the geometry and mechanics of generalized pseudo rigid bodies the geometry and the mechanics of generalized pseudo rigid bodies are studied the configuration space of a generalized pseudo rigid body is the linear group of non singular matrices with a positive determinant it admits the left and right actions of so n and the two sided action of so n so n the left and right so n actions on are both free so that is made into respective principal fiber bundles according to the left and right so n actions and further left and right connections can be defined on the respective fiber bundles however the two sided so n so n action is not free on and hence is not made into a principal fiber bundle with respect to this action in spite of this if is restricted to an open dense subset the isotropy subgroup at each point of is a finite discrete group so that the quotient space becomes a manifold and further one can define a connection on which will be called a bi connection the bi connection is used to reduce the pseudo rigid body system on with the so n so n symmetry though in the cotangent bundle reduction theorem and its variants one usually assumes that the action of a lie group on the configuration space is free or that the isotropy subgroup of the lie group is trivial the reduction procedure works well if the isotropy subgroup is not trivial but a finite group as an application of the reduction procedure relative equilibria are discussed in relation with the reduced hamilton and lagrange equations of motion a necessary and sufficient condition is given for a relative equilibrium in terms of an amended potential on the reduced phase space

qualitative analysis for a diffusive predator prey model with hunting cooperative abstract this paper investigates a predator prey model that adds a cooperation term to the attack rate of the predator population for the non spatial model the existence and stability of non negative equilibrium points and bifurcations are studied by choosing cooperation coefficient as control parameter our analytical results show that hunting cooperation can be beneficial to the predator population however as the cooperation coefficient increases hunting cooperation can also destabilize the model and promote a sudden collapse of the predator population for the spatial model the stability of positive constant steady state solution hopf bifurcation and turing instability are discussed it is obtained that when the predation diffusion is not smaller than the prey diffusion the spatial model can reserve the stable stability of the positive constant steady state solution it is noted that the model without hunting cooperative does not generate turing instability while the model with hunting cooperative may generate turing instability

a geometric setting for internal motions of the quantum three body system quantum mechanics for internal motions of the three body system is set up on the basis of the complex vector bundle theory the three body system is called a triatomic molecule in the born oppenheimer approximation the internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator this bundle is equipped with a linear connection which is a natural consequence of a geometric interpretation of the so called eckart condition the coupling of the internal motion with the rotation is understood naturally in terms of this connection the internal hamiltonian operator is obtained which includes the internal motion rotation coupling and a centrifugal potential the complex vector bundle for the triatomic molecule proves to be a trivial bundle though the geometric setting for the internal motion is independent of whether the bundle is trivial or not

multiple bifurcations in a delayed predator prey system with nonmonotonic functional response abstract a delayed predator prey system with nonmonotonic functional response is studied by using the normal form theory of retarded functional differential equations developed by faria and magalhaes the bifurcation analysis of the model indicates that there is a bogdanov takens singularity for any time delay value a versal unfolding of the model at the bogdanov takens singularity is obtained on the other hand it is shown that small delay changes the stability of the equilibrium of the model for some parameters and the system can exhibit hopf bifurcation as the time delay passes through some critical values

effect of weak prey in leslie gower predator prey model a leslie gower prey predator model incorporating partial prey protection is proposed prey densities are divided into two groups strong and weak the special dynamic of each group is considered the condition of existence and uniqueness of the equilibrium point in the positive octant is determined and by constructing a suitable lyapunov function it is shown that the unique equilibrium point is stable in the positive octant the effects of strong prey becoming weak prey is studied and it is shown that it has no influence on the persistent property of the system and can enhance the co existence of the prey predator

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

stable galerkin reduced order models for linearized compressible flow the galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations the numerical stability of galerkin models is shown to depend on the choice of inner product for the projection for the linearized euler equations a symmetry transformation leads to a stable formulation for the inner product boundary conditions for compressible flow that preserve stability of the reduced order model are constructed preservation of stability for the discrete implementation of the galerkin projection is made possible using a piecewise smooth finite element basis stability of the reduced order model using this approach is demonstrated on several model problems where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation

numerical analysis of the leray reduced order model abstract standard roms generally yield spurious numerical oscillations in the simulation of convection dominated flows regularized roms use explicit rom spatial filtering to decrease these spurious numerical oscillations the leray rom is a recently introduced regularized rom that utilizes explicit rom spatial filtering of the convective term in the navier stokes equations this paper presents the numerical analysis of the finite element discretization of the leray rom error estimates for the rom differential filter which is the explicit rom spatial filter used in the leray rom are proved these rom filtering error estimates are then used to prove error estimates for the leray rom finally both the rom filtering error estimates and the leray rom error estimates are numerically investigated in the simulation of the two dimensional navier stokes equations with an analytic solution

reduction of nonlinear embedded boundary models for problems with evolving interfaces embedded boundary methods alleviate many computational challenges including those associated with meshing complex geometries and solving problems with evolving domains and interfaces developing model reduction methods for computational frameworks based on such methods seems however to be challenging indeed most popular model reduction techniques are projection based and rely on basis functions obtained from the compression of simulation snapshots in a traditional interface fitted computational framework the computation of such basis functions is straightforward primarily because the computational domain does not contain in this case a fictitious region this is not the case however for an embedded computational framework because the computational domain typically contains in this case both real and ghost regions whose definitions complicate the collection and compression of simulation snapshots the problem is exacerbated when the interface separating both regions evolves in time this paper addresses this issue by formulating the snapshot compression problem as a weighted low rank approximation problem where the binary weighting identifies the evolving component of the individual simulation snapshots the proposed approach is application independent and therefore comprehensive it is successfully demonstrated for the model reduction of several two dimensional vortex dominated fluid structure interaction problems

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

optimal control for holonomic and nonholonomic mechanical systems with symmetry and lagrangian reduction in this paper we establish necessary conditions for optimal control using the ideas of lagrangian reduction in the sense of reduction under a symmetry group the techniques developed here are designed for lagrangian mechanical control systems with symmetry the benefit of such an approach is that it makes use of the special structure of the system especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems lagrangian reduction can do in one step what one can alternatively do by applying the pontryagin maximum principle followed by an application of poisson reduction the idea of using lagrangian reduction in the sense of symmetry reduction was also obtained by bloch and crouch proc 33rd cdc ieee 1994 pp 2584 2590 in a somewhat different context and the general idea is closely related to those in montgomery comm math phys 128 1990 pp 565 592 and vershik and gershkovich dynamical systems vii v arnold and s p novikov eds springer verlag 1994 here we develop this idea further and apply it to some known examples such as optimal control on lie groups and principal bundles such as the ball and plate problem and reorientation examples with zero angular momentum such as the satellite with moveable masses however one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of bloch krishnaprasad marsden and murray arch rational mech anal 1996 to appear the snakeboard is used to illustrate the method

minimal subspace rotation on the stiefel manifold for stabilization and enhancement of projection based reduced order models for the compressible navier stokes equations for a projection based reduced order model rom of a fluid flow to be stable and accurate the dynamics of the truncated subspace must be taken into account this paper proposes an approach for stabilizing and enhancing projection based fluid roms in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace attention is focused on the full non linear compressible navier stokes equations in specific volume form as a step toward a more general formulation for problems with generic non linearities unlike traditional approaches no empirical turbulence modeling terms are required and consistency between the rom and the navier stokes equation from which the rom is derived is maintained mathematically the approach is formulated as a trace minimization problem on the stiefel manifold the reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems including a problem involving laminar flow over an airfoil with a high angle of attack and a channel driven cavity flow problem

steady states of a predator prey model with prey taxis abstract we study the steady states of a predator prey model with prey taxis incorporating holling type ii functional response under the homogeneous neumann boundary condition the stability of equilibrium points and the existence of non constant steady states are investigated we obtain that the prey tactic sensitivity coefficient delays the stability of the unique positive constant solution but for other equilibrium points stability the prey tactic sensitivity coefficient does not influence on it furthermore we derive some sufficient conditions relative to the prey tactic sensitivity coefficient which confines the existence of steady states and find that even if the interaction coefficient is sufficiently large there also exist non constant positive steady states under some conditions

non constant positive steady states of a predator prey system with non monotonic functional response and diffusion this paper deals with non constant positive steady state solutions of a predator prey system with non monotonic functional response also called holling type iv interaction terms and diffusion under the homogeneous neumann boundary condition we first establish positive upper and lower bounds for such solutions and then study their non existence global existence and bifurcation

