implicit lyapunov functions and isochrones of linear systems lyapunov s direct method is based on a lyapunov function that explicitly depends on the system s state vector this article describes implicit lyapunov functions and corresponding stability theorems that are counterparts of the explicit case we also define isochrones of linear systems and describe their relationship to implicit lyapunov functions determining settling times is considered as an application

a note on uniform observability we prove that the classical inequality p sub t spl les spl oscr sub t sup 1 c sub t relating the variance of the kalman filter estimate the observability matrix and the controllability matrix is not true this inequality is the cornerstone of the asymptotic stability theory of the kalman filter for time varying systems we provide another inequality of the same type

closed characteristics on non compact hypersurfaces in r 2n viterbo demonstrated that any 2n 1 dimensional compact hypersur face m r 2n of contact type has at least one closed characteristic this result proved the weinstein conjecture for the standard symplectic space r 2n various extensions of this theorem have been obtained since all for compact hypersurfaces in this paper we consider non compact hypersurfaces m r 2n coming from mechanical hamiltonians and prove an analogue of viterbo s result the main result provides a strong connection between the top half homology groups hi m i n 2n 1 and the existence of closed characteristics in the non compact case including the compact case

on evaluating the bounded input bounded output stability integral for second order systems it is shown that the value of this integral is easily calculated by using the nastiest disturbance switch curve in spiral coordinates

symmetric closed characteristics on symmetric compact convex hypersurfaces in r 2n in this article let r2n be a compact convex hypersurface which is symmetric with respect to the origin where n 2 or 3 we prove that if carries exactly n geometrically distinct closed characteristics then all of them must be symmetric

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

multiple brake orbits on convex hypersurfaces under asymmetric pinch conditions abstract in this paper we give an example of an asymmetric convex compact hypersurface possessing precisely n brake orbits for h c 1 r 2 n r satisfying h p q h p q for all p q r n r n we prove multiplicity result of the brake orbits on h 1 1 pinched by not necessarily symmetric hypersurfaces

on the existence of periodic solutions for a class of symetric hamiltonian systems abstract hamiltonian systems are used to model the motion of discrete mechanical systems this paper establishes the existence of periodic solutions for a class of such systems the method developed to prove existence should be useful for other such problems keywords minimax technique ordinary differential equations and variational problem

solving high order ordinary differential equations with radial basis function networks this paper is concerned with the application of radial basis function networks rbfns for numerical solution of high order ordinary differential equations odes two unsymmetric rbf collocation schemes namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process are developed to solve high order odes directly and the latter is found to be considerably superior to the former good accuracy and high rate of convergence are obtained with the proposed indirect method

exponential difference operator approximation for the sixth order onsager equation an accurate easy to program method of solving the one dimensional form of the sixth order onsager equation is presented boundary conditions are included naturally aip

a closed analytic form for the time maximum disturbance isochrones of second order linear systems it is shown that each time maximum disturbance isochrone boundary of reachable set for second order linear systems with the most stressful bounded disturbance has a closed analytic form further it is shown that these isochrones can be constructed by purely geometric methods

on the normal exponential map in singular conformal metrics abstract brake orbits and homoclinics of autonomous dynamical systems correspond via maupertuis principle to geodesics in riemannian manifolds endowed with a metric which is singular on the boundary jacobi metric motivated by the classical yet still intriguing in many aspects problem of establishing multiplicity results for brake orbits and homoclinics as done in giambo et al 2005 2010 2011 and by the development of a morse theory in giambo et al 2014 for geodesics in such kind of metric in this paper we study the related normal exponential map from a global perspective

application of generalized differential quadrature rule in blasius and onsager equations the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to third order non linear differential equations of the blasius type and to sixth order linear onsager differential equations high 3rd order differential equations in fluid mechanics are dealt with without using point techniques the half space domain is simplified in a practical way accurate results are obtained for both kinds of problems the wide applicability of the gdqr in high order differential equations is manifested further through this work copyright 2001 john wiley sons ltd

