the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

shift operators for a dirac oscillator shift operators are obtained for a dirac oscillator these operators and other algebraic methods are used to determine energy eigenvalues and eigenkets expectation values matrix elements and coordinate space wave functions

connected simple systems transition matrices and heteroclinic bifurcations given invariant sets a b and c and connecting orbits a b and b c we state very general conditions under which they bifurcate to produce an a c connecting orbit in particular our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits and for which the conley index of the invariant sets is the same as that of a hyperbolic critical point our proof depends on the connected simple system associated with the conley index for isolated invariant sets

conformal invariance in a dirac oscillator the conformal invariance properties of a dirac oscillator are established a set of operators is constructed whose algebra shows that it can be considered as a conformal system the operators are then used to solve the problem using algebraic techniques the superconformal generalization of the algebra is also worked out and some consequences of these invariances for the properties of the model are mentioned

asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations abstract by using previous results of djafari rouhani b djafari rouhani ergodic theorems for nonexpansive sequences in hilbert spaces and related problems ph d thesis yale university 1981 part i pp 1 76 b djafari rouhani asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces j math anal appl 147 1990 465 476 b djafari rouhani asymptotic behaviour of almost nonexpansive sequences in a hilbert space j math anal appl 151 1990 226 235 for dissipative systems we study the weak and strong convergence of solutions to the following class of second order nonhomogeneous evolution equations u t c u t a u t f t a e t 0 u 0 u 0 sup t 0 u t where a is a monotone operator in a real hilbert space h c 0 and f r h is a given function our results extend previous results by morosanu g morosanu nonlinear evolution equations and applications editura academiei romane and d reidel publishing company bucharest 1988 g morosanu asymptotic behaviour of solutions of differential equations associated to monotone operators nonlinear anal 3 1979 873 883 and mitidieri e mitidieri some remarks on the asymptotic behaviour of the solutions of second order evolution equations j math anal appl 107 1985 211 221 who studied the case c 0 and f 0 by assuming that a is maximal monotone and a 1 0 as well as previous results by veron l veron problemes d evolution du second ordre associes a des operateurs monotones c r acad sci paris ser a 278 1974 1099 1101 l veron equations d evolution du second ordre associees a des operateurs maximaux monotones proc roy soc edinburgh sect a 75 1975 76 131 147 and the authors b djafari rouhani h khatibzadeh asymptotic behavior of solutions to some homogeneous second order evolution equations of monotone type j c inequal appl 2007 art id 72931 8 pp b djafari rouhani h khatibzadeh asymptotic behavior of bounded solutions to some second order evolution systems rocky mountain j math in press who studied the case f 0 some applications are also presented

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

asymptotic behavior of solutions of nonlinear volterra equations 0 recently baillon and clement 3 established mean ergodic theorems for solutions of nonlinear volterra equations in hilbert space in the present paper we study the asymptotic behavior of solutions of 1 1 by using nonlinear ergodic theorems the first mean ergodic theorem was given by baillon l and it was improved by several authors 3 8 141 163

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

chaos near a resonant inclination flip abstract horseshoes play a central role in dynamical systems and are observed in many chaotic systems however most points in a neighborhood of the horseshoe escape after finitely many iterations in this work we construct a new model by re injecting the points that escape the horseshoe we show that this model can be realized within an attractor of a flow arising from a three dimensional vector field after perturbation of an inclination flip homoclinic orbit with a resonance the dynamics of this model without considering the re injection often contains a cuspidal horseshoe with positive entropy and we show that for a computational example the dynamics with re injection can have more complexity than the cuspidal horseshoe alone

a strong convergence theorem for solutions to a nonhomogeneous second order evolution equation abstract in this paper we establish the strong convergence of possible solutions to the following nonhomogeneous second order evolution system u t c u t a u t f t a e t 0 u 0 u 0 sup t 0 u t to an element of a 1 0 with an exponential rate of convergence when f 0 where a is a general maximal monotone operator in a real hilbert space h c 0 is a real constant and f r h is a given function we show also that the curve u is almost nonexpansive and present some applications of our result

asymptotic behaviour of almost nonexpansive sequences in a hilbert space abstract the results in 9 11 12 for nonexpansive sequences are generalized to almost nonexpansive sequences in a hilbert space and using this notion a direct proof of a result of h brezis and f e browder 6 is given

asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces abstract by suitably modifying the methods used by b djafari rouhani ergodic theorems for non expansive sequences in hilbert spaces and related problems part i pp 1 76 thesis yale university and a new proof of the weak convergence theorems for non expansive sequences and curves in hilbert spaces preprint 1984 and by b djafari rouhani and s kakutani ergodic theorems for non expansive non linear operators in a hilbert space preprint 1984 we study the asymptotic behaviour of the quasi autonomous dissipative system du dt au where a is a monotone operator in a hilbert space h and l 1 0 h

isolating blocks and epsilon chains for maps abstract the relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored epsilon chains are used as a mathematical representation of computer generated pseudo orbits the question when is a pseudo orbit shadowed by a true orbit is discussed isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks the conley index is extended and applied to maps

nonunitary representations of the su 2 algebra in the dirac equation with a coulomb potential a novel realization of the classical su 2 algebra is introduced for the dirac relativistic hydrogen atom defining a set of operators that allow the factorization of the problem an extra phase is needed as a new variable in order to define the algebra we take advantage of the operators to solve the dirac equation using algebraic methods a similar path to the one used in the angular momentum case is used hence the radial eigenfunctions so calculated comprise nonunitary representations of the algebra one of the interesting properties of such nonunitary representations is that they are not labeled by integer nor by half integer numbers as occurs in the usual angular momentum representation