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

cotangent bundle reduction and poincare birkhoff normal forms in this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free lie group action the canonical coordinates enable us to compute poincare birkhoff normal forms of relative equilibria using standard algorithms the case of simple mechanical systems with symmetries is studied in detail as examples we compute poincare birkhoff normal forms for a lagrangian equilateral triangle configuration of a three body system with a morse type potential and the stretched out configuration of a double spherical pendulum

a matrix inequality associated with bounds on solutions of algebraic riccati and lyapunov equations a new proof is presented for the inequality tr xy leq parallel x parallel 2 cdot tr y this argument is valid under the condition that y be real symmetric nonnegative definite x may be any square matrix

bifurcation of steady state solutions in predator prey and competition systems we discuss steady state solutions of systems of semilinear reaction diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region it is easy to find solutions to the systems such that either u or v is identically zero such solutions correspond to the case where one of the species is extinct by using decoupling and global bifurcation theory techniques we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co exist

preserving lagrangian structure in nonlinear model reduction with application to structural dynamics this work proposes a model reduction methodology that preserves lagrangian structure and achieves computational efficiency in the presence of high order nonlinearities and arbitrary parameter dependence as such the resulting reduced order model retains key properties such as energy conservation and symplectic time evolution maps we focus on parameterized simple mechanical systems subjected to rayleigh damping and external forces and consider an application to nonlinear structural dynamics to preserve structure the method first approximates the system s lagrangian ingredients the riemannian metric the potential energy function the dissipation function and the external force and subsequently derives reduced order equations of motion by applying the forced euler lagrange equation with these quantities from the algebraic perspective key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness matri

reliable reduced order models for time dependent linearized euler equations development of optimal reduced order models for linearized euler equations is investigated recent methods based on proper orthogonal decomposition pod applicable for high order systems are presented and compared particular attention is paid to the link between the choice of the projection and the efficiency of the reduced model a stabilizing projection is introduced to induce a stable reduced order model at finite time even if the energy of the physical model is growing the proposed method is particularly well adapted for time dependent hyperbolic systems and intrinsically skew symmetric models this paper also provides a common methodology to reliably reduce very large nonsymmetric physical problems

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

on non selective harvesting of a multispecies fishery incorporating partial closure for the populations highlightswe describe a prey predator fishery model incorporating partial closure for the populations the criterion for the existence of interior equilibrium point local and global stability of the system are derived the existence of bionomic equilibrium is discussed the optimal system is derived and solved numerically using an iterative method with runge kutta fourth order scheme numerical simulations are carried out to verify the analytically proved results a prey predator type fishery model incorporating partial closure for the populations is described in this paper the proposed model deals with a problem of non selective harvesting of a prey predator system in which both the prey and the predator species obey logistic law of growth the predator dependent beddington deangelis type functional response is taken into consideration dynamic behavior of the system is analyzed partial closure for the populations is considered as a controlling instrument to regulate the harvesting of the populations a dynamic framework towards the optimal utilization of the resource is developed using pontryagin s maximum principle the optimal system is numerically solved using an iterative method with runge kutta fourth order scheme simulation results show that the optimal control scheme can achieve sustainable ecosystem results are analyzed with the help of graphical illustrations

s shaped global bifurcation curve and hopf bifurcation of positive solutions to a predator prey model where d is a bounded domain in r n 1 with smooth boundary d and a b c d m are positive constants problem 1 0 is known as the predator prey model with holling tanner type interactions where u and v represent the densities of the prey and predator respectively hence we are only interested in positive solutions by which we mean that both u and v are positive in d it is easy to see that a 1 is necessary for the existence of positive solutions to 1 0 and we shall also assume that d 1 the case d 1 is rather different and will not be discussed here we are mainly interested in studying the number of positive solutions of 1 0 and the stability of these solutions in particular we shall show that when b d c and m fall into certain range the solution set u v a of 1 0 forms a s shaped smooth curve and that hopf bifurcation occurs along this curve this not only confirms rigorously similar numerical observations on 1 0 made in 4 but also shows that the corresponding parabolic system

effect of prey refuge on spatiotemporal dynamics of the reaction diffusion system abstract the present investigation deals with the problem of a two dimensional diffusive predator prey model incorporating prey refuge in this investigation we determine the turing space in the spatial domain and perform extensive numerical simulation from both the mathematical and the biological points of view in order to study the effect of diffusion coefficients and other momentous parameters of the system in turing instability furthermore the stability behaviours of the proposed model in the absence of diffusion about the feasible equilibrium points are explored in particular the exact turing domain is delineated in the parameter space various spatial patterns in spatial domain through diffusion driven instability of the present model are depicted and analysed the results indicate that the effect of prey refuge plays a significant role on the control of pattern formation of the populations

stabilization of projection based reduced order models summary a rigorous method for stabilizing projection based linear reduced order models without significantly affecting their accuracy is proposed unlike alternative approaches this method is computationally efficient it requires primarily the solution of a small scale convex optimization problem furthermore it is nonintrusive in the sense that it operates directly on readily available reduced order operators these can be precomputed using any data compression technique including balanced truncation balanced proper orthogonal decomposition proper orthogonal decomposition or moment matching the proposed method is illustrated with three applications the stabilization of the reduction of the computational fluid dynamics based model of a linearized unsteady supersonic flow the reduction of a computational structural dynamics system and the stabilization of the reduction of a coupled computational fluid dynamics computational structural dynamics model of a linearized aeroelastic system in the transonic flow regime copyright 2012 john wiley sons ltd

energy balance and mass conservation in reduced order models of fluid flows abstract in this paper we investigate theoretically and computationally the conservation properties of reduced order models roms for fluid flows specifically we investigate whether the roms satisfy the same or similar energy balance and mass conservation as those satisfied by the navier stokes equations all of our theoretical findings are illustrated and tested in numerical simulations of a 2d flow past a circular cylinder at a reynolds number r e 100 first we investigate the rom energy balance we show that using the snapshot average for the centering trajectory which is a popular treatment of nonhomogeneous boundary conditions in roms yields an incorrect energy balance then we propose a new approach in which we replace the snapshot average with the stokes extension theoretically the stokes extension produces an accurate energy balance numerically the stokes extension yields more accurate results than the standard snapshot average especially for longer time intervals our second contribution centers around rom mass conservation we consider roms created using two types of finite elements the standard taylor hood th element which satisfies the mass conservation weakly and the scott vogelius sv element which satisfies the mass conservation pointwise theoretically the error estimates for the sv rom are sharper than those for the th rom numerically the sv rom yields significantly more accurate results especially for coarser meshes and longer time intervals

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

stratified reduction of classical many body systems with symmetry the centre of mass system for n bodies admits a natural so 3 action and thereby is stratified into strata according to the orbit types for the so 3 action the principal stratum consists of non singular configurations for which the isotropy subgroup is trivial so that it is made into an so 3 principal fibre bundle the strata of lower dimension consist of singular configurations at each of which the isotropy subgroup is not trivial practically singular configurations are collinear ones and simultaneous multiple collision classical lagrangian and hamiltonian systems are defined on the tangent and the cotangent bundles over each stratum respectively the euler lagrange and the hamilton equations for the n body system are derived on the variational principle according to the stratification of the centre of mass system the reduction procedure will be accordingly performed for the lagrangian and the hamiltonian systems with symmetry respectively by the rotational symmetry the euler lagrange and the hamilton equations are reduced to those defined on reduced bundles from the tangent and the cotangent bundles respectively

qualitative analysis of a predator prey model with holling type ii functional response incorporating a prey refuge abstract we study a predator prey model with holling type ii functional response incorporating a prey refuge under homogeneous neumann boundary condition we show the existence and non existence of non constant positive steady state solutions depending on the constant m 0 1 which provides a condition for protecting 1 m u of prey u from predation moreover we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions

cross diffusion induced instability and pattern formation for a holling type ii predator prey model this paper is devoted to studying a predator prey model with holling type ii functional response and cross diffusion subject to neumann boundary condition our main interest lies in the effects of cross diffusion on stability and stationary patterns more precisely the presented results show that cross diffusion can not only destabilize a uniform equilibrium which is stable for the kinetic and random diffusion reaction systems but also create spatial patterns even when the random diffusion fails to do so furthermore our results also reveal that in this kind of ecological system instability and stationary patterns can appear only when the predators rapidly move away from a large group of preys regardless of the speed that the preys keep away from the predators

