on homoclinic orbits for a class of noncoercive superquadratic hamiltonian systems abstract in this paper we prove the existence of homoclinic solutions for a class of noncoercive first order hamiltonian systems j x m t x u g t u x 0 by the minimax methods in critical point theory specially a generalized mountain pass theorem when u is a linear operator with adjoint u and g t y satisfies the superquadratic condition g t x y 2 as y uniformly in t and need not satisfy the global ambrosetti rabinowitz condition

existence of homoclinic solutions for a class of second order systems abstract in this paper we study the existence of nontrivial homoclinic solutions for a second order differential equation of the form q a q l t q w q t q 0 here we assume that a is a antisymmetric constant matrix l t is a continuous positive definite symmetric matrix valued function depending periodically on t and assume that potential w is asymptotically quadratic as q 0 and q

brownian motion with restoring drift micro canonical ensemble and the thermodynamic limit we take up the old problem of micro canonical conditioning in the context of diffusion starting with a potential the schrodinger operator with ground state is carried by a conjugation into the diffusion generator with invariant density the latter motion is made micro canonical by first conditioning the path to be periodic and then further conditioning on the empirical mean square or particle number the thermodynamics are then studied by taking while d remains fixed the problem in this form owes its inception to mckean vaninsky cite mv2 who obtained the following result for with they showed the same type of diffusion appears in the thermodynamic limit but with drift arising from the shifted potential being such that the limiting mean square equals d their method of proof predicts the same outcome for so long as d is smaller than the canonical mean square while if the matter was unresolved the purpose of this note is to show a type of phase transition takes place in this case the conditioning is overcome in the limit and one sees the original stationary diffusion on the line the proof employs an entropy inequality due to csiszar 1

a dependent data extension of wald s identity and its application to sequential test performance computation the development of an exponential transformation of measure not restricted to independent identically distributed data is presented for a class of semi markov processes known as markov additive ma process the ma measure transformation is applied to obtain a new form of the wald identity to demonstrate the utility of this extension to ma processes the application of sequential tests with markov data to error probability performance analysis is investigated in particular it is demonstrated how previous work dealing with robust error probability evaluation in the presence of uncertain serial dependency can be applied to sequential tests

an asymptotically least favorable chernoff bound for a large class of dependent data processes it is desired to determine the worst case asymptotic error probability performance of a given detector operating in an environment of uncertain data dependency a class of markov data process distributions is considered which satisfy a one shift dependency bound and agree with a specified univariate distribution within this dependency contamination class the distribution structure which minimizes the exponential rate of decrease of detection error probabilities is identified this is a uniform least favorability principle because the least favorable dependency structure is the same for all bounded memoryless detectors the error probability exponential rate criterion used is a device of large deviations theory the results agree well with previous results obtained using pitman s asymptotic relative efficiency are which is a more tractable small signal performance criterion in contrast to are large deviations theory is closely related to finite sample error probabilities via the finite sample chernoff bounds and other exponentially tight bounds and other approximations

genericity of the multibump dynamics for almost periodic duffing like systems in this paper we consider slowly oscillating perturbations of almost periodic du ng like systems i e systems of the form u u a t t w0 u t r u rn where w c2n rn r is superquadratic and a and are positive and almost periodic by variational methods we prove that if 0 is small enough then the system admits a multibump dynamics as a corollary we get that the system u u a t w0 u t r u rn admits multibump solutions whenever a a where a is an open dense subset of a c r r a is almost periodic and a t 0 t r

linear maps preserving the spectral radius abstract let x be a complex banach space if b x b x is a surjective linear map such that a and a have the same spectral radius for every a b x then c where is either an algebra automorphism or an antiautomorphism of b x and c is a complex constant such that c 1

homoclinic orbits and subharmonics for nonlinear second order difference equations abstract we show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the mountain pass theorem relying on ekeland s variational principle and the diagonal method and the homoclinic orbit as the limit of the subharmonics a completely new way is provided for dealing with the existence of solutions for difference equations

infinitely many homoclinic solutions for a class of subquadratic second order hamiltonian systems in this paper we mainly consider the existence of infinitely many homoclinic solutions for a class of subquadratic second order hamiltonian systems u l t u w u t u 0 where l t is not necessarily positive definite and the growth rate of potential function w can be in 1 3 2 using the variant fountain theorem we obtain the existence of infinitely many homoclinic solutions for the second order hamiltonian systems

