a discrete symmetry group g and another locally identical dynamical system with no residual symmetry we also study the converse mapping lifting a dynamical system without symmetry to a multiple cover which is equivariant under g this is done in r 3 for the two element rotation and inversion groups comparisons are done for the equations of motion the strange attractors that they generate and the branched manifolds that classify these strange attractors a dynamical system can have many inequivalent multiple covers all equivariant under the same symmetry group g these are distinguished by the value of a certain topological index many examples are presented a new global bifurcation the peeling bifurcation is described

experimental study of compressible turbulent mixing layers compressible turbulent mixing layers have been investigated experimentally using pressure mesurements schlieren photographs and velocity measurements with a two component laser doppler velocimeter system seven mixing layer cases were examined with relative mach numbers ranging from 0 40 to 1 97 which spans the region of significant compressibility effects both the spatial development and similarty of the mixing layers were considered

comparison of tests for embeddings it is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test the classical tests which depend on real numbers fractal dimensions lyapunov exponents averaged over an attractor are compared with a topological test that depends on integers the comparison can only be done for mappings into three dimensions we find that the classical tests fail to predict when a mapping is an embedding and when it is not we point out the reasons for this failure which are not restricted to three dimensions

validation case study prediction of compressible turbulent mixing layer growth rate a recently developed validation metric is applied to the example problem of growth rate prediction for a turbulent compressible mixing layer emphasis is placed on the validation exercise as a well defined process which must include quantification of numerical error evaluation of the level of confidence in the experimental data and estimation of model accuracy available data for compressible mixing layers are reexamined resulting in a new recommended data set for the study of mixing layer growth rate numerical solutions to the favre averaged navier stokes equations with various two equation turbulence closures and compressibility models are obtained and verified through formal solution verification procedures the validation metric results are presented for each of the models including local error estimates for detailed evaluation of model performance and global metrics that facilitate model performance comparisons

a unified rans les model large eddy simulation les is computationally extremely expensive for the investigation of wall bounded turbulent flows at high reynolds numbers a way to reduce the computational cost of les by orders of magnitude is to combine les equations with reynolds averaged navier stokes rans equations used in the near wall region a large variety of such hybrid rans les methods are currently in use such that there is the question of which hybrid rans les method represents the optimal approach the properties of an optimal hybrid rans les model are formulated here by taking reference to fundamental properties of fluid flow equations it is shown that unified rans les models derived from an underlying stochastic turbulence model have the properties of optimal hybrid rans les models the rest of the paper is organized in two parts first a priori and a posteriori analyses of channel flow data are used to find the optimal computational formulation of the theoretically derived unified rans les model and to show that this computational model which is referred to as linear unified model lum does also have all the properties of an optimal hybrid rans les model second a posteriori analyses of channel flow data are used to study the accuracy and cost features of the lum the following conclusions are obtained i compared to rans which require evidence for their predictions the lum has the significant advantage that the quality of predictions is relatively independent of the rans model applied ii compared to les the significant advantage of the lum is a cost reduction of high reynolds number simulations by a factor of 0 07 re 0 46 for coarse grids the lum has a significant accuracy advantage over corresponding les iii compared to other usually applied hybrid rans les models it is shown that the lum provides significantly improved predictions

lower upper symmetric gauss seidel method for the euler and navier stokes equations on developpe un schema de relaxation multigrille application a des ecoulements transsoniques

modeling dissipation equation in supersonic turbulent mixing layers with high density gradients in this work the ability of a ke model to compute mixing layers with significant density gradients is discussed it is shown that as in boundary layers deficiencies appear for compressible flows even if the ke model is improved by compressibility corrections a modification of the sige coefficient is proposed to compute efficiently compressible mixing layers with high density variations this new formulation leads to a separate modelling of the subsonic and supersonic effects the results are presented on mixing layer computations with density ratios between 1 7 velocity density co gradients and 7 velocity density cross gradients and the convective mach number evolves from 0 3 to 1 they show a clear improvement of the capability of the ke model to accurately compute high speed mixing layers

a class of representations of affine kac moody algebras in one class of representations of simple affine algebras there is a highest weight lambda while all other dominant weights are of the form lambda q delta where q is a positive integer and delta is a vector called the null vector these representations are enumerated and a simple recursive formula is given for their weight multiplicities the derivation of the formula illustrates some important features of affine algebras

k zeta enstrophy compressible turbulence model for mixing layers and wall bounded flows we extend the model of robinson et al to compressible turbulent flows the success of the in reproducing a variety of incompressible flows should provide a good basis for the current model as in the incompressible model the current model will be based on the exact compressible equations to ensure that the correct physics is incorporated

orbit depths of affine kac moody algebras since the number of weights in a highest weight representation of an affine algebra is infinite it is particularly useful for these algebras to classify weight systems in terms of weyl orbits it is shown that even though depths in affine orbits are unbounded any weight depth may be determined in a number of steps that is the same for all weights the concepts of root layers and families and partner roots are introduced and used in the procedure a projection of weight vectors onto a euclidean subspace is used to present a simple geometrical picture of orbits

a study of inviscid supersonic mixing layers using a second order total variational diminishing scheme direct numerical simulations are performed to investigate several aspects of ideal gas inviscid two dimensional temporally growing high speed mixing layers stability dynamics and interaction between shocks and vortical structures agreement with experimental results indicates that the growth rate of a mixing layer decreases as the mach number increases eddy shocklets are captured and found to rotate in the opposite direction to the rotation of the local large structure

the compressible hybrid rans les formulation using an additive operator in this work the compressible governing equations for the hybrid reynolds averaged large eddy simulations are formally derived by applying a hybrid filter to the navier stokes equations this filter is constructed by linearly combining the reynolds average rans and large eddy simulation les operators with a continuous blending function the derived hybrid equations include additional terms that represent the interactions between rans and les formulations the relevance of these terms is investigated in flat plate turbulent boundary layer simulations and indicate that these additional terms play a fundamental role in compensating for the turbulence that is neither modeled nor resolved in the transition region between rans and les results also show that when the additional terms are included the calculations are not very sensitive to the blending function implemented in the hybrid filter in the contrary when these terms are neglected and a step function is implemented in the hybrid filter nonphysical discontinuities are predicted in the flow statistics

natural labelling schemes for simple roots and irreducible representations of exceptional lie algebras the simple root system of each exceptional simple lie algebra is explicitly constructed in a variety of forms each construction is based on the natural embedding in the exceptional algebra of a classical semi simple algebra of the same rank the procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented dynkin diagrams the various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras for each such labelling scheme the modification rules for dealing with non standard representation labels are tabulated the connection between the natural labels and dynkin labels is given in detail and a comparison is made with the labels of b g wybourne and m j bowick 1977

computations of flow past a circular cylinder using a continuous turbulence model modeling is calculated with reynolds averaged navier stokes equations using a function that is uniquely determined by the ratio of the resolved small scale length to the integral length scale the unified treatment of reynolds averaged navier stokes large eddy simulation could be achieved with this continuous model by controlling the coarseness of the mesh the continuous reynolds averaged navier stokes large eddy simulation model is initially tested with flow past a circular cylinder at the reynolds number re 3900 both constant integral length scales and varied integral length scales based on the mixing length are used in the simulations the mesh dependence is also studied with both coarse and fine meshes the comparisons are made and analyzed between the results with different integral length scales and different mesh sizes these results are also compared with the experimental results and reynolds averaged navier stokes results the results demonstrate that with the continuous modeling the large eddy simulation like simulation can be achieved by solving reynolds averaged navier stokes equations alone the spatially varied integral length scale is necessary to capture the more accurate turbulence quantities for anisotropic wall bounded turbulent flow

a compressible navier stokes flow solver with scalar transport this paper concerns the extension of the gas kinetic scheme for the compressible navier stokes equations to the flow calculation with interfaces and mixing the objective of the current research targets mainly on the accurate capturing of navier stokes diffusive interfaces where the thickness can be resolved by the cell size firstly a new bgk ns scheme coupling with the level set type scalar function transport is constructed even though the scalar function is directly incorporated into the gas distribution function and it evolves according to the gas kinetic equation it likes more or less a color function which has no direct contribution for the time evolution of conservative flow variables such as mass momentum and energy due to the coupling of the scalar function into the gas kinetic formulation the governing equations for the scalar function turns out to be an advection diffusion equations and the diffusive coefficient can be controlled by the particle collision time which makes the current scheme suitable for the gas mixing problems with a controllable diffusion coefficients however for the non mixing or sharp interface problems such as the interface between gas and liquid the current method can be used as a scheme similar to the level set method where the interface location can be identified with a fixed level set value such as q 0 the current method is applied to a few examples from the simple square wave propagation and diffusion to the 3d rayleigh taylor instability the supersonic mixing layer and the shock helium bubble interaction case show clearly the convergence of the current navier stokes solver for the flow problems with mixing of components and interface once the interface thickness can be well resolved by the cell size in the case of shock hitting sf6 cylinder the computation predicts the experimental measure very well in the current scheme the schmidt number can be freely chosen according to the physical reality

the problem of spurious lyapunov exponents in time series analysis and its solution by covariant lyapunov vectors we briefly recall the methods for the determination of lyapunov exponents from time series data and their limitations one particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system leading to extra so called spurious lyapunov exponents several methods to identify the true ones have been proposed which do not give satisfactory results we show that the geometric information contained in covariant lyapunov vectors can be used to identify the true exponents we illustrate its use and its limitations by applying it to experimental nmr laser data this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

embeddings of low dimensional strange attractors topological invariants and degrees of freedom when a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding dependent we show that there are just three topological properties that depend on the embedding parity global torsion and knot type we discuss how they can change with the embedding finally we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings these results apply only to chaotic attractors of genus one which covers the majority of cases in which experimental data have been subjected to topological analysis this means that the conclusions drawn from previous analyses for example that the mechanism generating chaotic behavior is a smale horseshoe mechanism a reverse horseshoe a gateau roul e an s template branched manifold etc are not artifacts of the embedding chosen for the analysis

analysis and prediction of thin airfoil stall phenomena with hybrid turbulence methodology analysis and prediction of thin airfoil stall phenomena of a naca 64a006 airfoil are numerically investigated using large eddy simulation les reynolds averaged navier stokes rans hybrid methodology with a high order compact differencing scheme subsonic flow of m 0 17 with the high reynolds number of re 5 8 x 10 6 is considered and the angle of attack is varied from 4 0 to 11 0 deg

large eddy simulation inflow conditions for coupling with reynolds averaged flow solvers hybrid approaches using a combination of reynolds averaged navier stokes rans approaches and large eddy simulations les have become increasingly popular one way to construct a hybrid approach is to apply separate flow solvers to components of a complex system and to exchange information at the interfaces of the domains for the les flow solver boundary conditions then have to be defined on the basis of the reynolds averaged flow statistics delivered by a rans flow solver this is a challenge which also arises for instance when defining les inflow conditions from experimental data the problem for the coupled rans les computations is further complicated by the fact that the mean flow statistics at the interface may vary in time and are not known a priori but only from the rans solution the present study defines a method to provide les inflow conditions through auxiliary a priori les computations where an les inflow database is generated the database is modified to account for the unsteadiness of the interface flow statistics

global topology from an embedding an embedding of chaotic data into a suitable phase space creates a diffeomorphism of the original attractor with the reconstructed attractor although diffeomorphic the original and reconstructed attractors may not be topologically equivalent in a previous work we showed how the original and reconstructed attractors can differ when the original is three dimensional and of genus one type in the present work we extend this result to three dimensional attractors of arbitrary genus this result describes symmetries exhibited by the lorenz attractor and its reconstructions

geometrical classification scheme for weights of lie algebras every weight of a lie algebra belongs to one particular weyl orbit the depth within an orbit has been defined as the minimum number of weyl reflections associated with simple roots that are necessary to transform the weight into the dominant weight of the orbit a simple set of classification parameters that measures the depth and distinguishes between different weights of the same depth is introduced the parameters are geometrical in that they are related directly to the orientation of the weight vector they correspond to a particular member of the weyl group that transforms the weight into the dominant weight a simple rule is given for listing all the weights of an orbit in terms of the parameters

lessons from lesfoil project on large eddy simulation of flow around an airfoil a synopsis of important results obtained by seven partners in the brite euram project lesfoil sponsored by the european union is presented this project was devoted to assessing the feasibility of large eddy simulations les forthecomputationofthee owaroundanairfoil asatestcase theaerospatialea airfoilat re 2 10 6 and 13 3 deg angle of attack was chosen all partners performed simulations of this e ow with various methods most of which employed e nite volume schemes and various subgrid scale models of eddy viscosity type the key results of theindividual partners publicationsarepresented ina comparativeway itis demonstratedthatthiscone guration with realistic conditions for aeronautical e ows is extremely demanding for les one reason is the need to achieve a threshold resolution for the adequate discretization of the attached and mildly separating boundary layer the second and even more demanding requirement is the need for an adequate treatment of transition which in the present computations was achieved by increased resolution

a new flux splitting scheme a new flux splitting scheme is proposed the scheme is remarkably simple and yet its accuracy rivals and in some cases surpasses that of roe s solver in the euler and navier stokes solutions carried out in this study the scheme is robust and converges as fast as the roe splitting we propose an appropriately defined cell face advection mach number using values from the two straddling cells via associated characteristic speeds this interface mach number is then used to determine the upwind extrapolation for the convective quantities accordingly the name of the scheme is coined as the advection upstream splitting method ausm we also introduce a new pressure splitting which is shown to behave successfully yielding much smoother results than other existing pressure splittings of particular interest is the supersonic blunt body problem in which the roe scheme gives anomalous solutions the ausm produces correct solutions without difficulty for a wide range of flow conditions as well as grids

representation theory for strange attractors embeddings are diffeomorphisms between some unseen physical attractor and a reconstructed image different embeddings may or may not be equivalent under isotopy we regard embeddings as representations of the attractor review the labels required to distinguish inequivalent representations for an important class of dynamical systems and discuss the systematic ways inequivalent embeddings become equivalent as the embedding dimension increases until there is finally only one universal embedding in a suitable dimension

hybrid simulation approach for cavity flows blending algorithm and boundary treatment issues the maturation of high performance computer architectures and computational algorithms has prompted the development of a new generation of models that attempt to combine the robustness and efficiency offered by the reynolds averaged navier stokes equations with the higher level of modeling offered by the equations developed for large eddy simulation the application of a new hybrid approach is discussed where the transition between these equation sets is controlled by a blending function that depends on local turbulent flow properties as well as the local mesh spacing the utilization of local turbulence properties provides added control in specifying the regions of the flow intended for each equation set removing much of the burden from the grid generation process moreover the model framework allows for the combination of existing closure model equations avoiding the difficulty of formulating a single set of closure coefficients that perform well in both reynolds averaged and large eddy simulation modes simple modifications to common second order accurate reynolds averaged navier stokes algorithms are proposed to enhance the capturing of large eddy motions

extension and replacement bases for semisimple lie algebras two simple prescriptions are given for obtaining sets of orthogonal bases for semisimple lie algebras the first method allows one to obtain the irreducible representations of all the simple lie algebras starting from su 2 and dynkin s method for constructing representations from simple roots and highest weights the second method relates algebras of the same rank several examples are discussed a method is given for listing the bases that may be obtained from the prescriptions

high order fluxes for conservative skew symmetric like schemes in structured meshes developing high order non dissipative schemes is an important research task for both steady and unsteady flow computations we take as a starting point the built in de aliasing property of the discretized skew symmetric form for the non linear terms of the navier stokes equations recalled in kravchenko and moin 1 two families of high order locally conservative schemes matching this discretized skew symmetric form are considered and rewritten in terms of telescopic fluxes for both finite difference and finite volume approximations in the context of compressible flows the jameson s scheme 2 is shown to be the second order member of larger families of skew symmetric like centered schemes the fourth order finite volume and finite difference and the sixth order finite difference schemes which belong to this family are provided the proposed schemes are extended to shock capturing schemes either by modifying the jameson s artificial viscosity or by hybriding the centered flux with weno 3 fluxes an adapted interpolation is proposed to extend the use of the proposed schemes to non regular grids several tests are provided showing that the conjectured order is properly recovered even with irregular meshes and that the shock capturing properties allow us to improve the second order results for standard test cases the improvement due to fourth order is then confirmed for the estimation of the growth of two ts waves and three crow instability dimensional unstable modes for both confined and free shear flows the last application concerns the steady computation using the spalart allmaras model of a separated boundary layer it confirms that the use of a high order scheme improves the results even in this type of steady applications

representations of affine kac moody algebras and the affine scalar product simple procedures are given for finding all the dominant weights in a highest weight representation of an affine algebra for finding the weyl orbit of an arbitrary weight and for determining whether or not any given weight is in any given representation a simple definition of congruency is given that applies to all affine algebras the standard indefinite scalar product is generalized the generalization is used in the procedures

blending functions in hybrid large eddy reynolds averaged navier stokes simulations several blending functions for use in hybrid large eddy simulation les reynolds averaged navier stokes rans simulations of shock separated flows are tested the blending functions shift the turbulence closure from a k turbulence model near solid surfaces to a k subgrid closure away from the wall three distinct forms for the blending function are developed one that depends on the ratio of the von karman length scale and the taylor microscale another that depends on the ratio of the rans eddy viscosity to the subgrid eddy viscosity and a third which replaces the von karman length scale in the first form with the distance to the nearest wall comparisons are made for two cases mach 2 79 flow over a 20 deg compression corner and mach 2 88 flow over a 25 deg compression expansion corner inflow boundary conditions for all calculations employ the rescaling reintroducing procedure developed by xiao et al xiao x edwards j r hassan h a and baurle r a inflow boundary conditions for hybrid large eddy reynolds averaged navier stokes simulations aiaa journal v ol 41 no 8 2003 pp 1481 1489 for hybrid les rans simulations of wall bounded flows in general the blending function based on the von karman length scale gives the best results when compared with measured data the skin friction predictions show the highest sensitivity to the various blending functions

spurious lyapunov exponents in attractor reconstruction lyapunov exponents perhaps the most informative invariants of a complicated dynamical process are also among the most difficult to determine from experimental data in particular when using embedding theory to build chaotic attractors in a reconstruction space extra open quotes spurious close quotes lyapunov exponents arise that are not lyapunov exponents of the original system the origin of these spurious exponents is discussed and formulas for their determination in the low noise limit are given copyright ital 1998 ital the american physical society

inflow boundary conditions for hybrid large eddy reynolds averaged navier stokes simulations inflow boundary conditions are developed for hybrid large eddy simulation les reynolds averaged navier stokes approaches they are based on an extension of the rescaling reintroducing method developed for les to a hybrid scheme a blending function is used to shift the turbulence closure from a model near the wall to a subgrid scale model away from the wall the approach was tested for a flat plate and then applied to the study of a 25 deg compression expansion ramp for a mach number of 2 88 and a reynolds number of 3 24 10 7 m in general improvements over the model were noted in the recovery region the significance of this work is that it provides a way for les methods to address flows at a high reynolds number

when are projections also embeddings we study an autonomous four dimensional dynamical system used to model certain geophysical processes this system generates a chaotic attractor that is strongly contracting with four lyapunov exponents ensuremath lambda i that satisfy ensuremath lambda 1 ensuremath lambda 2 ensuremath lambda 3 l0 so the lyapunov dimension is d l 2 ensuremath mid ensuremath lambda 3 ensuremath mid ensuremath lambda 1 l3 in the range of coupling parameter values studied as a result it should be possible to find three dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior we study mappings into r 3 to determine which can be used as embeddings to reconstruct the dynamics we find dramatically different behavior in the two simplest mappings projections from r 4 to r 3 in one case the one parameter family of attractors studied remains topologically unchanged for all coupling parameter values in the other case during an intermediate range of parameter values the projection undergoes self intersections while the embedded attractors at the two ends of this range are topologically mirror images of each other