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

the mechanics and control for multi particle systems this paper deals with the mechanics and control for multi particle systems from a geometric point of view the centre of mass system is viewed as a principal fibre bundle with structure group so 3 the base space of which is called the internal or shape space a natural connection and a natural riemannian metric are both defined on the centre of mass system the equations of motion for the multi particle system are derived in the lagrangian formalism adapted to the bundle structure and then reduced with the conserved total angular momentum in contrast with this the control problem is studied with non holonomic constraints i e with the vanishing total angular momentum and equations of motion are determined for an optimally controlled multi particle system the resultant equations derived in each of the mechanical and control systems are to be compared

impact of the fear effect in a prey predator model incorporating a prey refuge abstract in this paper we investigate the influence of anti predator behaviour due to the fear of predators with a holling type ii prey predator model incorporating a prey refuge we first provide the existence and stability of equilibria of the model next we give the existence of hopf bifurcation and limit cycle in addition we study the impact of the fear effect on the model analytically and numerically and find that the fear effect can not only reduce the population density of predator at the positive equilibrium but also stabilize the system by excluding the existence of periodic solutions moreover we also find that prey refuge has great impact on the persistence of the predator

model reduction for compressible flows using pod and galerkin projection abstract we present a framework for applying the method of proper orthogonal decomposition pod and galerkin projection to compressible fluids for incompressible flows only the kinematic variables are important and the techniques are well known in a compressible flow both the kinematic and thermodynamic variables are dynamically important and must be included in the configuration space we introduce an energy based inner product which may be used to obtain pod modes for this configuration space we then obtain an approximate version of the navier stokes equations valid for cold flows at moderate mach number and project these equations onto a pod basis the resulting equations of motion are quadratic and are much simpler than projections of the full compressible navier stokes equations

proper orthogonal decomposition closure models for turbulent flows a numerical comparison this paper puts forth two new closure models for the proper orthogonal decomposition reduced order modeling of structurally dominated turbulent flows the dynamic subgrid scale model and the variational multiscale model these models which are considered state of the art in large eddy simulation together with the mixing length and the smagorinsky closure models are tested in the numerical simulation of 3d turbulent flow past a circular cylinder at re 1000 five criteria are used to judge the performance of the proper orthogonal decomposition reduced order models the kinetic energy spectrum the mean velocity the reynolds stresses the root mean square values of the velocity fluctuations and the time evolution of the pod coefficients all the numerical results are benchmarked against a direct numerical simulation based on these numerical results we conclude that the dynamic subgrid scale and the variational multiscale models are the most accurate

stability analysis of diffusive predator prey model with modified leslie gower and holling type iii schemes abstract the stability of a diffusive predator prey model with modified leslie gower and holling type iii schemes is investigated a threshold property of the local stability is obtained for a boundary steady state and sufficient conditions of local stability and un stability for the positive steady state are also obtained furthermore the global asymptotic stability of these two steady states are discussed our results reveal the dynamics of this model system

reduction of quantum systems on riemannian manifolds with symmetry and application to molecular mechanics this paper deals with a general method for the reduction of quantum systems with symmetry for a riemannian manifold m admitting a compact lie group g as an isometry group the quotient space q m g is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions if the action of the compact group g is free m is made into a principal fiber bundle with structure group g in this case reduced quantum systems are set up as quantum systems on the associated vector bundles over q m g this idea of reduction fails if the action of g on m is not free however the peter weyl theorem works well for reducing quantum systems on m when applied to the space of wave functions on m the peter weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on m which are interpreted as hilbert spaces for reduced quantum systems on q the concept of connection on a principal fiber bundle is generalized to be defined well on the

non existence of non constant positive steady states of two holling type ii predator prey systems strong interaction case abstract we prove the non existence of non constant positive steady state solutions of two reaction diffusion predator prey models with holling type ii functional response when the interaction between the predator and the prey is strong the result implies that the global bifurcating branches of steady state solutions are bounded loops

asymptotic behavior and multiplicity for a diffusive leslie gower predator prey system with crowley martin functional response abstract a diffusive predator prey system with modified leslie gower and crowley martin functional response is considered the extinction and permanence of the time dependent system are determined by virtue of the comparison principle then the sufficient and necessary conditions for the existence of coexistence states are obtained furthermore the stability uniqueness and exact multiplicity of coexistence states are investigated by means of the combination of the perturbation theory bifurcation theory and degree theory our results indicate that c 1 have an effect on the stability and exact multiplicity of coexistence states finally some numerical simulations are presented to verify and complement the theoretical results

stabilization of projection based reduced order models for linear time invariant systems via optimization based eigenvalue reassignment abstract a new approach for stabilizing unstable reduced order models roms for linear time invariant lti systems through an a posteriori post processing step applied to the algebraic rom system is developed the key idea is to modify the unstable eigenvalues of the rom system by moving these eigenvalues into the stable half of the complex plane it is demonstrated that this modification to the rom system eigenvalues can be accomplished using full state feedback a k a pole placement algorithms from control theory this approach ensures that the modified rom is stable provided the system s unstable poles are controllable and observable however the accuracy of the stabilized rom is not guaranteed to remedy this difficulty and guarantee an accurate stabilized rom a constrained nonlinear least squares optimization problem for the stabilized rom eigenvalues in which the error in the rom output is minimized is formulated this optimization problem is small and therefore computationally inexpensive to solve performance of the proposed algorithms is evaluated on two test cases for which roms constructed via the proper orthogonal decomposition pod galerkin method suffer from instabilities

on minimization on stiefel manifolds abstract the minimization of a smooth function f r kn r under the constraint that vectors x 1 x 2 x k r n k n form an orthonormal system seems to be a new and interesting global optimization problem with important theoretical and practical applications the set of feasible points determines a differentiable manifold introduced by stiefel in 1935 based on the nice geometric structure the optimality conditions are obtained by the global lagrange multiplier rule and global optimality conditions based on local information which make the advantages of using the riemannian geometry in difficult smooth optimization problems clear

entropy stable reduced order modeling of nonlinear conservation laws abstract reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model we introduce projection based hyper reduced models of nonlinear conservation laws which are globally conservative and inherit a semi discrete entropy inequality independently of the choice of basis and choice of parameters

stratified reduction of many body kinetic energy operators the center of mass system of many bodies admits a natural action of the rotation group so 3 according to the orbit types for the so 3 action the center of mass system is stratified into three types of strata the principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either so 2 or so 3 depending on whether the isotropy subgroup is isomorphic with so 2 or so 3 the stratum in question consists of collinear configurations or of a single configuration of the multiple collision it is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision the energy operator for nonsingular configurations has singularity at singular configurations however the singularity is not essential in the sense that both of the rotational and vibrational energy integrals have a finite value this can be proved by using the boundary conditions of wave functions at singular configurations for three body systems for simplicity it is shown in addition that the energy operator for collinear configurations has also singularity at the multiple collision but the singularity is not essential either in the sense that the kinetic energy integral is not divergent at the multiple collision reduction procedure is applied to the respective energy operators for the nonsingular and the collinear configurations to obtain respective reduced operators both of which are expressed in terms of internal coordinates

structure preserving galerkin pod reduced order modeling of hamiltonian systems abstract the proper orthogonal decomposition reduced order model pod rom has been widely used as a computationally efficient surrogate model in large scale numerical simulations of complex systems however when it is applied to a hamiltonian system a naive application of the pod method can destroy the hamiltonian structure in the reduced order model in this paper we develop a new reduced order modeling approach for hamiltonian systems which modifies the galerkin projection based pod rom so that the appropriate hamiltonian structure is preserved since the pod truncation can degrade the approximation of the hamiltonian function we propose to use a pod basis from shifted snapshots to improve the approximation to the hamiltonian function we further derive a rigorous a priori error estimate for the structure preserving rom and demonstrate its effectiveness in several numerical examples this approach can be readily extended to dissipative hamiltonian systems port hamiltonian systems etc

reduction of quantum systems with symmetry continuous and discrete reduction of dynamical systems is closely related with symmetry the purpose of this article is to show that fourier analysis both on compact lie groups and on finite groups serves as a reduction procedure for quantum systems with symmetry on an equal footing the reduction procedure is applied to systems of many identical particles lying in r3 which admit the action of a rotation group so 3 and of a symmetric or permutation group

optimality for a diffusive predator prey system in a spatially heterogeneous environment incorporating a prey refuge abstract in this paper we consider the optimal harvesting controls for a diffusive predator prey system which is formulated by prey refuge and modified leslie gower functional response the locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous neumann boundary condition the nonexistence and existence of nonconstant steady states are obtained by leray schauder degree theory also the optimal harvesting strategies to get maximum sustainable yield msy and maximum economic yield mey are correspondingly given