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

ergodic theorems for hybrid sequences in a hilbert space with applications abstract in this paper we introduce the notion of generalized hybrid sequences extending the notion of nonexpansive sequences introduced and studied in our previous work djafari rouhani 1981 1990 1990 1997 2002 2004 2002 2 3 4 5 6 7 8 and prove ergodic and convergence theorems for such sequences in a hilbert space h subsequently we apply our results to prove new fixed point theorems for generalized hybrid mappings first introduced in kocourek et al 2010 14 takahashi and takeuchi 2011 20 defined on arbitrary nonempty subsets of h

zeta functions periodic trajectories and the conley index abstract we use the conley index to produce a simple sufficient condition stated in terms of the betti numbers of the cohomology conley index for the existence of periodic orbits for flows which admit a cross section

index filtrations and the homology index braid for partially ordered morse decompositions abstract on a morse decomposition of an invariant set in a flow there are partial orderings defined by the flow these are called admissable orderings of the morse decomposition the index filtrations for a total ordering of a morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissable partial orderings of a morse decomposition it is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration the homology index braid of the corresponding admissable ordering of the morse decomposition is obtained by passing to homology in the chain complex braid keywords morse decomposition conley index index filtration

asymptotic behavior of bounded solutions to a nonhomogeneous second order evolution equation of monotone type abstract by using previous results of djafari rouhani b djafari rouhani ergodic theorems for nonexpansive sequences in hilbert spaces and related problems ph d thesis yale university 1981 part i pp 1 76 b djafari rouhani asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces j math anal appl 147 1990 465 476 b djafari rouhani asymptotic behaviour of almost nonexpansive sequences in a hilbert space j math anal appl 151 1990 226 235 for dissipative systems we study the asymptotic behavior of solutions to the following system of second order nonhomogeneous evolution equations u t a u t f t a e t 0 u 0 u 0 sup t 0 u t where a is a maximal monotone operator in a real hilbert space h and f r h is a suitable function we investigate weak and strong convergence theorems for solutions to this system our results extend previous results of morosanu g morosanu nonlinear evolution equations and applications editura academiei romane and d reidel publishing company bucharest 1988 g morosanu asymptotic behaviour of solutions of differential equations associated to monotone operators nonlinear anal 3 1979 873 883 and mitidieri e mitidieri some remarks on the asymptotic behaviour of the solutions of second order evolution equations j math anal appl 107 1985 211 221 who studied the case f 0 by assuming that a 1 0 as well as previous results of the authors b djafari rouhani h khatibzadeh asymptotic behavior of bounded solutions to some second order evolution systems rocky mountain j math in press who studied the homogeneous case without this additional assumption

asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a hilbert space abstract we extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in rouhani and kim j nonlinear convex anal 4 2003 175 for a commutative semigroup to a general semitopological semigroup study the asymptotic behavior of such curves in a hilbert space and apply our results to study the asymptotic behavior for almost orbits of semigroups of asymptotically nonexpansive type mappings over a general semitopological semigroup in a hilbert space since in this case the semigroup is not directed the proofs in rouhani and kim j nonlinear convex anal 4 2003 175 do not extend to this case and new methods have to be used for the proofs in addition to rouhani j math anal appl 147 1990 465 j math anal appl 151 1990 226 nonlinear anal 44 2001 627 nonlinear anal 51 2002 735 and rouhani and kim j nonlinear convex anal 4 2003 175 our results extend and unify many previously known results j math anal appl 127 1987 206 proc amer math soc 104 1988 431 nonlinear anal 29 1997 539 topol methods nonlinear anal 5 1995 305 j math anal appl 105 1985 514 nonlinear anal 26 1996 1411 pacific j math 126 1987 277 j math anal appl 206 1997 451 chinese ann math 6 1996 729 nonlinear anal 14 1990 69 j math anal appl 85 1982 172 proc amer math soc 81 1981 253 proc amer math soc 97 1986 55 canad j math 44 1992 880 j math anal appl 142 1989 242 proc amer math soc 117 1993 385

detection and computation of high codimension bifurcations in diffuse predator prey systems abstract in this paper we consider reaction diffusion model with modified leslie gower and holling type ii functional response and by varying simultaneously three model parameters our model exhibits different types of complex dynamics this allowed us to delimit several bifurcation surfaces with higher codimension corresponding to turing turing transcritical turing bogdanov taken turing hopf andronov turing saddle node moreover by varying at least three bifurcation parameters we show that small variations in the ratio of the diffusion coefficients can significantly alter bifurcation structure finally the paper ends with the emergence of spatio temporal patterns via numerical simulations these simultaneous parameter variations leaded to spatio temporal local and global bifurcations which are always catastrophic our results give new insights about how simultaneous changes in environmental and life history parameters drive different distribution dynamics of predator prey populations this only underscores the importance of including global bifurcations in the analysis of food chain models such results can be used to improve ecological decision making on species conservation

asymptotic behavior of the solutions of non autonomous systems in banach spaces abstract in this paper we study the weak and strong convergence of the integral solution of the following nonlinear evolution inclusion u t a t u t f t t 0 u 0 x 0 where a t t 0 is a family of m accretive operators and f is a multi function which extends some results in this area