approximate primal solutions and rate analysis for dual subgradient methods in this paper we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the lagrangian dual of a primal convex possibly nondifferentiable constrained optimization problem our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods such as the recent resource allocation problems in large scale networks for such problems we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions these algorithms use a constant stepsize in view of its simplicity and practical significance we provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions these estimates are given per iteration thus providing a basis for analyzing the trade offs between the desired level of error and the selection of the stepsize value our analysis relies on the slater condition and the inherited boundedness properties of the dual problem under this condition it also relies on the boundedness of subgradients which is ensured by assuming the compactness of the constraint set

a natural classification of vibration modes of polygonal ducts based on group theoretic analysis in this work the natural frequencies and mode shapes of polygonal ducts are obtained by a finite element analysis via a group theoretic approach we show that the group theoretic techniques provide a natural setting for the solution of the problem via optimum utilization of the symmetry in the problem the simplifications and insights due to the inherent symmetry is intuitively obvious in simple symmetric structures however a systematic exploitation of this symmetry in more complicated structures is less obvious and is possible perhaps only via useful theorems from group representation theory and the associated projection operator theory we illustrate all the main steps with the help of an example of dihedral groups this example is directly applicable to the problem addressed in this paper the group theoretic approach splits up the original problem into independent subproblems and thereby affects significant computational savings more importantly the approach captures the role of symmetry in the problem and provides insights which are otherwise not obvious the multiplicity of the natural frequencies arising out of the inherent symmetry is determined a priori and neatly separated into different subproblems this suggests a neat physical classification of the mode shapes on a symmetry basis which is unique to this approach the values of the natural frequencies and the mode shapes for triangular square and pentagonal ducts are shown to be in good agreement with the existing results reported in the literature the classification of mode shapes on this symmetry basis is used in the analysis of the polygonal ducts with number of sides varying up to 16

a primal dual approach to inexact subgradient methods for optimization problems with computationally demanding objective functions and subgradients inexact subgradient methods ixs have been introduced by using successive approximation schemes within subgradient optimization methods au et al 1994 in this paper we develop alternative solution procedures when the primal dual information of ixs is utilized this approach is especially useful when the projection operation onto the feasible set is difficult we also demonstrate its applicability to stochastic linear programs

universal unfoldings of group invariant equations which model second and third harmonic resonant capillary gravity waves abstract an investigation is made into the free surface capillary gravity waves caused by the interaction of the fundamental mode and a harmonic of either one half or one third its wavelength the problem is studied by casting it as an integro differential equation in a suitable function space this equation is found to be invariant under certain group actions the infinite dimensional problem is then reduced by means of the classical lyapunov schmidt procedure to a set of finite equations known as the bifurcation equations although it is not possible to calculate the bifurcation equations exactly the symmetries inherent in the problem enable us to make quite strong statements about the structure of the bifurcation equations in particular it allows us to replace them by a simpler set of equations the solution set of which is qualitatively the same as that of the original equations we then discuss unfoldings of the bifurcation equations this is a procedure whereby a finite set of new parameters is adjoined to the equations these possess the property that any perturbation of the equations can be represented by a certain choice of these parameters their physical significance is discussed and the bifurcaion diagrams are interpreted in the context of the original hydrodynamical problem

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

approximation of symmetry breaking bifurcations for the rayleigh convection problem the objective of this work is to analyze a numerical approximation of the rayleigh convection problem for a fluid confined on the lateral surface of a cylinder and heated uniformly from below the numerical scheme presented combines a finite element method in the axial direction and a fourier spectral method in the angular direction by the group representation theory it is shown that this scheme allows the numerical realization of symmetry breaking bifurcations error estimates in energy norms with the optimal rate of convergence are obtained

algebraic multigrid for higher order finite elements two related approaches for solving linear systems that arise from a higher order finite element discretization of elliptic partial differential equations are described the first approach explores direct application of an algebraic based multigrid method amg to iteratively solve the linear systems that result from higher order discretizations while the choice of basis used on the discretization has a significant impact on the performance of the solver results indicate that amg is capable of solving operators from both poisson s equation and a first order system least squares fosls formulation of stoke s equation in a scalable manner nearly independent of basis order p for 3

dual subgradient method with averaging for optimal resource allocation abstract a dual subgradient method is proposed for solving convex optimization problems with linear constraints as novelty the recovering of primal solutions can be avoided instead the optimal convergence rate for the whole sequence of primal dual iterates is obtained this is due to the primal dual averaging strategies which are incorporated into the iterative scheme we apply our dual subgradient method with averaging to optimal resource allocation within a multi agent environment the proposed dual subgradient method naturally corresponds to a distributed process of production price adjustments and effectively leads to a market equilibrium