closed characteristics on compact convex hypersurfaces in r 2n for any given compact c 2 hypersurface in r 2n bounding a strictly convex set with nonempty interior in this paper an invariant n is defined and satisfies n n 2 1 where a denotes the greatest integer which is not greater than a r the following results are proved in this paper there always exist at least n geometrically distinct closed characteristics on if all the geometrically distinct closed characteristics on are nondegenerate then n n if the total number of geometrically distinct closed characteristics on is finite there exists at least an elliptic one among them and there exist at least n 1 of them possessing irrational mean indices if this total number is at most 2 n 2 there exist at least two elliptic ones among them

a new obstruction to embedding lagrangian tori n let 2n be endowed with the canonical symplectic form a i 1 dxi dy i and consider a lagrangian embedding of a compact manifold j l r 2 a a well known result due to g r o m o v g states that the cohomology class of a in h z e 2 l is non zero we identify l with its image in r2 an equivalent assertion is that the liouville form 2 y 7 1 xldy i pulls back to a closed form j 2 on l whose cohomology class does not vanish g r o m o v s proof uses subtle properties of holomorphic maps into c he shows that there is always a holomorphic disk a in c with boundary in l on such a disk a is positive hence sa a soa 2 0 in this paper we shall only consider the case l t and give an elementary proof of gromov s result elementary should be understood in the sense of number theory that is without using holomorphic functions in fact our result is somewhat more precise and can be stated as follows

irrationally elliptic closed characteristics on compact convex hypersurfaces in r6 abstract in this paper let r 6 be a compact convex hypersurface which carries exactly three geometrically distinct closed characteristics we prove that at least two of them must be irrationally elliptic

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

multiple brake orbits in bounded convex symmetric domains abstract in this paper we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in r n for n 2

functions on the sphere with critical points in pairs and orthogonal geodesic chords abstract using an estimate on the number of critical points for a morse even function on the sphere s m m 1 we prove a multiplicity result for orthogonal geodesic chords in riemannian manifolds with boundary that are diffeomorphic to euclidean balls this yields also a multiplicity result for brake orbits in a potential well

an index theory and existence of multiple brake orbits for star shaped hamiltonian systems soit h r 2n r une fonction continument differentiable et j o i ou i est la matrice identite n n on considere le systeme hamiltonien x jh x on etudie l existence de solutions periodiques sur une hypersurface compacte donnee h 1 1

equivariant morse theory for starshaped hamiltonian systems soit une hypersurface en forme d etoile dans r 2n le probleme de trouver des caracteristiques fermees de peut se reduire classiquement a un probleme variationnel ceci conduit a etudier une fonctionnelle s 1 equivariante sur un espace de hilbert

resonance identity for symmetric closed characteristics on symmetric convex hamiltonian energy hypersurfaces and its applications in this paper we establish a new resonance identity for symmetric closed characteristics on symmetric compact convex hypersurface in r2n when there exist only finitely many geometrically distinct symmetric closed characteristics as its applications some interesting results about the stability and multiplicity of symmetric closed characteristics are obtained and also we prove that if is c generic it carries infinitely many symmetric closed characteristics

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

the existence of two closed geodesics on every finsler 2 sphere in this paper we prove that for every finsler metric on s2 there exist at least two distinct prime closed geodesics

time maximum disturbance design for stable linear systems a model predictive scheme it is known that the most stressful bounded time maximum disturbance for stabilized linear systems is of bang bang type this bang bang disturbance can often be implemented with a switch set of current states for second order systems the isochrones as the level sets determine the value of the disturbance index in a state space setting we suggest an efficient way to construct the time maximum disturbance from the information of the isochronal wave front using it in a model predictive scheme this overcomes the shortcomings of the original switch set which are constructed through time backward computation and only available for first and second order systems simulation results show how the isochrones evolve and can be utilized in synthesizing the time maximum disturbance for linear systems of second and then higher order for third order systems the associated l sub spl infin gain of the time maximizing disturbance is found

