the general properties of discrete time competitive dynamical systems abstract for strongly competitive discrete time dynamical systems on a strongly ordered topological vector space it is proved that any or limit set is unordered and lies on some invariant hypersurface with codimension one which generalizes hirsch s results for competitive autonomous systems of ordinary differential equations to competitive maps in a very general framework and implies the sarkovskii s theorem for planar strongly competitive and monotone maps we also show that the definition of competitiveness gives more restriction on homeomorphic property

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

the structure of symmetric attractors we consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems we show for example that the symmetry of an attractor cannot in general be an arbitrary subgroup of the group of symmetries in addition there are group theoretic restrictions on the symmetry of connected components of a symmetric attractor the symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time average of a chaotic dynamical system

a remark on stable subharmonic solutions of time periodic reaction diffusion equations abstract this paper is concerned with a time periodic reaction diffusion equation it is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation such solutions are called subharmonic solutions in this paper for any domain time period 0 and integer n 2 we construct an example of a time periodic reaction diffusion equation on with a minimal period which possesses a stable solution of minimal period n

discrete order preserving semigroups and stability for periodic parabolic differential equations on etudie la stabilite globale des solutions des equations paraboliques dans une forme non auto adjointe avec des coefficients periodiques dans le temps la structure mathematique est celle des semigroupes preservant l ordre a temps discret s n n e n sur un reseau de banach

convergence to periodic solutions in periodic quasimonotone reaction diffusion systems abstract the author studies the periodic time dependent quasimonotone reaction diffusion systems u i t d i u i f i t x u 1 u n t 0 x u i n i u i 0 t 0 x u i x 0 u 0 i x x formula i 1 2 n in a proper banach space satisfying i fi uj 0 for all 1 i j n ii f t x u is periodic in t of period 0 and iii fi t x u fi t x u for all 0 1 and i 1 2 n it is proved that every solution of such a system converges to a periodic solution the same result also holds if we replace iii by iii f t x u is analytic and every periodic solution is liapounov stable

the star systems x and a proof of the c1ω stability conjecture for flows abstract in this paper we give a new proof for the c 1 stability conjecture for flows our approach is considerably different from that of hayashi or wen and a good deal of liao s obstruction sets theory is used

monotone dynamical systems with dense periodic points abstract in this paper we prove a recent conjecture by m hirsch which says that if f is a discrete time monotone dynamical system with f a homeomorphism on an open connected subset of a finite dimensional vector space and the periodic points of f are dense in then f is periodic

the universal regular quotient of the chow group of points on projective varieties let x be a projective variety of dimension n de ned over an algebraically closed eld k for x irreducible and non singular matsusaka ma constructed an abelian variety alb x and a morphism a x alb x called the albanese variety and mapping respectively depending on the choice of a base point on x which is universal among the morphisms to abelian varieties see lang la serre se for other constructions over the eld of complex numbers the existence of alb x and a was known before and has a purely hodge theoretic description see igusa i for the hodge theoretic construction incidentally the terminology albanese variety was introduced by a weil for reasons explained in his commentary on the article 1950a of volume i of his collected works see w one of which is that the paper alb of albanese de nes it for a surface as a quotient of the group of 0 cycles of degree 0 modulo an equivalence relation let chn x deg 0 denote the chow group of 0 cycles of degree 0 on x modulo rational equivalence when x is irreducible and non singular a remarkable feature of the albanese morphism a is that it factors through a regular homomorphism u chn x deg 0 alb x that is a homomorphism which when composed with the cycle map c x chn x deg 0 gives an algebraic morphism this follows immediately from the fact that an abelian variety does not contain any rational curve thus one can reformulate matsusaka s theorem as the invent math 135 595 664 1999 doi 10 1007 s002229900890

observing the symmetry of attractors abstract we show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space equivariant projections have long been used but they can give misleading results if used improperly and have been considered untrustworthy we find conditions under which an equivariant projection generically shows the correct symmetry of the attractor