stability and bifurcation of a ratio dependent prey predator system with cross diffusion abstract this paper is purported to investigate a ratio dependent prey predator system with cross diffusion in a bounded domain under no flux boundary condition the asymptotical stabilities of nonnegative constant solutions are investigated to this system moreover without estimating the lower bounds of positive solutions the existence multiplicity of positive steady states are considered by using fixed points index theory bifurcation theory energy estimates and the differential method of implicit function and inverse function

analysis of a model predator prey system with refuges abstract we study a model of a predator prey system with refuges proposed in mcnair theoret population biol 29 1986 38 63 we show that all solutions are ultimately bounded and eventually enter a triangular region in the first quadrant moreover two hopf bifurcations occur and using singularity theory one can show that the two branches of periodic solutions join finally we show that the periodic solutions produced by the hopf bifurcation are unique and hence limit cycles which are globally asymptotically stable from the first quadrant with an equilibrium point deleted

stratified dynamical systems and their boundary behaviour for three bodies in space with insight into small vibrations the centre of mass system for many particles is stratified into strata by the rotation group action the principal stratum consists of nonlinear configurations the collinear configurations form a lower dimensional stratum classical mechanics for many particles with nonlinear configurations and for those with collinear configurations are set up on the tangent or cotangent bundles over respective strata and can be reduced by the use of rotational symmetry a question arises as to how a many body system behaves in a neighbourhood of a collinear configuration the system may make a vibration to bend its collinear configuration which is a motion taking place across the boundary of the principal stratum this paper deals with the behaviour of those boundaries for three bodies in space the equations of motion for small vibrations as boundary behaviour at a collinear configuration will be given as a limit of those equations of motion for nonlinear configurations by means of two key facts that the isotropy subgroup may act non trivially on the tangent space at the collinear configuration and that vibrations take place in a constant plane in space further the perturbation of small vibrations is studied by applying moser s averaging method to show that small vibrations may give rise to finite rotations after a sufficiently large number of periods

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

existence of positive solutions for a prey predator model with refuge and diffusion abstract we are concerned with a prey predator reaction diffusion model with monotonic functional response and specific refuge size we discuss both existence and nonexistence of positive solutions of the model by using fixed point index theory and bifurcation theory as the main argument tools we also discuss bifurcation solutions of which stability is established by using spectrum analysis methods moreover we analyze the effects of refuge size on the dynamics of the model

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

nontrivial solutions of predator prey systems with small diffusion on considere le systeme elliptique non lineaire u x au bu 2 cuv pour xed d v x ev fv 2 guv d region bornee de r n on etudie comment la famille des solutions non negatives de ce systeme change quand les parametres du systeme changent

global bifurcation analysis and pattern formation in homogeneous diffusive predator prey systems abstract the dynamics of a general diffusive predator prey system is considered existence and nonexistence of non constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non constant steady state solutions are studied

coexistence theorems of steady states for predator prey interacting systems on donne des conditions necessaires et suffisantes pour l existence des solutions positives des etats stationnaires pour des systemes predateur proie sous des conditions aux limites de dirichlet sur r n

bifurcation branch of stationary solutions in a general predator prey system with prey taxis abstract in this paper a general reaction diffusive predator prey system with prey taxis subject to the homogeneous neumann boundary condition is considered firstly we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique and then choosing the prey tactic sensitivity coefficient as bifurcation parameter we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey tactic sensitivity is repulsive moreover we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions our results show that prey taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns

manopt a matlab toolbox for optimization on manifolds optimization on manifolds is a rapidly developing branch of nonlinear optimization its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms in particular optimization on manifolds is well suited to deal with rank and orthogonality constraints such structured constraints appear pervasively in machine learning applications including low rank matrix completion sensor network localization camera network registration independent component analysis metric learning dimensionality reduction and so on the manopt toolbox available at www manopt org is a user friendly documented piece of software dedicated to simplify experimenting with state of the art riemannian optimization algorithms by dealing internally with most of the differential geometry the package aims particularly at lowering the entrance barrier

rotational vibrational energy spectra of triatomic molecules near relative equilibria complete hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry j math phys 44 4411 2003 in this article the full hamiltonian is specialized for a rigid and for a semirigid molecule for the rigid molecule all internal coordinates are fixed at constants so that the hamiltonian operator comes to take an ordinary matrix form and accordingly the schrodinger equation becomes an algebraic eigenvalue equation the eigenvalues then provide rotational energy spectra of the rigid molecule for the semirigid molecule the full hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter indeed the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondege