sil nikov saddle node interaction near a codimension 2 bifurcation laser with injected signal abstract we describe a sil nikov saddle node interaction in the proximity of a hopf saddle node bifurcation both from numerical and theoretical viewpoints inspired by numerical observations on a three dimensional model of a laser with injected signal we describe the interaction near the codimension 2 point using a return map controlled by the parameters of the bifurcation the theoretical predictions compare well with numerical experiments finally we discuss the reasons for the existence of the sil nikov saddle node bifurcation in the laser with injected signal and correspondingly the characteristics of other systems that might display this bifurcation

centre manifold reduction and the stuart landau equation for fluid motions the centre manifold reduction to derive the stuart landau equation is examined a double expansion in terms of the fourier series and linear eigenfunctions is introduced in hydrodynamic equations a centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the stuart landau equation through a formal expansion in linear eigenfunctions the latter equation is shown to be equivalent with the one derived by the method of multiple scales numerical coefficients involved in the quintic stuart landau equation are evaluated for plane poiseuille flow convection in a vertical slot and rayleigh benard convection in all the cases the coefficients converge as a dimension of the ode system increases and approach the numerical values obtained by the method of multiple scales

piecewise constant galerkin method for a class of cauchy singular integral equations of the second kind in l2 abstract in this work we present piecewise constant galerkin method for a class of cauchy singular integral equations of the second kind with constant coefficients in l 2 0 1 c using a sequence of orthogonal finite rank projections we prove the existence and uniqueness theorems for the cauchy integral equation and the approximate equation respectively we perform the error analysis for which we give new and improved estimates for the rates of convergence numerical example illustrates the theoretical results

on the complexity of sphere decoding in digital communications sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications in some cases the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure another claim is that although worst case complexity is exponential the expected complexity of the algorithm is polynomial herein we study the expected complexity where the problem size is defined to be the number of symbols jointly detected and our main result is that the expected complexity is exponential for fixed signal to noise ratio snr contrary to previous claims the sphere radius which is a parameter of the algorithm must be chosen to ensure a nonvanishing probability of solving the detection problem this causes the exponential complexity since the squared radius must grow linearly with problem size the rate of linear increase is however dependent on the noise variance and thus the rate of the exponential function is strongly dependent on the snr therefore sphere decoding can be efficient for some snr and problems of moderate size even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size

statistical precoding with decision feedback equalization over a correlated mimo channel the decision feedback df transceiver combining linear precoding and df equalization can establish point to point communication over a wireless multiple input multiple output channel matching the df transceiver design parameters to the channel characteristics can improve system performance but requires channel knowledge we consider the fast fading channel scenario with a receiver capable of tracking the channel state variations accurately while the transmitter only has long term channel distribution information the receiver design problem given channel state information is well studied in the literature we focus on transmitter optimization which amounts to designing a statistical precoder to assist the channel tailored df equalizer we develop a design framework that encompasses a wide range of performance metrics common cost functions for precoder optimization are analyzed thereby identifying a structure of typical cost functions transmitter design is approached for typical cost functions in general and we derive a precoder design formulation as a convex optimization problem two important subclasses of cost functions are considered in more detail first we explore a symmetry of df transceivers with a uniform subchannel rate allocation and derive a simplified convex optimization problem which can be efficiently solved even as system dimensions grow second we explore the tractability of a certain class of mean square error based cost functions and solve the transmitter design problem with a simple algorithm that identifies the convex hull of a set of points in r2 the behavior of df transceivers with optimal precoders is investigated by numerical means

superconvergence of the iterated galerkin methods for hammerstein equations in this paper the well known iterated galerkin method and iterated galerkin kantorovich regularization method for approximating the solution of fredholm integral equations of the second kind are generalized to hammerstein equations with smooth and weakly singular kernels the order of convergence of the galerkin method and those of superconvergence of the iterated methods are analyzed numerical examples are presented to illustrate the superconvergence of the iterated galerkin approximation for hammerstein equations with weakly singular kernels

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions the equivalence between the method of centre manifold and the method of multiple scales was examined when they are applied to the weakly nonlinear stability analysis for fluid motions the relationship between the landau equations of the seventh order which were reduced from the navier stokes equations by applying these two methods is clarified how we should define the disturbance amplitude to obtain the complete equivalence between them is discussed

complex dynamics in the 1 3 spatial resonance abstract the interaction between two steady state bifurcations with spatial wave numbers in the ratio 1 3 is considered periodic boundary conditions are assumed the resulting o 2 equivariant normal form truncated at third order exhibits a number of global bifurcations that may result in complex dynamics the origin of this behavior is elucidated with the help of careful numerical simulations and analysis of appropriate return maps the results generalize to other 1 n resonances

resonant mode interactions in rayleigh benard convection department of physics university of california berkeley california 94720 received 5 february 1998 in systems with midplane re ection symmetry the dominant spatial resonance is the 1 3 resonance numeri cal continuation is used to study this and other 1 2k11 resonances in two dimensional convection betweenno slip perfectly conducting horizontal plates periodic boundary conditions are imposed in the horizontal these resonances in uence the process of wave number selection at moderate rayleigh numbers through thegeneration of hybrid solutions and thereby modify the eckhaus picture of wave number selection unlike thebetter known mixed modes the hybrid solutions have the same symmetry as a pair of primary rolls both hybridand symmetry breaking mixed modes are computed and their linear stability properties with respect to pertur bations preserving different spatial periods are determined a complete description of the effects of midplanere ection on wave number selection emerges only steady solutions are considered and the prandtl number is xed ats510 s1063 651x 98 05709 2 pacs number s 47 20 ky 05 45 1bi introduction

assignment problems a golden anniversary survey abstract having reached the 50th golden anniversary of the publication of kuhn s seminal article on the solution of the classic assignment problem it seems useful to take a look at the variety of models to which it has given birth this paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years the intention here is not to identify every such paper of which there have been hundreds nor to identify the best solution procedure for each variation rather the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

transceiver optimization for multiuser mimo systems we consider the uplink of a multiuser system where the transmitters as well as the receiver are equipped with multiple antennas each user multiplexes its symbols by a linear precoder through its transmit antennas we work with the system wide mean squared error as the performance measure and propose algorithms to find the jointly optimum linear precoders at each transmitter and linear decoders at the receiver we first work with the case where the number of symbols to be transmitted by each user is given we then investigate how the symbol rate should be chosen for each user with optimum transmitters and receivers the convergence analysis of the algorithms is given and numerical evidence that supports the analysis is presented

optimality of beamforming in fading mimo multiple access channels we consider the sum capacity of a multi input multi output mimo multiple access channel mac where the receiver has the perfect channel state information csi while the transmitters have either no or partial csi when the transmitters have partial csi it is in the form of either the covariance matrix of the channel or the mean matrix of the channel for the covariance feedback case we mainly consider physical models that result in single sided correlation structures for the mean feedback case we consider physical models that result in in phase received signals under these assumptions we analyze the mimo mac from three different viewpoints first we consider a finite sized system we show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices in the covariance and mean feedback models respectively also we find the conditions under which beamforming is optimal for all users second in the covariance feedback case we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system in the mean feedback case we show through simulations that this is not necessarily true third we consider the asymptotic case where the number of users is large we show that in both no and partial csi cases beamforming is asymptotically optimal in particular in the case of no csi we show that a simple form of beamforming which may be characterized as an arbitrary antenna selection scheme achieves the sum capacity in the case of partial csi we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity finally we generalize our covariance feedback results to double sided correlation structures in the appendix

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

on optimum power allocation for the v blast a unified analytical framework for optimum power allocation in the unordered v blast algorithm and its comparative performance analysis are presented compact closed form approximations for the optimum power allocation are derived based on average total and block error rates the choice of the criterion has little impact on the power allocation and overall the optimum strategy is to allocate more power to lower step transmitters and less to higher ones high snr approximations for optimized average block and total error rates are given the snr gain of optimization is rigorously defined and studied using analytical tools including lower and upper bounds high and low snr approximations the gain is upper bounded by the number of transmit antennas for any modulation format and type of fading channel while the average optimization is less complex than the instantaneous one its performance is almost as good at high snr a measure of robustness of the optimized algorithm is introduced and evaluated the optimized algorithm is shown to be robust to perturbations in individual and total transmit powers based on the algorithm robustness a pre set power allocation is suggested as a low complexity alternative to the other optimization strategies which exhibits only a minor loss in performance over the practical snr range

heteroclinic cycles and periodic orbits for the o 2 equivariant 0 1 2 mode interaction we study the quadratic normal form describing the generic interaction of fourier modes of wavenumbers 0 1 and 2 under the symmetry group o 2 of rotations and reflections in the case that the homogeneous quadratic terms preserve energy the sum of the squares of absolute values of the complex variables this system is a generalization of the 1 2 mode interaction studied by dangelmayr g dangelmayr steady state mode interactions in the presence of o 2 symmetry dyn stab syst 1 2 1986 159 185 armbruster et al d armbruster j guckenheimer p holmes heteroclinic cycles and modulated travelling waves in systems with o 2 symmetry physica d 29 1988 257 282 and others and its restriction to the 1 2 subspace is a degenerate case of that system it displays all the classes of fixed points periodic orbits standing and travelling waves invariant tori modulated travelling waves and heteroclinic cycles found in the 1 2 interaction as well as new heteroclinic cycles connecting pure and mixed modes chaotic cycles and strange periodic orbits we describe the key dynamical features show that the degenerate 1 2 case possesses a second organizing center at which bifurcation curves coalesce provide representative bifurcation sets and diagrams for the 1 2 and 0 1 2 systems and use a conservative limit to understand the periodic orbits in the latter system 2005 elsevier 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

the 1 2 spatial resonance on a hexagonal lattice in two layered rayleigh benard problems this paper investigates pattern formation resulting from the interaction between steady modes with wavenumbers in the ratio 1 2 a two layered rayleigh benard problem with infinite extent is examined two layers are separated by a non deformable and thin dividing plate this physical set up is known to provide an exact 1 2 resonance between critical modes restricting the wavevectors of interacting modes on a hexagonal lattice we derive 12 dimensional amplitude equations up to cubic order among steady solutions of the equations the stable ones are hexagons and mixed hexagons travelling waves corresponding to relative equilibria are also found to be stable depending on parameter values

instabilities induced by a weak breaking of a strong spatial resonance abstract through multiple scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady state pattern forming instabilities in the case that the wavelengths of the instabilities are nearly in the ratio 1 2 in the case of exact 1 2 resonance the amplitude equations are odes here they are pdes we discuss the stability of spatially periodic solutions to long wavelength disturbances by including these modulational effects we are able to explore the relevance of the exact 1 2 results to spatially extended physical systems for parameter values near to this codimension two bifurcation point these new instabilities can be described in terms of reduced normal form pdes near various secondary codimension two points the robust heteroclinic cycle in the odes is destabilised by long wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle an analytic expression giving the approximate period of this orbit is derived

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

probing a subcritical instability with an amplitude expansion an exploration of how far one can get abstract we explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite wavelength instability we do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers this is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic laminar state we apply the method to two model equations a subcritical generalization of the swift hohenberg equation and a novel extension of the kuramoto sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate from infinity and that can nevertheless be probed perturbatively we explore the reliability and robustness of such an expansion with a particular focus on the use of these methods for determining the existence and approximate properties of finite amplitude stationary solutions such methods obviously are to be used with caution the expansions are often only asymptotic approximations and if they converge their radius of convergence may be small nevertheless expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results

resonant triad interactions in symmetric systems abstract we analyse resonant triad interactions in symmetric systems the case of coupling at quadratic order in amplitudes and the case of coupling at cubic order are considered we study modulated travelling waves and heteroclinic cycles in amplitude equations and we discuss their stability types we describe possible dynamical regimes and spatial patterns arising from triad interactions of long waves in rapidly rotating hagen poiseuille flow

optimal precoder design for correlated mimo communication systems using zero forcing decision feedback equalization we consider the design of the precoder for a multi input multi output mimo communication system equipped with a decision feedback equalizer dfe receiver for such design problems perfect knowledge of the channel state information csi at both the transmitter and the receiver is usually required however in the environment of wireless communications it is often difficult to provide sufficiently timely and accurate feedback of csi from the receiver to the transmitter for such designs to be practically viable in this paper we consider the optimum design of a precoder for a wireless communication link having m transmitter antennas and n receiver antennas m n in which the channels are assumed to be flat fading and may be correlated we assume that full knowledge of csi is available at the receiver at the transmitter however only the first and second order statistics of the channels are available our goal here is to come up with an efficient design of the optimal precoder for such a mimo system by minimizing the average arithmetic mean squared error mse of zero forcing zf decision feedback detection subject to a constraint on the total transmitting power applying some of the properties of the matrix parameters this nonconvex optimization problem can be transformed into a convex geometrical programming problem which can then be efficiently solved using an interior point method the performance of the mimo system equipped with this optimum precoder and a zf dfe has also been found to be comparable and in some cases superior to that of v blast which necessitates optimally ordered successive interference cancellation based on the largest postdetection signal to noise ratio snr in terms of tradeoff between performance and implementation simplicity our new system is certainly an attractive alternative

on the nonlinear dynamics of a saline boundary layer formed by throughflow near the surface of a porous medium we consider gravitational instability of saline boundary layers observed at the subsurface of salt lakes this boundary layer is the result of the convective transport induced by the evaporation at the horizontal surface of a confined porous medium when this upward transport is balanced by salt dispersion a steady state boundary layer is formed however this boundary layer can be unstable when perturbed this results in complex groundwater motion and density fields the aim of this paper is to investigate the existence of finite amplitude solutions describing these resulting patterns both the number of solutions and their structure their stability and their dependency on the system rayleigh and peclet numbers for this purpose we construct a low dimensional dynamical system a reduced model by projecting the nonlinear model equations onto a relatively small set of eigenfunctions of the problem linearized at criticality the galerkin projection approach is complicated by the fact that the problem under consideration is non self adjoint due to the existing evaporation this implies that the eigenfunctions do not form an orthogonal set and therefore the adjoint eigenfunctions are used for the projection the reduced model is constructed in such a way that it is capable of providing solutions in the strongly nonlinear regime as well convergence of these solutions towards the fully nonlinear model results is shown by means of direct numerical simulations further the reduced model seems to partly capture the complex nonlinear behavior as seen in hele shaw experiments by wooding et al r a wooding s w tyler i white p a anderson convection in groundwater below an evaporating salt lake 2 evolution of fingers or plumes water resour res 33 6 1997 1219 1228 the physical transition mechanism that explains the occurrence of some observed bifurcation types is presented as well

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

superconvergence of some projection approximations for weakly singular integral equations using general grids this paper deals with superconvergence phenomena in general grids when projection based approximations are used for solving fredholm integral equations of the second kind with weakly singular kernels four variants of the galerkin method are considered they are the classical galerkin method the iterated galerkin method the kantorovich method and the iterated kantorovich method it is proved that the iterated kantorovich approximation exhibits the best superconvergence rate if the right hand side of the integral equation is nonsmooth all error estimates are derived for an arbitrary grid without any uniformity or quasi uniformity condition on it and are formulated in terms of the data without any additional assumption on the solution numerical examples concern the equation governing transfer of photons in stellar atmospheres the numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp and confirm the superiority of the iterated kantorovich scheme

new type of complex dynamics in the 1 2 spatial resonance abstract the amplitude equations describing the 1 2 spatial resonance in o 2 symmetric systems are re examined complex dynamical behavior associated with a new class of global connections is identified and analyzed these connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by armbruster et al physica d 29 1988 257 no longer exist the resulting behavior is characteristic of systems with o 2 z 2 symmetry in which the symmetry z 2 is weakly broken

the application of centre manifolds to amplitude expansions ii infinite dimensional problems abstract the theory of centre manifolds for infinite dimensional systems is described with emphasis on the practical computational aspects of applying the theory to near critical problems and in particular to computation of the centre manifold the calculations are illustrated by detailed analyses of two specific problems

a picture of the global bifurcation diagram in ecological interacting and diffusing systems abstract this paper concerns global phenomena of pattern formation in stationary reaction diffusion equations possessing turing s diffusion induced instability which appear typically in mathematical biology global bifurcation diagrams with respect to two diffusion parameters are presented by integrating two complementary approaches analytical and numerical simple and double bifurcation analysis using group theoretic methods for the compact lie group d x results from singular perturbations when one diffusion constant is sufficiently small and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large are the main analytical results a new numerical method is used to trace all bifurcating branches we examine interrelations between those local and semi global results under the light of global pictures obtained by numerical studies a variety of interesting and new diagrams near double eigenvalues are observed coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions

statistical framework for optimization in the multi user mimo uplink with zf dfe we consider performance optimization in the uplink of a multiuser multiantenna communication system each user multiplexes data onto several independently encoded data streams which are spatially precoded and conveyed over a fading narrowband multiple input multiple output mimo channel all users data streams are decoded successively at the receiving base station using zero forcing decision feedback equalization zf dfe we target the joint optimization of a decoding order and linear precoders for all users based on long term channel information for a class of general mimo channel models including the separable correlation and double scattering models we show that the choice of precoder for a certain user does not affect the performance of the others this leads to a particularly straightforward characterization of general user utility regions as a polyblock or a convex polytope if time sharing is allowed we formulate the decoding ordering problem under transmit correlated rayleigh fading as a linear assignment problem enabling the use of existing efficient algorithms combining decoding ordering with single user precoder optimization by means of alternating optimization we propose an efficient iterative scheme that is verified numerically to converge fast and perform close to optimally successfully reaping the benefits of both precoding and ordering in the mimo uplink

analysis of decision feedback detection for mimo rayleigh fading channels and the optimization of power and rate allocations for an uncoded k transmit n receive antenna coherent narrow band communication system employing a decorrelating decision feedback detector d dfd the exact average over channel realizations joint error probability jep as well as the average per symbol error probabilities seps are derived without making any simplifying assumptions on error propagation it is proved that the diversity orders of the jep and the sep of every symbol is limited by error propagation to n k 1 based on our exact error probability analysis however we suggest an optimization of jep over nonnegative quadrature amplitude modulation qam constellation sizes rates and average powers across transmitters which yield significant improvements over the usual equal power and equal rate assignment in fact the jep of such an optimized design has the much improved diversity order of n which is also the diversity order obtained through the optimum maximum likelihood ml detector moreover it is seen that these simple optimized designs can achieve a significant fraction of the spl epsi outage capacity even without outer codes it is also known but only through simulations that when the symbols are detected in certain channel realization dependent orders it is possible to improve substantially over fixed order detection in the case of the equal rate and equal power assignment we provide an analysis for a recently proposed channel dependent ordering rule and show that it does not provide an improvement of the diversity order of the jep beyond n k 1 another ordering rule that was proposed earlier to maximize the worst case post detection signal to noise ratio snr under the perfect feedback assumption is shown to be optimal under a more compelling criterion that does not involve that simplifying assumption while efficiently computable this ordering rule is seen to perform almost as well as the optimal channel dependent ordering rule that minimizes the conditional jep and hence the jep nevertheless a multiple input multiple output mimo system with an optimized rate and power allocation and a fixed order of detection is not only less complex but also has a significantly lower jep than that of the equal power equal rate system where transmitters are detected in a channel dependent order optimal or otherwise

degenerate 1 2 steady state mode interaction mhd flow in a vertical slot abstract degenerate bifurcations resulting from 1 2 mode interaction between quasi neutral steady modes in mhd natural convection have been investigated on a weakly nonlinear basis in the limit of small prandtl number and small magnetic prandtl number two types of nonlinearly degenerate situations are found to take place for relatively low hartmann numbers 4 0432 and 8 7701 the amplitude equations of the quintic order are derived by means of the center manifold reduction the entire picture of the bifurcation is obtained for the actual numerical values of the coefficients in the amplitude equations a fully nonlinear numerical analysis of the hydrodynamic equations is also carried out the local characteristics predicted by the amplitude equations are found to be justified

hexagons and triangles in the rayleigh benard problem quintic order equations on a hexagonal lattice on a weakly nonlinear basis we revisit the pattern formation problem in the boussinesq convection for which nonlinear terms of the quadratic order are known to vanish from amplitude equations it