convergence on cooperative cascade systems with length one abstract motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by smith and sontag this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one first by fixing a solution of a base system converging to an equilibrium we establish both the nonordering of limit sets and the limit set dichotomy for the solutions of the cascade system combining these tools with the idea of limiting equation we then prove the sequential limit set trichotomy and hence the quasiconvergence in generic meaning the generic convergent result is finally obtained by improving the limit set dichotomy

the lyapunov exponents of generic volume preserving and symplectic maps we show that the integrated lyapunov exponents of c1 volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding oseledets splitting is trivial all lyapunov exponents are equal to zero or else dominated uniform hyperbolicity in the projective bundle almost everywhere we deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold almost every orbit either is projectively hyperbolic or has all lyapunov exponents equal to zero similarly for a residual subset of all c1 symplectic diffeomorphisms on any compact manifold either the diffeomorphism is anosov or almost every point has zero as a lyapunov exponent with multiplicity at least 2 finally given any set s gl d satisfying an accessibility condition for a residual subset of all continuous s valued cocycles over any measure preserving homeomorphism of a compact space the oseledets splitting is either dominated or trivial the condition on s is satisfied for most common matrix groups and also for matrices that arise from discrete schr nodinger operators

a formula for a symmetry detective detectives are used to determine the symmetry of attractors in finite dimensional systems we derive an explicit formula for a symmetry detective for each representation of a finite group

k dimensional invariant cones of random dynamical systems in rn with applications abstract we investigated a linear random dynamical system which strongly preserves a cone c of dimension k abbr k cone in r n under some general assumptions it is shown that such system admits a measurable family of k dimensional subspaces and a measurable family of n k dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of r n we further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems as well as the linear competitive cooperative tridiagonal systems this generalizes the floquet theory for these deterministic non autonomous or time periodic systems to the random systems

some characterizations of domination we show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates then we consider sets in rm gl d mathbb r with the property that any cocycle with values in has a dominated splitting we characterize these sets in terms of existence of invariant multicones thus extending a two dimensional result by avila bochi and yoccoz we give an example showing how these multicones can fail to have convexity properties

a dynamical proof of the multiplicative ergodic theorem we shall give a proof of the following result of oseledec in which gl d denotes the space of invertible d d real matrices denotes any norm on the space of d d matrices and log t max 0 log t for t 0

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

a proof of stability conjecture for three dimensional flows we give a proof of the c1 stability conjecture for three dimensional flows i e prove that there exists a hyperbolic structure over the q set for the structurally stable three dimensional flows mafn s proof for the discrete case motivates our proof and we find his perturbation techniques crucial in proving this conjecture we have overcome several new difficulties e g the change of period after perturbation the ergodic closing lemma for flows the existence of dominated splitting over q y where 3 is the set of singularities for the flow the discontinuity of the contracting rate function on singularities etc based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems i e we create unstable saddle connections if there are accumulations of periodic orbits on the singularities

on generalizations of the albanese variety in this note the relation between two generalizations of the classical albanese variety namely the one for smooth non complete varieties by faltings and wustholz and the one for singular varieties by esnault srinivas and viehweg is investigated over the field of complex numbers

detection of symmetry of attractors from observations i theory abstract barany et al 1993 proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps we use an idea in barany et al to re cast definitions of symmetry detectives assuming that we only have access to equivariant observations from the system detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space this paper generalises and develops their suggestion among the generalisations we consider is the use of nonpolynomial detectives and we show using the notion of prevalence of hunt et al 1992 that our detectives and the detectives of barany et al 1993 dellnitz et al 1994 and golubitsky and nicol 1995 give the correct symmetry of attractors almost certainly in a measure theoretic sense we show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this we show how one can find the symmetry of an attractor from examination of a poincare section in part ii of this article ashwin and tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with s4 symmetry

spatial period multiplying instabilities of hexagonal faraday waves abstract a recent faraday wave experiment with two frequency forcing reports two types of superlattice patterns that display periodic spatial structures having two separate scales physica d 123 1998 99 these patterns both arise as secondary states once the primary hexagonal pattern becomes unstable in one of these patterns so called superlattice two the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2 3 from the original scale of the hexagons in contrast the time averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale 3 larger than the original scale and apparently has 60 rotation symmetry we present a symmetry based approach to the analysis of this bifurcation taking as our starting point only the observed instantaneous symmetry of the superlattice two pattern presented in physica d 123 1998 99 and the subharmonic nature of the secondary instability we show a that a pattern with the same instantaneous symmetries as the superlattice two pattern can bifurcate stably from standing hexagons b that the pattern has a spatio temporal symmetry not reported in physica d 123 1998 99 and c that this spatio temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time averaged pattern but not for the apparent 60 rotation symmetry the approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns and does not rely on the primary pattern having small amplitude

dynamics alternatives and generic convergence for c1 smooth strongly monotone discrete dynamical systems abstract for c 1 smooth strongly monotone discrete time dynamical systems we prove dynamics alternatives which concludes that any compact orbit is either asymptotic to a linearly stable cycle or manifestly unstable for this purpose we improve several properties of the exponential separation for continuous maps the generic convergence to cycles is obtained as a by product of the dynamics alternatives

on thec1stability conjecture for flows we give a proof of thec1stability conjecture of palis and smale for flows which has reduced to proving thatc1structural stability implies axiom a the proof is based on the fundamental work of liao and mane and on the recent powerfulc1connecting lemma of hayashi

detecting the symmetry of attractors abstract this paper addresses the issue of how to determine numerically the symmetry of an attractor for dynamical systems the symmetries of attractors in phase space are related to patterns in the time average of the solution our approach to this question proceeds in two parts first we prove a general theorem based on group theoretic and differential topological ideas which states that generically the symmetry of a thickened attractor can be computed from the symmetries of a point in an auxilliary space this theorem proceeds by integrating an equivariant mapping over the thickened attractor once this is done the numerical computation of symmetries reduces to showing that a certain nonnegative number is zero numerically demonstrating that this number is zero can be difficult thus the second part of the algorithm is to consider how this number varies with parameters and noting that sudden jumps towards zero can be associated with increases in symmetry the paper is divided into two parts in the first we prove the general theorem and in the second we illustrate how the numerical techniques work on several examples including discrete dynamical systems with tetrahedral symmetry in r 3 and systems of three coupled cells in high dimensions the integral mentioned previously is difficult to compute for such examples we assume that an ergodic theorem is valid and that symmetries can be computed using a time average we compare both of these methods on the low dimensional examples as well as detect points of symmetry creation for a reaction diffusion equation on an interval this technique can also be used in principle to compute the symmetries of an attractor in an experiment from a time series

symmetry groups of attractors for maps equivariant under the action of a finite group on n the possible symmetries of fixed points are known and correspond to the isotropy subgroups this paper investigates the possible symmetries of arbitrary possibly chaotic attractors and finds that the necessary conditions of melbourne dellnitz golubitsky 15 are sufficient at least for continuous maps

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

a rapidly converging lower bound for the joint spectral radius via multiplicative ergodic theory abstract we use ergodic theory to prove a quantitative version of a theorem of m a berger and y wang which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices the proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded

phase translation group actions on strongly monotone skew product semiflows we establish a convergence property for pseudo bounded forward orbits of strongly monotone skew product semiflows with invariant phasetranslation group actions the results are then applied to obtain global convergence of certain chemical reaction networks whose associated systems in reaction coordinates are monotone as well as the dynamics of certain reactiondiffusion systems in time recurrent structure including periodicity almost periodicity and almost automorphy

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

nonsingular star flows satisfy axiom a and the no cycle condition we give an affirmative answer to a problem of liao and mane which asks whether for a nonsingular flow to loose the stability it must go through a critical element bifurcation more precisely a vector field s on a compact boundaryless manifold is called a star system if s has a c 1 neighborhood mathcal u in the set of c 1 vector fields such that every singularity and every periodic orbit of every x in mathcal u is hyperbolic we prove that any nonsingular star flow satisfies axiom a and the no cycle condition

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

dynamics of the periodic type k competitive kolmogorov systems abstract for time periodic dissipative and irreducible type k competitive kolmogorov systems it is proved that there is a canonically defined countable family f of unordered disjoint invariant sets with the property that for every persistent trajectory whose limit set is not a cycle there exists a unique trajectory in some element of f such that these two trajectories are asymptotic and the corresponding points in these two trajectories are k related

parameter uncertainties in models of equivariant dynamical systems abstract we examine the advantages of using equivariant versus nonequivariant phenomenological models for chaotic systems that possess an inversion symmetry numerical experiments used the minimum description length mdl criteria for model selection they indicated that the selection between equivariant versus nonequivariant models should be made on an a priori basis we also examine the relationship between the optimal truncation accuracy of the parameters of the model as predicted by the mdl criteria and the true uncertainty in the parameters of the model numerical experiments indicate that the true uncertainties are poorly approximated by the optimal truncation accuracy they also indicate that the true uncertainties may be useful for determining the presence of an inversion symmetry in reconstructed attractors

on the dynamics of dominated splitting let f m m be a c 2 dieomorphism of a compact surface we give a complete description of the dynamics of any compact invariant set having dominated splitting in particular we prove a spectral decomposition theorem for the limit set l f under the assumption of dominated splitting moreover we describe all the bifurcations that these systems can exhibit and the dierent types of dynamics that could follow for small c r perturbations

convergence to equilibrium on invariant d hypersurfaces for strongly increasing discrete time semigroups abstract convergence to equilibrium for every positive semi orbit of strongly increasing discrete time semigroups on strongly ordered spaces is proved the domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered banach space which is continuously imbedded into a banach lattice the semigroup is assumed to be order compact and every positive semi orbit is assumed to be order bounded the crucial hypothesis is the ljapunov stability of all equilibria it is also proved that the set of equilibria is a simply ordered arc the key tools are invariant d hypersurfaces

three dimensional competitive lotka volterra systems with no periodic orbits the following conjecture of m l zeeman is proved if three interacting species modeled by a competitive lotka volterra system can each resist invasion at carrying capacity then there can be no coexistence of the species indeed two of the species are driven to extinction it is also proved that in the other extreme if none of the species can resist invasion from either of the others then there is stable coexistence of at least two of the species in this case if the system has a fixed point in the interior of the positive cone in r3 then that fixed point is globally asymptotically stable representing stable coexistence of all three species otherwise there is a globally asymptotically stable fixed point in one of the coordinate planes of r3 representing stable coexistence of two of the species

vector fields with the oriented shadowing property abstract we give a description of the c 1 interior int 1 orientsh of the set of smooth vector fields on a smooth closed manifold that have the oriented shadowing property a special class b of vector fields that are not structurally stable is introduced it is shown that the set int 1 orientsh b coincides with the set of structurally stable vector fields an example of a field of the class b belonging to int 1 orientsh is given bibliography 18 titles

asymptotic behaviour of comparable skew product semiflows with applications the almost 1 cover lifting property of omega limit sets is established for nonmonotone skew product semiflows which are comparable to uniformly stable eventually strongly monotone skew product semiflows these results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of odes reaction diffusion systems differential systems with time delays and semilinear parabolic equations

detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of josephson junctions symmetry breaking and symmetry increasing bifurcations in a system of globally coupled josephson junctions which possesses sn symmetry have been investigated by constructing a detective for the group sn the permutation group of n elements in addition with help of this detective several bifurcations are followed numerically in two parameters an algorithm to obtain a detective for any finite subgroup of o n is given

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

continuity properties of the lower spectral radius 2014 london mathematical society the lower spectral radius or joint spectral subradius of a set of real d times d matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set the lower spectral radius arises naturally in connection with a number of topics including combinatorics on words the stability of linear inclusions in control theory and the study of random cantor sets in this article we apply some ideas originating in the study of dominated splittings of linear cocycles over a dynamical system to characterize the points of continuity of the lower spectral radius on the set of all compact sets of invertible d times d matrices as an application we exhibit open sets of pairs of 2 times 2 matrices within which the analogue of the lagarias wang finiteness property for the lower spectral radius fails on a residual set and discuss some implications of this result for the computation of the lower spectral radius

c 1 stably expansive flows abstract in this paper the c 1 interior of the set of vector fields whose integrated flows are expansive is characterized as the set of vector fields without singularities satisfying both axiom a and the quasi transversality condition and it is proved that the above vector fields possessing the shadowing property must be structurally stable as a corollary there exists a non empty c 1 open set of vector fields whose integrated flows do not have the shadowing property

symmetry and convergence properties for non negative solutions of nonautonomous reaction diffusion problems nonautonomous parabolic equations of the form u t u f u t on a symmetric domain are considered using the moving hyperplane method it is proved that any bounded nonnegative solution symmetrises as t this is then used to show that for nonlinearities periodic in t any non negative bounded solution approaches a periodic solution

exponential separation and principal floquet bundles for linear parabolic equations on general bounded domains nondivergence case we consider the dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients under such minimal regularity assumptions we establish the existence of a principal floquet bundle exponentially separated from a complementary invariant bundle as a special case of our main theorem assuming the coefficients are time periodic we obtain a new result on the existence of a principal eigenvalue of an associated time periodic parabolic eigenvalue problem we also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time periodic uniformly parabolic operators finally we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity

convergence and stability for essentially strongly order preserving semiflows abstract this paper is concerned with a class of essentially strongly order preserving semiflows which are defined on an ordered metric space and are generalizations of strongly order preserving semiflows for essentially strongly order preserving semiflows we prove several principles which are analogues of the nonordering principle for limit sets the limit set dichtomy and the sequential limit set trichotomy for strongly order preserving semiflows then under certain compactness hypotheses we obtain some results on convergence quasiconvergence and stability in essentially strongly order preserving semiflows finally some applications are made to quasimonotone systems of delay differential equations and reaction diffusion equations with delay and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

an extension of the fuglede commutativity theorem modulo the hilbert schmidt class to operators of the form _ _ we study the operators a x y m xn and a x m xn which map the algebra of all bounded linear operators on a separable hilbert space to itself where m j and n are separately commuting sequences of normal operators we prove that 1 when m 2 the hilbert schmidt norms of a x and a x are equal finite or infinite 2 for m 2 3 if a x and a x are hilbert schmidt operators then their hilbert schmidt norms are equal 3 if a a have the property that for each x a x 0 implies a x 0 then for each x if a x is a hilbert schmidt operator then a 2 x is also and the latter has the same hilbert schmidt norm as a2 x note that fuglede s theorem is immediate from 1 in the case mi 2 m1 n2 and n1 i m2 the proofs employ the duality between the trace class and the class of all bounded linear operators and unlike the early proofs of fuglede s theorem they are free of complex function theory let h denote a separable finite dimensional complex hilbert space let l h d k h d c d f h c2 c1 11 ilp 11 11 denote respectively the class of all bounded linear operators acting on h the compact operators the schatten p class the finite rank operators the hilbert schmidt class the trace class the c norm and the operator norm definition let k mn n 1 and k nn m 1 be two separately commuting sequences of normal operators define a a l h l h by

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory this series of papers is concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part of the series focuses on the development of general theory first the notions of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations for general positive random dynamical systems in ordered banach spaces are introduced which extend the classical notions of principal floquet subspaces principal lyapunov exponents and exponential separations for strongly positive deterministic systems in strongly ordered banach to general positive random dynamical systems in ordered banach spaces under some quite general assumptions it is then shown that a positive random dynamical system in an ordered banach space admits a family of generalized principal floquet subspaces a generalized principal lyapunov exponent and a generalized exponential separation we will consider in the forthcoming part s the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations including random leslie matrix models random cooperative systems of ordinary differential equations and random parabolic equations institute of mathematics and computer science wroc law university of technology wybrzeze wyspianskiego 27 pl 50 370 wroc law poland url http www im pwr wroc pl mierczyn index html e mail address mierczyn pwr wroc pl department of mathematics and statistics auburn university auburn university al 36849 usa e mail address wenxish auburn edu 2010 mathematics subject classification primary 37h15 37l55 37a30 secondary 15b52 34f05 35r60

detection of symmetry of attractors from observations ii an experiment with s 4 symmetry abstract in part i of this paper we adapt the detectives of barany et al 1993 for symmetries of attractors of symmetric dynamical systems to apply to experiments as an illustration of this theory we consider the dynamics of a system of four forced coupled electronic oscillators with full permutation symmetry s4 on varying parameters in the circuit we obtain attractors with many different symmetry types and detect these using the methods described in part i

on the finite dimensional dynamical systems with limited competition the asymptotic behavior of dynamical systems with limited competition is investigated we study index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to a nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits applications are made to time periodic ordinary differential equations and reaction diffusion equations

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

robust transitive singular sets for 3 flows are partially hyperbolic attractors or repellers inspired by lorenz remarkable chaotic flow we describe in this paper the structure of all c 1 robust transitive sets with singularities for flows on closed 3 manifolds they are partially hyperbolic with volume expanding central direction and are either attractors or repellers in particular any c 1 robust attractor with singularities for flows on closed 3 manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has positive lyapunov exponent at every orbit showing that any c 1 robust attractor resembles a geometric lorenz attractor

the generalised berger wang formula and the spectral radius of linear cocycles using ergodic theory we prove two formulae describing the relationships between different notions of joint spectral radius for sets of bounded linear operators acting on a banach space the first formula was previously obtained by v s shulman and yu v turovski using operator theoretic ideas the second formula shows that the joint spectral radii corresponding to several standard measures of noncompactness share a common value when applied to a given precompact set of operators this result may be seen as an extension of classical formulae for the essential spectral radius given by r nussbaum a lebow and m schechter both results are obtained as a consequence of a more general theorem concerned with continuous operator cocycles defined over a compact dynamical system as a byproduct of our method we answer a question of j e cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle

quasi shadowing and limit quasi shadowing for quasi partially hyperbolic strings of flows abstract in this paper we study the quasi shadowing property for quasi partially hyperbolic pseudo orbit made by quasi partially hyperbolic strings we also investigate the corresponding l p limit and asymptotic quasi shadowing properties as an application we prove that the set of points with periodic center leaves is dense in the non wondering set for some partially hyperbolic flows

hyperbolicity and stability for hamiltonian flows abstract we prove that a hamiltonian star system defined on a 2d dimensional symplectic manifold m d 2 is anosov as a consequence we obtain the proof of the stability conjecture for hamiltonians this generalizes the 4 dimensional results in bessa et al 2010 5

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

convergence in smooth strongly monotone flows defined by semilinear parabolic equations on considere des equations paraboliques semi lineaires dy dt ay f y ou a est un operateur sectoriel sur un banach x ayant une resolvante compacte et f est une c 2 application de l espace des puissances fractionnaires x 0 1 dans x soit y t une solution qui est quasi convergente mais non convergente alors il existe un voisinage u de y 0 y 0 avec la propriete suivante si z u y 0 a une trajectoire bornee et soit z y 0 soit z y 0 alors la solution partant de z converge vers un equilibre