construction of nearly orthogonal nedelec bases for rapid convergence with multilevel preconditioned solvers this paper presents a systematic approach to constructing high order tangential vector basis functions for the multilevel finite element solution of electromagnetic wave problems the new bases allow easy computation of a preconditioner to eliminate or at least weaken the indefiniteness of the system matrix and thus reduce the condition number of the system matrix when these bases are used in multilevel solutions where the multilevels correspond to the order of the basis functions the resulting p multilevel ilu preconditioned conjugate gradient method mpcg provides an optimal rate of convergence we first derive an admissible set of vectors of order p and decompose this set into two subspaces rotational and irrotational gradient we then reduce the number of vectors by making them orthogonal to all previously constructed lower order bases the remaining vectors are made mutually orthogonal in both the vector space and in the range space of the curl operator the resulting vector basis functions provide maximum orthogonality while maintaining tangential continuity of the field the zeroth order space is further decomposed using a scalar vector formulation to eliminate convergence problems at extremely low frequencies numerical experiments show that number of iterations needed for the solution by mpcg is basically constant regardless of the order of the basis or of the matrix size computational speed is improved by several orders of magnitude due to the fast matrix solution of mpcg and to the high accuracy of the higher order bases

variable target value subgradient method polyak s subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution in this paper we extend the convergence properties of the polyak s subgradient algorithm with a fixed target value to a more general case with variable target values then a target value updating scheme is provided which finds an optimal solution without prior knowledge of the optimal objective value the convergence proof of the scheme is provided and computational results of the scheme are reported

the resonance of modes in the problem of two dimensional capillary gravity waves abstract we report here our numerical computation of two dimensional capillary gravity waves on surface of irrotational flow of incompressible inviscid fluid in previous papers 1 4 we reported rather detailed numerical experiment these computations were however limited to solutions near double bifurcation points of mode 1 2 and 1 3 here mode n m means the linear and or nonlinear interaction of waves of wave number m and those of wave number n in the present paper we report what we found numerically near bifurcation points of mode 1 4 and 2 3 as in the case of modes 1 2 and 1 3 our numerical simulation shows much more solutions than were mathematically predicted in other works

systematic description of imperfect bifurcation behavior of symmetric systems abstract a systematic method is presented for describing experimental curves of force vs strain of a system with regular polygonal dihedral group symmetry subject to bifurcation behavior with an aim toward overcoming the following problems 1 it is difficult to judge whether the system is undergoing bifurcation or not 2 the perfect behavior of the system cannot be known due to the presence of initial imperfections 3 those curves are often qualitatively different from bifurcation diagrams predicted by mathematics the tools employed are the asymptotic theory for imperfect bifurcation such as the koiter law and the stochastic theory of initial imperfections the former theory is extended in this paper to the system with regular polygonal symmetry to present asymptotic laws for recovering perfect curves with reference to the experimental ones these laws are formulated for physically observable displacements instead of the variables in the mathematical bifurcation diagrams in order to make them readily applicable to the experimental curves the stochastic theory is combined with an asymptotic law to develop a means to identify the multiplicity of the bifurcation point the systematic method for describing the experimental curves developed in this manner is applied to the bifurcation analysis of regular polygonal truss domes to testify its validity furthermore this method is applied to the shear behavior of cylindrical sand specimens to show that they in fact are undergoing bifurcation and in turn to demonstrate the importance of a viewpoint of bifurcation in the study of shear behavior of materials the need of a dual viewpoint of bifurcation and plasticity in the study of constitutive relationship of materials is emphasized to conclude the paper

a skeleton structure of self replicating dynamics abstract self replicating patterns srp have been observed in several chemical reaction models such as the gray scott gs model as well as in physical experiments watching these experiments computational and physical is like watching the more familiar coarsening processes but in reverse the number of unit localized patterns increases until they fill the domain completely self replicating dynamics then can be regarded as a transient process from a localized trigger to a stable stationery or oscillating turing pattern since it is a transient process it is very difficult to give a suitable definition to characterize srp it cannot be described in terms of well studied structures such as the attractor or a singular saddle orbit for a dynamical system in this paper we present a new point of view to describe the transient dynamics of srp over a finite interval of time we focus our attention on the basic mechanism causing srp from a global bifurcational point of view and take our clues from two model systems including the gs model a careful analysis of the anatomy of the global bifurcation diagram suggests that the dynamics of srp is related to a hierarchical structure of limit points of folding bifurcation branches in parameter regions where the branches have ceased to exist thus the skeleton structure mentioned in the title refers to the remains of bifurcation branches the aftereffects of which are manifest in the dynamics of srp one of the natural and important problems is about the existence of an organizing center from which the whole hierarchical structure of limit points emerges in our setting the numerics suggests a strong candidate for that i e bogdanov takens turing btt singularity together with the presence of a stable critical point and so this indicates a universality of the above structure in the class of equations sharing this characteristic

local and global aspects of the 1 n mode interaction for capillary gravity waves abstract the structure of the solutions of the capillary gravity wave problem in a neighbourhood of a 1 n mode interaction with n 4 are considered extending the results of jones and toland in this important case these results are based on the known general form of the bifurcation equations which arises due to the symmetry of the problem and also the universal unfolding of a pitchfork bifurcation it is shown how the higher order terms of the bifurcation equation unfold a secondary pitchfork bifurcation which occurs in the truncated bifurcation equations obtained by ignoring the higher order terms the global effect of these local mode interactions is then considered and it is shown how repeated mode interactions result in secondary loops of solutions being formed which intersect three primary branches of solutions an example of this behaviour in the capillary gravity wave problem is given

large scale convex optimization via saddle point computation this article proposes large scale convex optimization problems to be solved via sad dle points of the standard lagrangian a recent approach for saddle point computation is specialized by way of a specific perturbation technique and unique scaling method to convex optimization problems with differentiable objective and constraint funct ions in each iteration the update directions for primal and dual variables are determined by gradients of the lagrangian these gradients are evaluated at perturbed points that are generated from current points via auxiliary mappings the resulting algorithm suits massively parallel computing though in this article we consider only a serial implementation we test a version of our code embedded within gams on 16 nonlinear problems which are mainly large these models arise from multistage optimization of economic systems for larger problems with adequate precision requirements our implementation appears faster than minos

singular secondary bifurcation a bifurcation problem is analyzed for a brussellator boundary value problem which is a typical reaction diffusion system the bifurcation parameter lambda is proportional to the length of the system we employ previously developed perturbation methods to analyze the secondary bifurcation of steady solutions that arise from the splitting of multiple primary bifurcation points the resulting bifurcation equations are nonlinear algebraic equations to determine the amplitudes of the solutions the coefficients in these equations depend upon the system parameters in the brussellator problem such as the two prescribed constant reactants a and b for critical values of these parameters the solutions of the algebraic system are singular and hence the perturbation method is invalid for parameter values at and near these critical values a new perturbation method is employed yielding new branches of steady solutions and possible tertiary bifurcations to time periodic solutions thus the analysis of singulari

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

bifurcation hierarchy of a rectangular plate abstract the mechanism of the complex recursive bifurcation behavior of a four sides simply supported rectangular plate is investigated such complex behavior is due to the hidden symmetry of the plate associated with the periodic nature of the solutions the group theoretic bifurcation theory is employed to arrive at a lattice of subgroups which expresses the rule for the behavior this rule is shown to suffer degeneration due to the restriction by the boundaries compared with that of geometrical symmetry the governing equation of this plate is discretized by means of galerkin s method with the use of the double fourier series as shape functions the tangential stiffness matrix of the plate is shown to be block diagonalized by appropriately permuting the order of the fourier series following the rule presented the bifurcation analysis of the plate is carried out to assess the validity of the rule and to demonstrate the merit of the block diagonalization as a result of this the vital role of the bifurcation rule in the proper understanding and successful analysis of the complex bifurcation behavior has been demonstrated

the bifurcation and secondary bifurcation of capillary gravity waves the bifurcation and secondary bifurcation of capillary gravity waves is analysed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two the existence and multiplicity of solutions is seen via the implicit function theorem to be a special case of the secondary bifurcation phenomena which occur when a double eigenvalue splits under perturbation into two simple eigenvalues in the presence of a symmetry in the problem

multi p preconditioners a class of algebraic multi p methods for solving the p version of the finite element equations is first presented and discussed it is then shown how these multi p methods can be used as preconditioners for the conjugate gradient cg method in particular it is shown that given any preconditioner mp to cg a multi p preconditioner bp based on mp can be constructed which leads to a smaller condition number and hence faster convergence numerical experiments on representative problems indicate that the condition numbers after multi p preconditionings are in fact independent of p the numerical results also show greater efficiency for the preconditioned conjugate gradient pcg method with the multi p preconditioners in terms of number of iterations and cpu time when compared with two sophisticated linear equation solvers 1 a direct frontal solver specially designed for the p version of the finite element analysis 2 a highly tuned pcg code in itpack2c http www netlib org itpack 1994 preliminary comparisons of the number of iterations are also made with rockits solvers international inc boulder co 1994 a new commercial code used for the p version

a class of convergent primal dual subgradient algorithms for decomposable convex programs in this paper we develop a primal dual subgradient algorithm for preferably decomposable generally nondifferentiable convex programming problems under usual regularity conditions the algorithm employs a lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal dual optimal solutions several classical types of penalty functions are shown to satisfy these specified properties a geometric convergence rate is established for the algorithm under some additional assumptions this approach has three principal advantages firstly both primal and dual solutions are available which prove to be useful in several contexts secondly the choice of step sizes which plays an important role in subgradient optimization is guided more determinably in this method via primal and dual information thirdly typical subgradient algorithms suffer from the lack of an appropriate stopping criterion and so the quality of the solution obtained after a finite number of steps is usually unknown in contrast by using the primal dual gap the proposed algorithm possesses a natural stopping criterion

the classification of bifurcations with hidden symmetries we set up a singularity theoretic framework for classifying one parameter steady state bifurcations with hidden symmetries this framework also permits a non trivial linearization at the bifurcation point many problems can be reduced to this situation for instance the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with neumann or dirichlet boundary conditions we formulate an appropriate equivalence relation with its associated tangent spaces so that the usual methods of singularity theory become applicable we also present an alternative method for computing those matrix valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems this result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an n dimensional rectangle under neumann boundary conditions

pattern generation in space and aspect a survey is presented of theories for the generation and maintenance of spatial pattern in reaction diffusion equations and their generalizations applications are selected from the biological sciences and physical chemistry special emphasis is placed on nonlocal interaction as manifested by the inclusion of terms involving higher derivatives or integrals it is stressed that traditional ideas of spatial pattern generation can usefully be extended to the study of pattern in general descriptive aspect variables particularly in understanding ecological diversity and heterogeneity

transitions between nonsymmetric and symmetric steady states near a triple eigenvalue we examine the existence of nonuniform steady state solutions of a certain class of reaction diffusion equations our analysis concentrates on the case where the first bifurcation is near a triple eigenvalue we derive the conditions for a continuous transition between nonsymmetric and symmetric solutions when the bifurcation parameter progressively increases from zero finally we give an example of a four variables model which presents the possibility of a triple eigenvalue

bifurcation hierarchy of symmetric structures abstract a group theoretic method for the analysis of bifurcation behavior of regular polygonal symmetric structures is described possible bifurcation paths and points of these structures are categorized in terms of dihedral and cyclic groups which express the symmetry in particular we offer a complete description of those double bifurcation points which occur due to group symmetry the type the number and the stability of bifurcation paths branching at these points are determined by deriving bifurcation equations the existence of a potential function plays a substantial role for the existence of bifurcation paths as a result of these all possible bifurcation process can be known a priori as a natural consequence of bifurcation hierarchy before actual numerical analysis an implementation of this method in numerical analysis shows its validity and usability

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

rate analysis of inexact dual first order methods application to dual decomposition we propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex problems the complicating constraints are moved into the cost using the lagrange multipliers the dual problem is solved by inexact first order methods based on approximate gradients for which we prove sublinear rate of convergence in particular we provide a complete rate analysis and estimates on the primal feasibility violation and primal and dual suboptimality of the generated approximate primal and dual solutions moreover we solve approximately the inner problems with a linearly convergent parallel coordinate descent algorithm our analysis relies on the lipschitz property of the dual function and inexact dual gradients further we combine these methods with dual decomposition and constraint tightening and apply this framework to linear model predictive control obtaining a suboptimal and feasible control scheme

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

a descent algorithm for nonsmooth convex optimization this paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable the algorithm can be implemented and may be considered a modification of the e subgradient algorithm and lemarechal s descent algorithm also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems a convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below limited computational experience with the algorithm is also reported

a lagrangian heuristic based branch and bound approach for the capacitated network design problem the capacitated network design problem is a multicommodity minimal cost network flow problem with fixed charges on the arcs and is well known to be np hard the problem type is very common in the context of transportation networks telecommunication networks etc in this paper we propose an efficient method for this problem based on a lagrangian heuristic within a branch and bound framework the lagrangian heuristic uses a lagrangian relaxation to obtain easily solved subproblems and solves the lagrangian dual by subgradient optimization it also includes techniques for finding primal feasible solutions the lagrangian heuristic is then embedded into a branch and bound scheme that yields further primal improvements special penalty tests and cutting criteria are developed the branch and bound scheme can either be an exact method that guarantees the optimal solution of the problem or be a quicker heuristic the method has been tested on networks of various structures and sizes computational comparisons between this method and a state of the art mixed integer code are presented the method is found to be capable of generating good feasible solutions to large scale problems within reasonable time and data storage limits

on the choice of step size in subgradient optimization abstract this paper recommends some procedures for the selection of step sizes in the context of subgradient optimization the first of these procedures is developed in detail in this study and is a theoretically convergent scheme this method has two phases the first phase is designed to accelerate the solution procedure towards an optimal solution while the second phase helps to close in on an optimal solution a second technique recommended is a simple minded scheme which although not theoretically convergent seems to be computationally very efficient these two methods are shown to compare favorably with held wolfe and crowder s scheme for prescribing step sizes we also suggest some modifications of the latter scheme to make it computationally more efficient

inexact subgradient methods with applications in stochastic programming in many instances the exact evaluation of an objective function and its subgradients can be computationally demanding by way of example we cite problems that arise within the context of stochastic optimization where the objective function is typically defined via multi dimensional integration in this paper we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods we refer to this new class of methods as inexact subgradient algorithms with relatively mild conditions imposed on the approximations we show that the inexact subgradient algorithms inherit properties associated with their traditional i e exact counterparts within the context of stochastic optimization the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi gradient methods additionally we study methods in which steplengths may be defined adaptively in a manner that reflects the improvement in the objective function approximations as the iterations proceed we illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs

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

lagrangian relaxation via ballstep subgradient methods we exhibit useful properties of ballstep subgradient methods for convex optimization using level controls for estimating the optimal value augmented with simple averaging schemes they asymptotically find objective and constraint subgradients involved in optimality conditions when applied to lagrangian relaxation of convex programs they find both primal and dual solutions and have practicable stopping criteria up until now similar results have only been known for proximal bundle methods and for subgradient methods with divergent series stepsizes whose convergence can be slow encouraging numerical results are presented for large scale nonlinear multicommodity network flow problems

global bifurcation phenomena of traveling wave solutions for some bistable reaction diffusion systems on considere le systeme bistable e u t e 2 u xx f u v v t v xx g u v ou e est petit on etudie les solutions onde de propagation qui tendent vers deux etats stables quand x

modified stiffness iteration to pinpoint multiple bifurcation points abstract a modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented the iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones two different transformation matrices are employed to modify the stiffness matrix one is a non orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix the other is an orthogonal one which is chosen with reference to the symmetry of the system under consideration the use of the non orthogonal transformation matrix is a key idea termed stiffness amplification method in this paper whereas the orthogonal matrix is based on block diagonalization which is becoming popular in group theoretic analysis of symmetric structures extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points

a mathematical mechanism for instabilities in stripe formation on growing domains abstract a cascade process involving stripe splitting in reaction diffusion systems with isotropically growing one dimensional domains is studied such cascades propagating from a smaller domain to a larger domain have been proposed as an answer to the criticism that the turing mechanism lacks robustness because many stable patterns can coexist on a large domain and therefore the final state is very sensitive to the initial conditions in contrast if the system starts with a small domain very few stable patterns are possible which limits the sensitivity to the initial conditions in order to show the validity and limitations of this scenario we clarify the underlying mathematical mechanism that drives various types of stripe splitting via a reduction from partial differential equations to ordinary differential equations as well as investigating global arrangements of the set of n mode stripe branches with d n symmetry of the stripe locations the mathematical simplification above allows us to reveal how each n mode stripe branch is destabilized as the domain grows and to characterize the associated eigenprofiles that actually determine the manner of splitting at the infinitesimal level we find that all the d n symmetry breaking instabilities typically occur simultaneously up to leading order before the saddle node point of the n mode stripe branch is reached the instability with the largest real part is of the alternate type every other peak splits at the infinitesimal level a symmetry preserving instability appears at the saddle node point which drives the simultaneous type of splitting i e mode doubling due to competition between these two types of instabilities the problem depends subtly on the growth speed alternate splitting typically arises for slow growth and simultaneous splitting for fast growth for intermediate growth rates the manner of splitting becomes mixed and sensitive to fluctuations