index iteration theories for periodic orbits old and new abstract in this paper we give a survey of the index iteration theories developed for periodic solution orbits of hamiltonian systems in treating various problems from 1950s up to now including some recent developments and some of their applications to periodic solution problems

lusternik schnirelmann theory and closed reeb orbits we develop a variant of lusternik schnirelmann theory for the shift operator in equivariant floer and symplectic homology our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated as an application we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere the unit cotangent bundle to the sphere and some other contact manifolds we also show that the lower conley zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of long and zhu this is the second combinatorial ingredient in the proof of the multiplicity results

convex hamiltonian energy surfaces and their periodic trajectories in this paper we introduce symplectic invariants for convex hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations in particular we show that they can be used to prove multiplicity results for the number of periodic trajectories

on the average indices of closed geodesics on definit la notion d indices moyens de geodesiques fermees on demontre une relation entre ces indices pour une variete compacte simplement connexe dotee d une metrique admissible

resonance identities for closed characteristics on compact star shaped hypersurfaces in r2n resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding hamiltonian systems in this paper we establish two new resonance identities for closed characteristics on every compact star shaped hypersurface in r2n when the number of geometrically distinct closed characteristics on is finite which extend those identities established by c viterbo in 1989 for star shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non degenerate and that by w wang x hu and y long in 2007 for strictly convex hypersurfaces in r2n

seifert conjecture in the even convex case in this paper we prove that there exist at least n geometrically distinct brake orbits on every c2 compact convex symmetric hypersurface in 2n satisfying the reversible condition n with n diag in in as a consequence we show that if the hamiltonian function is convex and even then seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integern 2014 wiley periodicals inc

hyperbolic closed characteristics on compact convex smooth hypersurfaces inr2n in this paper we prove that on every compactc2hypersurface inr2nbounding a convex set with non empty interior either there exists a sequence of variationally visible hyperbolic closed characteristcs with their minimal periods tending to infinity or there exists at least one variationally visible nonhyperbolic closed characteristic

iteration theory of l index and multiplicity of brake orbits abstract in this paper we first establish the bott type iteration formulas and some abstract precise iteration formulas of the maslov type index theory associated with a lagrangian subspace for symplectic paths as an application we prove that there exist at least n 2 1 geometrically distinct brake orbits on every c 2 compact convex symmetric hypersurface in r 2 n satisfying the reversible condition n furthermore if all brake orbits on this hypersurface are nondegenerate then there are at least n geometrically distinct brake orbits on it as a consequence we show that there exist at least n 2 1 geometrically distinct brake orbits in every bounded convex symmetric domain in r n furthermore if all brake orbits in this domain are nondegenerate then there are at least n geometrically distinct brake orbits in it in the symmetric case we give a positive answer to the seifert conjecture of 1948 under a generic condition

stability of closed geodesics on finsler 2 spheres abstract in this paper we prove that on every finsler 2 dimensional sphere either there exist infinitely many prime closed geodesics or there exist at least two irrationally elliptic prime closed geodesics

analytical mini max methods for hamiltonian brake orbits of prescribed energy on considere des caracterisations explicites de solutions periodiques de systemes hamiltoniens autonomes d ordre 2 de la forme q v q q t r n ou le potentiel v satisfait les conditions suivantes v c 2 r n v x v o 0 v est strictement convexe sur r n

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

brake type closed characteristics on reversible compact convex hypersurfaces in r2n abstract in this paper let be a c 3 compact convex in r 2 n satisfying the reversible condition n with n diag i n i n we prove that if there are exactly n geometrically distinct closed characteristics on and all of them are nondegenerate then all of them must be brake orbits up to a suitable translation of time moreover for n 2 or 3 we prove that if there are exactly n geometrically distinct closed characteristics on then all of them must be brake orbits up to a suitable translation of time

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious