perturbed kepler problem in quaternionic form a new technique for solving the perturbed kepler problem is presented as its approach is essentially algebraic it can be easily computerized and carried out to any perturbation order it also eliminates the notorious difficulty of small divisors and is well suited to deal with commensurable orbital periods as demonstrated by its explanation of kirkwood gaps

on the palais smale condition abstract we present a new sufficient assumption weaker than the classical ambrosetti rabinowitz condition which guarantees the boundedness of ps sequences moreover we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded ps sequences we also revise the costa magalhaes condition 8 to obtain cerami condition as a consequence some existence results derived by minimax methods were proved finally we establish the existence of positive solution under the subcritical polynomial growth condition while the strong superlinear condition is only required along an unbounded sequence in other words a certain degraded oscillation is allowed

chaos synchronization in gap junction coupled neurons depending on temperature the modified hodgkin huxley mhh equations exhibit a variety of dynamical behaviors including intrinsic chaotic firing we analyze synchronization in a large ensemble of mhh neurons that are interconnected with gap junctions by evaluating tangential lyapunov exponents we clarify whether the synchronous state of neurons is chaotic or periodic then we evaluate transversal lyapunov exponents to elucidate if this synchronous state is stable against infinitesimal perturbations our analysis elucidates that with weak gap junctions the stability of the synchronization of mhh neurons shows rather complicated changes with temperature we however find that with strong gap junctions the synchronous state is stable over the wide range of temperature irrespective of whether synchronous state is chaotic or periodic it turns out that strong gap junctions realize the robust synchronization mechanism which well explains synchronization in interneurons in the real nervous system

information cascade with marginal stability in a network of chaotic elements abstract a newly discovered cascade process of clusters is studied in a network of chaotic elements it is shown that the splitting of clusters and the synchronization of elements are balanced in a class of partially ordered states where marginal stability is sustained over an interval of the bifurcation parameters the partition information creation in bit space shows an avalanche process of information which leads to the anomalous behavior of power spectra roughly fitted by a power law form of the wavenumber lyapunov spectra have accumulation at null exponents analogous with those studied in fluid turbulence models

the flow of the n body problem near a simultaneous binary collision singularity and integrals of motion on the collision manifold we study the flow for the n body problem near a simultaneous binarycollision singularity we blow up the singularity to an invariant collision manifold we show the existence of integrals of motion on the collision manifold we integrate the flow on the collision manifold explicitly in terms of elementary functions we give a complete description of the flow near the singularity

on the c8 3 regularisation of simultaneous binary collisions in the collinear 4 body problem abstract a conjecture on the nature of the simultaneous binary collision is explored in the collinear 4 body problem it is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than c 8 3 differentiable with respect to initial conditions through a blow up of the singularity this loss of differentiability is investigated and a new proof of the c 8 3 regularity is provided in the process it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics by utilising recent work on transitions near such objects and their normal forms an asymptotic series of the transition past the singularity is explicitly computed it becomes remarkably apparent that the finite differentiability at 8 3 is due to the inability to construct a set of integrals local to the simultaneous binary collision the finite differentiability is shown to be independent of the value of the masses

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

cluster splitting bifurcation in a system of coupled maps abstract we consider cluster splitting bifurcations in a system of globally coupled maps as coupling parameter decreases at these transitions the number of clusters i e groups of elements with identical dynamics increases we demonstrate that different cascades of cluster splitting can occur depending on statistics of redistribution of the oscillators between new born clusters

spatiotemporal chaos in a coupled map lattice with unstable couplings we study the complex spatiotemporal behaviour of a coupled map lattice with a one humped chaotic map and an unstable laplacian coupling bifurcations are numerically investigated and interpreted using low dimensional approximations corresponding to the relevant degrees of freedom of the infinite dimensional system varying the control parameter we find different phases in the chaotic domain such as localized chaos or propagating chaos spatiotemporal intermittency and transient chaos according to our results unstable coupling leads to a number of new features including the effects that i the first bifurcation becomes discontinuous and ii the chaotic regime sets in sooner than for a single map

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

multiple solutions for a class of nonlinear sturm liouville problems when nonlinearities are not odd abstract this article concerns a sturm liouville problem whose nonlinearity is not odd a variational method is used to obtain the existence of infinitely many solutions which are distinguished by their nodal properties however an example shows some nodal classes can have no solutions while others do

multiple critical points of invariant functionals and applications abstract this paper deals with some multiplicity results of periodic orbits of hamiltonian systems and for solution of a non linear dirichlet problem these results follow from an abstract theorem of lusternik schnirelman type as applied to an invariant equation of the form lu delta f u 0 in a hilbert space x l sub 2 omega r sub n where l is an unbounded self adjoint operator and f is a c sub 1 strictly convex function author

symplectic regularization of binary collisions in the circular n 2 sitnikov problem we present a brief overview of the regularizing transformations of the kepler problem and we relate the euler transformation with the symplectic structure of the phase space of the n body problem we show that any particular solution of the n body problem where two bodies have rectilinear dynamics can be regularized by a linear symplectic transformation and the inclusion of the euler transformation into the group of symplectic local diffeomorphisms over the phase space as an application we regularize a particular configuration of the restricted circular n 2 body problem

multiple solutions for a class of nonlinear sturm liouville problems on the half line abstract a class of nonlinear sturm liouville problems is considered these problems admit zero as a trivial solution variational techniques and approximation arguments are used to obtain the existence of nontrivial solutions with any prescribed number of nodes results are also obtained in higher dimensions via investigation of the set of radial solutions of appropriate partial differential equations

the multiscale character of evoked cortical activity abstract both the architecture and the dynamics of the brain have characteristic features at different spatial scales however the existence nature and function of dynamical interdependencies between such scales have not been investigated we studied the multiscale properties of functional magnetic resonance imaging fmri data acquired while human subjects viewed a visual image traditional region of interest analysis of this data set revealed evoked activity in primary and extrastriate visual cortex wavelet transform in the spatial domain provides a multiscale representation of this evoked brain activity studying the correlation structure of this representation revealed strong and novel interdependencies in these data within and between different spatial scales we found that such correlations are stronger than those evident in the original data and comparable in magnitude to those obtained after gaussian smoothing however analysis of the data in the wavelet domain revealed additional structure such as positive correlations strong anti correlations and phase lagged interdependencies statistical significance of these effects was inferred through nonparametric bootstrap techniques we conclude that the spatial analysis of functional neuroimaging data in the wavelet domain provides novel information which may reflect complex spatiotemporal neuronal activity and information encoding it also affords a quantitative means of testing hierarchical and multiscale models of cortical activity

preservation of the maslov index along bifurcating branches of solutions of first order systems in rn abstract in this paper we study bifurcation from simple eigenvalues for systems of differential equations we prove the existence of global bifurcating branches of solutions on which the maslov index of suitable associated linear systems is preserved

multiple solutions of semilinear elliptic problems in a ball in this paper lower bounds for the number of solutions of semilinear elliptic problems in a ball of rn are given its hypotheses are only related to the behavior of the nonlinearities at and at 0 global assumptions are never made for example oddness is never required for the proof of multiplicity results

remarks on the mean field dynamics of networks of chaotic elements abstract fluctuations of the mean field of a globally coupled dynamical systems are discussed the origin of hidden coherence is related with the instability of the fixed point solution of the self consistent perron frobenius equation collective dynamics in globally coupled tent maps are re examined both with the help of direct simulation and the perron frobenius equation collective chaos in a single band state and bifurcation against initial conditions in a two band state are clarified with the return maps of the mean field lyapunov spectra and also the newly introduced lyapunov exponent for the perron frobenius equation future problems on the collective dynamics are discussed

globally coupled circle maps abstract a network of circle maps is investigated successive transitions are found among coherent ordered partially ordered po and turbulent phases in the ordered phase two cluster attractors with similarity have a large basin volume which are reduced to a unique circle map through a scaling transformation with two parameters the po phase is characterized by the increase of partition complexity in the phase chaotic itinerancy is observed whose dynamics consists of quasistationary high dimensional chaotic motion low dimensional attractor ruins and switching among them dependence of the diffusion constant on the size and the nonlinearity clearly deviates from that by a random phase approximation suggesting remaining correlation among elements even in the turbulent phase

cluster synchronization in an ensemble of neurons interacting through chemical synapses in networks of periodically firing spiking neurons that are interconnected with chemical synapses we analyze a cluster state where an ensemble of neurons are subdivided into a few clusters in each of which neurons exhibit perfect synchronization to clarify stability of the cluster state we decompose linear stability of the solution into two types of stabilities stability of a mean state and stabilities of clusters computing floquet matrices for these stabilities we clarify the total stability of the cluster state for any type of neuron and any strength of interaction even if the size of networks is infinitely large first we apply this stability analysis to investigating synchronization in the large ensemble of integrate and fire if neurons in one cluster state we find the change of stability of a cluster which elucidates that in phase synchronization of if neurons occurs with only inhibitory synapses then we investigate entrainment of two clusters of if neurons with different excitability if neurons with fast decaying synapses show low entrainment capability which is explained by a pitchfork bifurcation appearing in the two cluster state with change of synapse decay time constant second we analyze a one cluster state of hodgkin huxley hh neurons and discuss the difference in synchronization properties between if neurons and hh neurons

multiplicity results for a superlinear dirichlet problem on considere le probleme de dirichlet v f v v t0 1 ou t r v x max v x o et b b 1 o est une boule ouverte unite dans r n n 2 ici 0 1 est une valeur propre positive correspondant a la premiere valeur propre 1 du probleme r v v n 1 r v v ou 0r1 v 0 v 1 0

existence of radial solutions with prescribed number of zeros for elliptic equations and their morse index abstract in this paper we consider radially symmetric solutions of the nonlinear dirichlet problem u f x u 0 in where is a ball in r n n 3 and f satisfies some appropriate assumptions we prove existence of radially symmetric solutions with k prescribed number of zeros moreover when f x u k x u p 1 u using the uniqueness result due to tanaka 2008 21 we verify that these solutions are non degenerate and we prove that their radial morse index is exactly k

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

chaotic itinerancy and power law residence time distribution in stochastic dynamical systems chaotic itinerant motion among varieties of ordered states is described by a stochastic model based on the mechanism of chaotic itinerancy the model consists of a random walk on a half line and a markov chain with a transition probability matrix the stability of attractor ruin in the model is investigated by analyzing the residence time distribution of orbits at attractor ruins it is shown that the residence time distribution averaged over all attractor ruins can be described by the superposition of truncated power law distributions if the basin of attraction for each attractor ruin has a zero measure this result is confirmed by simulation of models exhibiting chaotic itinerancy chaotic itinerancy is also shown to be absent in coupled milnor attractor systems if the transition probability among attractor ruins can be represented as a markov chain

existence of multiple solutions for quasilinear and semilinear elliptic equations has been studied by many authors cf l 3 4 161 under the assumption that g r r is a continuous function such that g t t closes finite many eigenvalues when t varies from 0 to a the existence of one nontrivial solution for 1 2 was proved by amann and zehnder 3 and thews 21 the existence of two nontrivial solutions was studied by castro and lazer lo and ambrosetti and mancini 4 most of the multiplicity results for problem 1 2 are based on the fact that 1 2 can be rewritten as

collision ejection manifold and collective analytic continuation of simultaneous binary collisions in the planarn body problem abstract we study simultaneous binary collision sbc singularities oflbinaries in the planarn body problem 2l n we introduce the generalized levi civita transformation and follow it by a new transformation which we call the projective transformation near a sbc singularity we use this transformation to show the following near a sbc singularity 1 in the generalized levi civita variables near a sbc singularity the collection of collision and ejection orbits together with the singularity form a real analytic submanifold which we call the collision ejection ce manifold 2 let r c r e be the collection of sbc sbe orbits in phase space i e in the original variables then both r cand r eare real analytic 3 each collision orbit corresponds to a unique ejection orbit together they form a real analytic orbit in the generalized levi civita variables which we call a collision ejection orbit 4 let c r c e r e be a submanifold of initial conditions that end start in a simultaneous binary collision ejection singularity we show that c e can be chosen to be a real analytic submanifold of codimension 1 in r c r e and that the correspondence in item 3 above defines a real analytic section mapping from c to e 5 collision and ejection orbits can be collectively analytically continued i e each collision ejection orbit can be written as a convergent power series int1 3 with coefficients that depend real analytically on the initial conditions in c 6 sbc orbits ofj 7 a single binary collision singularity in then body problem is real analytic block regularizable we also give the asymptotic behaviour of collision and ejection orbits

partition complexity in a network of chaotic elements a network of chaotic elements is investigated with the use of globally coupled maps elements can split into clusters with synchronized oscillation in a partially ordered phase the clustering has an inhomogeneous tree structure like replica symmetry breaking in spin glass the clustering has large variety by attractors and strongly depends on initial conditions the variety of clusterings is characterized by the distribution of partitions originated in spin glass theory the qualitative similarity and quantitative disagreement of the given attractors with a spin glass are clarified

from collective oscillations to collective chaos in a globally coupled oscillator system abstract various types of collective behaviors are discovered in globally coupled ginzburg landau oscillators when the coupling is sufficiently weak the oscillators are either in complete synchrony or their phases are scattered completely randomly for finite coupling new collective behaviors emerge such as splitting of the population into a small number of clusters or their fusion into a continuous stringlike distribution in the phase plane low dimensional chaotic dynamics arises from the coupled motion of 3 point clusters chaotic motion is also exhibited by fused clusters which is of extremely high dimension possibly proportional to the system size as is implied from its lyapunov analysis in the latter type of chaos the motion of the string is in some cases characterized by repeated stretching and foldings it is argued how this kind of coherent behavior seen on a collective level does not contradict the high dimensionality of the corresponding chaotic attractor

existence of ground states with exponential decay for semi linear elliptic equations in ofrn abstract in this paper we prove the existence of a positive solution tending exponentially to zero at infinity for a class of semi linear elliptic equations with nonconstant coefficients in ofr n

uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball abstract the following dirichlet problem 1 1 u k x f u 0 in b u 0 on b is considered where b x r n x 1 n 2 k c 2 0 1 and k r 0 for 0 r 1 f c 1 r s f s 0 for s 0 assume moreover that f satisfies the following sublinear condition f s s f s for s 0 a sufficient condition is derived for the uniqueness of radial solutions of 1 1 possessing exactly k 1 nodes where k n it is also shown that there exists k c 0 1 such that 1 1 has three radial solutions having exactly one node in the case n 3

the kustaanheimo stiefel map the hopf fibration and the square root map on r3 and r4 abstract we study the kustaanheimo stiefel map ksm from u r 4 0 to x r 3 0 and the principal circle bundle p u x s 1 that it induces we show that the ksm is the appropriate generalization of the squaring map z z 2 z c and not quaternion multiplication in that the ksm induces a principal circle bundle on s 3 s 2 namely the hopf fibration while quaternion squaring is degenerate because the dimension of the fibers is not constant we construct two square root branches from the upper and lower half of r 3 to r 3 x 1 where x 1 is the nonpositive x 1 axis in r 3 and resembles the cut used to define the standard complex square root branches z we glue these two branches together we introduce what we like to call ks cylindrical coordinates with a 2 dimensional axis of rotation we also introduce what we call ks torical and spherical coordinates we use the ks cylindrical coordinates to define the full square root map on an s 1 cover of r 3 given by r 3 s 1 where is an equivalence relation on x 1 s 1

a multiplicity result for a class of strongly indefinite asymptotically linear second order systems we prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with dirichlet boundary conditions the key idea for the proof is to bring together the classical shooting method and the maslov index of the linear hamiltonian systems associated to the asymptotic limits of the given nonlinearity

borsuk ulam theorems for arbitrary s1 actions and applications abstract an s to the first power version of the borsuk ulam theorem is proved for a situation where fix s to the first power may be nontrivial the proof is accomplished with the aid of a new relative index theory applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an s to the first power symmetry minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by hamilton s equations in these arguments one obtains the solutions of the differential equations as critical points of an associated lagrangian by minimaxing the lagrangian over appropriate classes of sets intersection theorems such as are proved in this paper play a crucial role in this process in addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an s to the first power symmetry group

multiple solutions of semilinear dirichlet problems on a domain withs1 group actions abstract let be a bounded domain with a smooth boundary such that there exists an s 1 group of isometric linear transformations on in the present paper we consider the multiple existence of solutions of the problem formula where g c 1 r and h l 2

how fast elements can affect slow dynamics a chain of coupled chaotic elements with different time scales is studied in contrast with the adiabatic approximation we find that correlations between elements are transferred from faster to slower elements when the differences in the time scales of the elements lie within a certain range for such correlations to occur three features are essential strong correlations among the elements a bifurcation in the dynamics of the fastest element by changing its control parameter and cascade propagation of the bifurcation by studying coupled rossler equations we demonstrate that fast elements can affect the dynamics of slow elements when these conditions are satisfied the relevance of our results to biological memory is briefly discussed

on the multiplicity of radial solutions to superlinear dirichlet problems in bounded domains in this paper we are concerned with the existence and multiplicity of nodal solutions to the dirichlet problem associated to the elliptic equation u q x g u 0 in a ball or in an annulus in rn the nonlinearity g has a superlinear and subcritical growth at infinity while the weight function q is nonnegative in 0 1 and strictly positive in some interval r1 r2 0 1 by means of a shooting approach together with a phase plane analysis we are able to prove the existence of infinitely many solutions with prescribed nodal properties

mean field fluctuation of a network of chaotic elements remaining fluctuation and correlation in the large size limit fluctuation of the mean field is studied for a network of chaotic elements with the use of globally coupled maps as the size n is increased the variance of the mean field decreases up to a crossover size nc but then stays constant where the remaining variance is roughly proportional to 1nc the distribution of the mean field approaches a gaussian form up to nc but keeps some deviation for n nc the breakdown of the law of large numbers implies some hidden coherence among elements confirmed by finite mutual information and lyapunov spectra the breakdown is also supported by the instability of the self consistent solution for the perron frobenius operator the anomalous effect of additional noise to our system is also reported results from other globally coupled maps suggest that the law of large numbers is generally broken in a turbulent phase with local stretching and contraction but is satisfied for a map with stretching only

uniqueness and bifurcation for solutions of nonlinear sturm liouville eigenvalue problems a class of nonlinear sturm liouville problems is considered these problems admit zero as a trivial solution and the nonlinear operator linearized about zero has a purely continuous spectrum 0 uniqueness of solutions with a prescribed number of nodes is obtained by a shooting argument moreover combining with the previous existence results we prove there are infinitely many continua of solutions which are distinguished by nodal properties bifurcate from the line of trivial solutions at the lowest point of the continuous spectrum

bifurcation theory and radial solutions to elliptic boundary value problems abstract using basic tools from general bifurcation theory we explain through three well known problems how to establish the existence of multiple radial solutions to elliptic boundary value problems

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

study of a high dimensional chaotic attractor the nature of a very high dimensional chaotic attractor in an infinite dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high dimensional attractor in the phase space we introduce two complementary bases from which the attractor is observed one the lyapunov basis composed of the lyapunov vectors and the another the fourier basis composed of the fourier modes we introduce the exterior subspaces on the basis of the lyapunov vectors and observe the chaotic motion projected onto these exteriors it is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace goes out of the attractor the origin of such a phenomenon is attributed to more fundamental features of our attractor which become manifest when the attractor is observed from the lyapunov basis a counterpart of the phenomenon can be observed also on the fourier basis because there is a statistical one to one correspondence between the lyapunov vectors and the fourier modes in particular a statistical property of the high pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

existence of nontrivial solutions of semilinear elliptic equations soit cr n un domaine borne a frontiere lisse et g r r une application continue telle que g 0 0 on etudie l existence des solutions non triviales du probleme aux valeurs limites de la forme u g u dans u 0

complex spatiotemporal patterns in two lattice models with instability abstract in this paper we consider two lattice models i the discretized version of a simple nonlinear partial differential equation pde with a singular term and ii a coupled map lattice cml with unstable coupling both systems are defined by simple iteration rules involving the nearest neighbour grid points of the corresponding one dimensional lattice we show numerically that these models result in an apparently stochastic spatiotemporal behaviour and exhibit interesting regimes as the function of the parameters of the models in particular we find that by varying the relative weight b of the singular term in the pde the total width of the solution scales with b the cml leads to an intermittent behaviour between structured and turbulent regimes

on multiple solutions of a nonlinear dirichlet problem we prove that a semilinear elliptic boundary value problem has five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues we also prove that if the region is a ball the semilinear elliptic problem has two solutions that change sign and are nonradial