infinitely many solutions for diffusion equations without symmetry abstract we consider the following diffusion system t u x u b t x x u v x u h v t x u v t v x v b t x x v v x v h u t x u v t x r r n which is an unbounded hamiltonian system in l 2 r r n r 2 m z u v r r n r m r m b c r r n r n v c r n r and h c 1 r r n r 2 m r suppose that h b and v depend periodically on t and x and that h t x z is superquadratic in z as z without a symmetry assumption on h we establish the existence of infinitely many geometrically distinct solutions via a variational approach

a global condition for quasi random behavior in a class of conservative systems devaney has shown that an autonomous hamiltonian system in dimension 4 with an orbit homoclinic to a saddle focus equilibrium admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds in this paper we deal with two classes of saddle focus systems lagrangian systems defined on a two manifold in the presence of a gyroscopic force and fourth order systems arising in water wave theory we first establish by a standard variational method the existence of a homoclinic orbit then under a weak nondegeneracy condition we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic our condition is much easier to check than transversality for example it is automatically satisfied for gyroscopic systems on a two torus for topological reasons 1996 john wiley sons inc

hitting probabilities and large deviations let fyngn2z be a sequence of random variables in r d and let a r d then pfyn 2 a for some ng is the hitting probability of the set a by the sequence fyng we consider the asymptotic behavior as m 1 o fpfyn 2 ma some ng pfhitting mag whenever 1 the probability law of yn n satises the large deviation principle and 2 the central tendency of yn n is directed away from the given set a for a particular function i w e showpfyn 2 ma some n ge m i a

moderate deviations for empirical measures of markov chains lower bounds we obtain upper bounds for moderate deviations of empirical measures of markov chains with general state space in the topology under the minimal assumption of geometric ergodicity the regeneration split chain method and a result on projective systems are employed in the proof

on spectrally bounded linear maps on b x abstract let x be a complex banach space and denote by b x the algebra of all linear bounded operators on x for t b x denote by r t its spectral radius we prove that if b x b x is a linear and surjective map such that i given any a b b x there exists a positive real number m a b such that r a b m a b r a b for all complex numbers satisfying 1 ii r t 0 implies r t 0 then there exist a linear functional f b x c a nonzero complex number c and either a bounded invertible linear map u x x such that t c u t u 1 f t i t b x or a bounded invertible linear map u x x such that t c u t u 1 f t i t b x also the linear functional f satisfies the property that given any a b b x there exists n a b 0 such that f a f b n a b r a b for all complex numbers satisfying 1

large deviation bounds for functionals of viterbi paths in a number of applications the underlying stochastic process is modeled as a finite state discrete time markov chain that cannot be observed directly and is represented by an auxiliary process the maximum a posteriori map estimator is widely used to estimate states of this hidden markov model through available observations the map path estimator based on a finite number of observations is calculated by the viterbi algorithm and is often referred to as the viterbi path it was recently shown in and see also and that under mild conditions the sequence of estimators of a given state converges almost surely to a limiting regenerative process as the number of observations approaches infinity this in particular implies a law of large numbers for some functionals of hidden states and finite viterbi paths the aim of this paper is to provide the corresponding large deviation estimates

branching random walk in random environment phase transitions for local and global growth rates let n be the infinite particle system on whose evolution is as follows at each unit of time each particle independently is replaced by a new generation the size of a new generation descending from a particle at sitex has distributionf x and each of its members independently jumps to sitex 1 with probability 1 h 2 h 0 1 the sequence f x is i i d with uniformly bounded second moment and is kept fixed during the evolution the initial configuration 0 is shift invariant and ergodic two quantities are considered

infinitely many solutions for a class of nonlinear dirac equations without symmetry abstract this paper is concerned with the multiplicity of solutions for the nonlinear dirac equations i k 1 3 k k u v x a u u f u x u without symmetry assumption on f we establish the existence of infinitely many geometrically distinct solutions for superquadratic as well as asymptotically quadratic nonlinearities via variational approach

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

infinitely many homoclinic orbits for the second order hamiltonian systems with general potentials abstract in this paper we study the following nonperiodic second order hamiltonian systems u t l t u t r t u t r where l t may not be uniformly positive definite for all t r under more general conditions on r u t u we prove that the above system has infinitely many homoclinic orbits provided r is even in u

homoclinic solutions of hamiltonian systems with symmetry abstract we prove the existence of infinitely many homoclinic solutions for a first order hamiltonian system symmetric with respect to an action of a compact lie group by means of variational methods we make no convexity assumption on the hamiltonian

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

connecting orbits for some classes of almost periodic lagrangian systems abstract the existence is proved by means of variational arguments of infinitely many heteroclinic solutions connecting possibly degenerate equilibria for a class of almost periodic lagrangian system an analogous multiplicity result is then established for homoclinic solutions of systems with an almost periodic singular potential

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

existence and exponential decay of homoclinics in a nonperiodic superquadratic hamiltonian system abstract this paper deals with existence and exponential decay of homoclinic orbits in the first order hamiltonian system z j h z t z where the hamiltonian function h t z is nonperiodic in t r and superquadratic in z r 2 n with certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

fast homoclinic solutions for a class of damped vibration problems in this paper we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu t q t u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t our approach is variational and it is based on the critical point theory we prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function moreover some open problems proposed by zhang and ruan are resolved our theorems appear to be the first such result

periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear hamiltonian system in this paper we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear hamiltonian systems with impulsive conditions via variational methods however without impulses there is no homoclinic or periodic solution for the system considered in this paper moreover our results can be used to study the existence of periodic and homoclinic solutions of difference equations

generic type results for chaotic dynamics in equations with periodic forcing terms abstract we prove that a class of equations containing the classical periodically forced pendulum problem displays the main features of chaotic dynamics for a set of forcing terms open and dense in suitable spaces the approach is based on global variational methods

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

first order elliptic systems and the existence of homoclinic orbits in hamiltonian systems and h r x r 2 r is smooth by a homoclinic or a doubly asymptotic orbit to y where y is a periodic solution of 1 we mean a solution x y of 1 such that i x 0 y t l 0 as i tl oo finding homoclinic orbits in systems like 1 can be quite difficult in the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect in some situation this can be done by melnikov s method recently some progress has been made by applying variational methods coti zelati and ekeland showed in 3 using dual variational methods that 1 has at least one homoclinic orbit if h satisfies some convexity and growth assumptions in their proof the convexity assumption enters in a crucial way already in the set up of the variational problem variational methods have been used by benci and giannoni 2 who studied homoclinic orbits on compact riemannian manifolds and by p rabinowitz in 9 who proved existence of heteroclinic orbits on the n dimensional torus as well as their multiplicity in 2 and 9 only special hamiltonian of lagrangian type are studied the associated variational problem is then at least semi definite the

approximation of partial sums of arbitrary i i d random variables and the precision of the usual exponential upper bound this paper quantifies the degree to which exponential bounds can be used to approximate tail probabilities of partial sums of arbitrary i i d random variables the introduction of a single truncation allows the usual exponential upper bound to apply usefully whenever the summands are arbitrary i i d random variables more specifically let n be a fixed natural number and let z z 1 z 2 z n be arbitrary i i d random variables we construct a function f z n derived from the probability of occurrence of one or more large summands plus an upper bound of exponential type such that for some constant c 0 independent of z n and a and all real a c f z 2 n a formula math zj na 2f z n a furthermore examples show that the upper and lower bounds are achievable

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

solution set splitting at low energy levels in schrodinger equations with periodic and symmetric potential abstract the time independent superlinear schrodinger equation with spatially periodic and positive potential admits sign changing two bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set moreover we provide a recipe to explicitly verify the condition and we carry out the calculation in dimension one for a specific class of potentials

markov additive processes and perron frobenius eigenvalue inequalities we present a method for proving perron frobenius eigenvalue inequalities the method is to apply jensen s inequality to the change in a random evolution over a regenerative cycle of the underlying finite state markov chain one of the primary benefits of the method is that it readily gives necessary and sufficient conditions for strict inequality it also gives insights into some of the conjectures of j e cohen ney and nummelin s hypothesis 2 arises here as a condition for strict inequality and we explore its ramifications in detail for a special family of markov additive processes which we call fluid models this leads to a connection between hypothesis 2 and the condition p t p irreducible which arose in the work of cohen friedland kato and kelly

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

a global condition for periodic duffing like equations we study duffing like equations of the type q qa t w q with a e c r r periodic we prove that if the stable and unstable manifolds to the origin do not coincide then the system exhibits positive topological entropy

homoclinic solutions for some second order non autonomous hamiltonian systems without the globally superquadratic condition abstract this paper deals with the existence of homoclinic solutions for the following second order non autonomous hamiltonian system hs q v q t q f t where v c 1 r r n r v t q k t q w t q is t periodic in t f is aperiodic and belongs to l 2 r r n under the assumptions that k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w t q is not globally superquadratic on q and some additionally reasonable assumptions we give a new existence result to guarantee that hs has a homoclinic solution q t emanating from 0 the homoclinic solution q t is obtained as a limit of 2 k t periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the mountain pass theorem recent results in the literature are generalized and significantly improved