universal mechanism for the intermittent route to strange nonchaotic attractors in quasiperiodically forced systems to examine the universality for the intermittent route to strange nonchaotic attractors snas we investigate the quasiperiodically forced henon map ring map and toda oscillator which are high dimensional invertible systems in these invertible systems dynamical transition to an intermittent sna occurs via a phase dependent saddle node bifurcation when a smooth torus collides with a ring shaped unstable set we note that this bifurcation mechanism for the appearance of intermittent snas is the same as that found in a simple system of the quasiperiodically forced noninvertible logistic map hence the intermittent route to snas seems to be universal in the sense that it occurs through the same mechanism in typical quasiperiodically forced systems of different nature

on the existence of weak solutions of perturbated systems with critical growth sur un domaine borne de r n on considere le probleme de dirichlet pour le systeme elliptique a u g u f sur et u 0 ou a u est un operateur quasi lineaire et la perturbation g u depend de u et de d u on etudie des conditions suffisantes sur a u et sur g u pour l existence de solutions faibles

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

asymptotical stability of probabilistic boolean networks with state delays this paper devotes to establishing a bridge between asymptotical stability of a probabilistic boolean network pbn and a solution to its induced equations which are induced from the pbn s transition matrix by utilizing the semitensor product technique routinely the dynamics of a pbn with coincident state delays can be equivalently converted into that of a higher dimensional pbn without delays subsequently several novel stability criteria are derived from the standpoint of equations solution the most significant finding is that a pbn is globally asymptotically stable at a predesignated one point distribution if and only if a vector obtained by adding 1 at the bottom of this distribution is the unique nonnegative solution to pbn s induced equations moreover the influence of coincident state delays on pbn s asymptotical stability is explicitly analyzed without consideration of the convergence rate consequently such bounded state delays are verified to have no impact on pbn s stability albeit delays are time varying based on this worthwhile observation the time computational complexity of the aforementioned approach can be reduced by removing delays directly furthermore this universal procedure is summarized to reduce the time complexity of some previous results in the literature to a certain extent two examples are employed to demonstrate the feasibility and effectiveness of the obtained theoretical results

observability of boolean control networks with state time delays this brief deals with the problem of the observability for the boolean control networks with time delays in states first using semi tensor product of matrices and the matrix expression of logic the boolean control networks with state delays can be converted into discrete time delay dynamics then the observability of the boolean control networks via two kinds of inputs is investigated by giving necessary and sufficient conditions finally examples are given to illustrate the efficiency of the obtained results

a finite convergence criterion for the discounted optimal control of stochastic logical networks stochastic logical networks slns are discrete time stochastic dynamical systems with boolean or multivalued logical state variables the discounted infinite horizon optimal control problem for sln is addressed in this paper by resorting to the equivalent markov decision process description the infinite horizon optimization problem is presented in algebraic form then using the increasing dimension technique an improved finite convergence criterion which can find the optimal stationary policy is derived for value iteration approach to demonstrate the theoretical value of this approach it is applied to the optimization problems of the human machine game and the p53 mdm2 gene network

stabilization of boolean control networks under aperiodic sampled data control a novel technique is provided for the global stability analysis of boolean control networks bcns under aperiodic sampled data control asdc the sampling period is allowed to be taken from a lim

brief paper controllability of boolean control networks with time delays in states this paper investigates the controllability of boolean networks with time invariant delays in states after a brief introduction on converting the logic dynamics to discrete time delay systems the controllability via two kinds of controls is studied one kind of control is generated by boolean control networks another kind of control is free boolean sequences in both cases necessary and sufficient conditions of the controllability of boolean control networks are proved finally examples are given to illustrate the efficiency of the obtained results

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

solvability of perturbed elliptic equations with critical growth exponent for the gradient sur un domaine borne cr n on considere le probleme de dirichlet a u g u f u 0 ou a u t 1 n 1 d i a i x u d u definit un operateur pseudomonotone d un espace de sobolev h 0 1 p dans son espace dual et la perturbation g u g x u d u satisfait une condition de croissance

basin bifurcations in quasiperiodically forced coupled systems we study the effect of quasiperiodic forcing on a system of coupled identical logistic maps upon a variation of system parameters a variety of different dynamical regimes can be observed including phenomena such as bistability and multistability at the bifurcation to bistability in a manner reminiscent of attractor expansion at interior crises there is an abrupt change in the size of attractor basins in the bistable region attractor basins undergo additional bifurcations wherein holes and islands are created within the basins when system parameters change these can be understood by examining critical surfaces for the coupled system

state feedback stabilization for boolean control networks state feedback stabilization for boolean control networks is investigated in this technical note based on the algebraic representation of logical dynamics in terms of the semi tensor product of matrices a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller and a general control design approach is proposed when global stabilization is feasible via state feedback instead of designing the logical form of a stabilizing feedback law directly we first construct its algebraic representation and then convert the algebraic representation back to the logical form an example is worked out to illustrate the proposed design procedure

existence results for some quasilinear parabolic equations on considere le probleme de cauchy pour une equation dont la partie principale est un operateur parabolique quasilineaire le terme non lineaire additionnel a au plus une croissance quadratique par rapport au gradient de la solution on demontre l existence d une solution bornee faible sous des hypotheses de regularite minimales en supposant l existence d une sous solution et d une supersolution pour le probleme

bubbling route to strange nonchaotic attractor in a nonlinear series lcr circuit with a nonsinusoidal force we identify an unconventional route to the creation of a strange nonchaotic attractor sna in a quasiperiodically forced electronic circuit with a nonsinusoidal square wave force as one of the quasiperiodic forces through numerical and experimental studies we find that bubbles appear in the strands of the quasiperiodic attractor due to the instability induced by the additional square wave type force the bubbles then enlarge and get increasingly wrinkled as a function of the control parameter finally the bubbles get extremely wrinkled while the remaining parts of the strands of the torus remain largely unaffected resulting in the creation of the sna we term this the bubbling route to the sna we characterize and confirm this creation from both experimental and numerical data using maximal lyapunov exponents and their variance poincar e maps fourier amplitude spectra and spectral distribution functions we also strongly confirm the creation of a sna via the bubbling route by the distribution of the finite time lyapunov exponents

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

holder continuity of the inverse ofp laplacian abstract in this note we study the inverse operator p 1ofp lapalcian on a bounded domain rn we show that p 1 w 1 p w1 p0 is holder continuous and is a compact operator fromv q s tow1 p0 s 0 p q p the conjugate of critical sobolev exponent as an application we study existence of positive solutions of two nonlinear elliptic equations

brief paper an integer programming approach to optimal control problems in context sensitive probabilistic boolean networks a boolean network is one of the models of biological networks such as gene regulatory networks and has been extensively studied in particular a probabilistic boolean network pbn is well known as an extension of boolean networks but in the existing methods to solve the optimal control problem of pbns it is necessary to compute the state transition diagram with 2 n nodes for a given pbn with n states to avoid this computation an integer programming based approach is proposed for a context sensitive pbn cs pbn which is a general form of pbns in the proposed method a cs pbn is transformed into a linear system with binary variables and the optimal control problem is reduced to an integer linear programming problem by a numerical example the effectiveness of the proposed method is shown

lower semicontinuity of functionals via the concentration compactness principle in this paper we show the weak lower semicontinuity of some classes of functionals using the concentration compactness principle of p l lions these functionals involve an integral term and we do not know whether it can be handled by the de giorgi theorem the semicontinuity result allows us to prove several existence results for quasilinear elliptic equations

the birth of strange nonchaotic attractors abstract a mechanism is described for the development of strange nonchaotic attractors from two frequency quasiperiodic attractors in quasiperiodically driven maps this mechanism is intimately tied to the phenomenon of torus doubling the transition to strange nonchaotic behavior occurs when a period doubled torus collides with its unstable parent torus as the collision is approached the period doubled torus becomes extremely wrinkled ultimately becoming fractal at the collision the lyapunov exponent remains negative through the collision these collisions are shown to be a new type of attractor merging crisis the new feature is the possibility of nonchaotic attractors taking part in the crisis this mechanism is illustrated via numerical and analytical studies of a quasiperiodically driven logistic map

lyapunov based stability and construction of lyapunov functions for boolean networks this paper investigates the lyapunov based stability analysis and the construction of lyapunov functions for boolean networks bns and establishes a new framework of lyapunov theory for bns via the semitensor product of matrices first we study how to define a lyapunov function for bns a proper form of pseudo boolean functions is found and the concept of strict lyapunov functions is thus given it is shown that a pseudo boolean function in the proper form can play the role of lyapunov functions for bns based on which several lyapunov based stability results are obtained second we study how to construct a lyapunov function for bns and propose two methods for this problem one is a definition based method and the other is a structure based one third the existence of strict lyapunov functions is studied and a converse lyapunov theorem as well as a necessary and sufficient condition are obtained for the asymptotical stability finally as an application the obtained results are applied to the s

on reducible state variables of logical control networks abstract reducible state variables are important for the study of reduced order observer and control design of logical control networks lcns this paper studies the number of reducible state variables of lcns based on the upper bound of reducible state variables several sufficient necessary conditions are presented for the number of reducible state variables in addition using the regular subspace theory the coordinate transformation as well as the swap matrix are constructed and some new criteria are proposed to calculate reducible state variables

regularizing effect for some p laplacian systems abstract we study existence and regularity of weak solutions for the following p laplacian system p u a 1 u r 2 u f u w 0 1 p p u r w 0 1 p where is an open bounded subset of r n n 2 p v div v p 2 v is the p laplacian operator for 1 p n a 0 r 1 0 p 1 and f belongs to a suitable lebesgue space in particular we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions

asymptotic behaviour of linear dirichlet parabolic problems with variable operators depending on time in varying domains abstract we study the asymptotic behaviour of the solutions of linear parabolic dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously in the limit problem it appear the h limit of the operators and as it is usual in the homogenization of dirichlet problems a new term of order zero we also obtain a corrector result

strange non chaotic attractor in a quasiperiodically forced circle map abstract we show that in the quasiperiodically forced circle map strange non chaotic attractors can appear for non linearities far from the border of chaos the destruction of a two frequency quasiperiodic torus connected with the appearance of a strange non chaotic attractor is described in detail this strange non chaotic attractor is characterized by logarithmically slow diffusion of the phase it is shown that in this regime the high order phase locking states disappear and the rotation number varies rather smoothly with the parameters

local lipschitz continuity of the inverse of the fractional p laplacian holder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains let p 1 s 0 1 and rn an arbitrary bounded open set in the first part we consider the inverse s p p s 1 of the fractional p laplace operator p s with the dirichlet boundary condition we show that in the singular case p 1 2 the operator s p is locally lipschitz continuous on l and that global lipschitz continuity cannot be achieved we use this result to show that in the case n sp if 2nn 2s p 2 0 q npn sp 2 and are small constants then the nonlinear problem psu u qu u npn sp 2u hin u 0 on rn has at least one weak solution in the second part of the paper we prove that the operator p s generates a nonlinear submarkovian semigroup ss p t t 0 on l2 if p 2 and sp 0 such that for every t 0 and u v lq q 2 we have ss p t u ss p t v l c s t s u v lq s where s s and s are explicit constants depending only on n s p q

characterization of noise induced strange nonchaotic attractors received 17 january 2006 revised manuscript received 30 march 2006 published 17 july 2006strange nonchaotic attractors snas were previously thought to arise exclusively in quasiperiodic dynami cal systems a recent study has revealed however that such attractors can be induced by noise in nonquasi periodic discrete time maps or in periodically driven ows in particular in a periodic window of such asystem where a periodic attractor coexists with a chaotic saddle nonattracting chaotic invariant set none ofthe lyapunov exponents of the asymptotic attractor is positive small random noise is incapable of causingcharacteristic changes in the lyapunov spectrum but it can make the attractor geometrically strange bydynamically connecting the original periodic attractor with the chaotic saddle here we present a detailed studyof noise induced snas and the characterization of their properties numerical calculations reveal that thefractal dimensions of noise induced snas typically assume fractional values in contrast to snas in quasip eriodically driven systems whose dimensions are integers an interesting nding is that the uctuations of the nite time lyapunov exponents away from their asymptotic values obey an exponential distribution thegenerality of which we are able to establish by a theoretical analysis using random matrices we suggest apossible experimental test we expect noise induced snas to be general doi 10 1103 physreve 74 016203 pacs number s 05 45 gg

stability and set stability in distribution of probabilistic boolean networks we propose a new concept stability in distribution sd of a probabilistic boolean network pbn which determines whether the probability distribution converges to the distribution of the target state namely a one point distributed random variable in a pbn stability with probability one stability in the stochastic sense and sd are equivalent the sd is easily generalized to subset stability i e to set stability in distribution ssd we prove that the transition probability from any state to an invariant subset or to a fixed point is nondecreasing in time this monotonicity is an important property in establishing stability criteria and in calculating or estimating the transient period we also obtain a verifiable necessary and sufficient condition for sd of pbns with independently and identically distributed switching we then show that sd problems of pbns with markovian switching and pbn synchronizations can be recast as ssd problems of markov chains after calculating the largest invariant subset of a markov chain in a given set by the newly proposed algorithm we propose a necessary and sufficient condition for ssds of markov chains the proposed method and results are supported by examples

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

bifurcations and transitions in the quasiperiodically driven logistic map abstract we discuss several bifurcation phenomena that occur in the quasiperiodically driven logistic map this system can have strange nonchaotic attractors snas in addition to chaotic and regular attractors on snas the dynamics is aperiodic but the largest lyapunov exponent is nonpositive there are a number of different transitions that occur here from periodic attractors to snas from snas to chaotic attractors etc we describe some of these transitions by examining the behavior of the largest lyapunov exponent distributions of finite time lyapunov exponents and the invariant densities in the phase space

birth of strange nonchaotic attractors due to interior crisis abstract we study the interior crisis in the period 3 window of the quasiperiodically forced logistic map two routes from quasiperiodicity to chaos involving strange nonchaotic attractors sna are discovered along one route we observe a sudden widening of the sna this is similar to the interior crisis in chaotic systems along the other route we find a direct transition from an invariant curve to a strange nonchaotic attractor exactly at the interior crisis point this is a new mechanism of the appearance of strange nonchaotic attractors beyond the interior crisis the temporal behavior can be described as a crisis induced intermittency whose scaling behavior is discussed

global robust stability and stabilization of boolean network with disturbances abstract based on semi tensor product of matrices global robust stability and stabilization of boolean network bn with disturbances are investigated firstly the definition of global robust stability for bn with disturbances is presented by using the algebraic state space representation of disturbed bn some necessary and sufficient criteria are obtained to ensure the global robust stability with respect to w r t a fixed point or w r t a limit cycle in contrast if a given disturbed bn is not globally robust stable w r t a fixed point or w r t a limit cycle system can achieve stability by a matrix transformation technique however it is a difficult task to find such a suitable matrix transformation in this paper a pinning state feedback control design is proposed to find a suitable transformation the obtained results are well illustrated by numerical example

non autonomous perturbations for a class of quasilinear elliptic equations on r abstract this paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on r involving the p laplacian with a non autonomous perturbation the first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain pass argument the special features of the problem here is the complex structure of the linear part which in particular oblige to work into the space w 1 p r then one faces problems in the convergence of the sequences of derivatives u n u

fractalization of a torus as a strange nonchaotic attractor fractalization of a torus and its transition to chaos in a quasiperiodically forced logistic map is reinvestigated in relation to a strange nonchaotic attractor with the aid of a functional equation for the invariant curve the existence of a fractal torus in an interval in parameter space is confirmed by the length and the number of extrema of the torus attractor as well as the fourier mode analysis mechanisms of the onset of a fractal torus and the transition to chaos are studied in connection with the intermittency

the transformation between the galois nlfsrs and the fibonacci nlfsrs via semi tensor product of matrices in this paper a novel way is proposed to investigate the transformation between galois nonlinear feedback shift registers nlfsrs and fibonacci nlfsrs first the galois nlfsrs and fibonacci nlfsrs are regarded as two boolean networks bns and the corresponding algebraic equations are obtained based on semi tensor product stp of matrices then the definition of absolute equivalence is given to investigate the transformation between these two kinds of nlfsrs furthermore some interesting results are presented to achieve the transformation finally an example is provided to illustrate the effectiveness of obtained results

intermittency route to strange nonchaotic attractors in a non skew product map whether strange nonchaotic attractors snas can typically arise in non skew product maps has been a crucial question for more than two decades recently it was shown that snas arise in a particular non skew product map related to quasiperiodically driven continuous dynamical systems r badard chaos solitons fractals 28 1327 2006 chaos 18 023127 2008 in the present paper we derive badard s non skew product map from a periodically driven continuous dynamical system with spatially quasiperiodic potential and investigate onset mechanisms of snas in the map in particular we focus on a transition route to intermittent snas where snas appear after pair annihilations of stable and unstable fixed points located on a ring shaped invariant curve then the mean residence time and rotation numbers have a logarithmic singularity finally we discuss the existence of snas in a special class of non skew product maps

optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a hardy potential abstract we deal with the following parabolic problem u t u u p u x 2 f u 0 in 0 t u x t 0 on 0 t u x 0 u 0 x x where r n n 3 is a bounded regular domain such that 0 or r n p 1 0 and f 0 u 0 0 are in a suitable class of functions there are deep differences with respect to the heat equation 0 the main features in the paper are the following if 0 there exists a critical exponent p such that for p p there is no nontrivial local solution p is optimal in the sense that if p p there exists solution for suitable data if we consider the cauchy problem i e r n we find the same phenomenon about the critical power p as above moreover there exists a fujita type exponent f p in the sense that independently of the initial datum for 1 p f any solution blows up in a finite time respect to an integral norm this is a major difference with respect to the heat equation 0

boundary crisis in quasiperiodically forced systems abstract we study boundary crisis in quasiperiodically forced dissipative systems using the henon map as a characteristic example the crisis is due to a homoclinic tangency of the stable and unstable manifolds of an accessible invariant circle of saddle type on the basin boundary of the attractor numerical evidence shows that the type of boundary crisis can change as the saddle circle loses its accessibility due to the quasiperiodic forcing this implies the existence of codimension two double crisis vertices where a curve of boundary crisis and a curve of interior crisis meet we argue that bifurcation points of higher codimension must exist in the full parameter space

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

the p dirichlet to neumann operator with applications to elliptic and parabolic problems abstract in this paper we investigate the dirichlet to neumann operator associated with second order quasi linear operators of p laplace type for 1 p which acts on the boundary of a bounded lipschitz domain in r d for d 2 we establish well posedness and holder continuity with uniform estimates of weak solutions of some elliptic boundary value problems involving the dirichlet to neumann operator by employing these regularity results of weak solutions of elliptic problems we show that the semigroup generated by the negative dirichlet to neumann operator on l q enjoys an l q c 0 smoothing effect and the negative dirichlet to neumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and order preserving semigroup moreover we establish convergence in large time with decay rates of all trajectories of the semigroup and in the singular case 1 e 2 d d 2 p 2 for some e 0 we give upper estimates of the finite time of extinction

state feedback stabilization for probabilistic boolean networks abstract a probabilistic boolean network pbn is a discrete time system composed of a family of boolean networks bns between which the pbn switches in a stochastic fashion studying control related problems in pbns may provide new insights into the intrinsic control in biological systems and enable us to develop strategies for manipulating complex biological systems using exogenous inputs this paper investigates the problem of state feedback stabilization for pbns based on the algebraic representation of logic functions a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller and a control design method is proposed when the presented condition holds it is shown that the controller designed via the proposed procedure can simultaneously stabilize a collection of pbns that are composed of the same constituent bns

set stability and set stabilization of boolean control networks based on invariant subsets this study addresses the set stability of boolean networks bns and set stabilization of boolean control networks bcns set stability determines whether a bn converges to a given subset whereas set stabilizability addresses the issue of whether a bcn can be stabilized to a given subset many problems can be viewed as special cases of set stability and set stabilization including synchronization partial stability and partial stabilization problems the concepts of invariant subset and control invariant subset are introduced then algorithms for the largest invariant subset and the largest control invariant subset contained in a given subset are proposed based on the invariant subsets obtained the necessary and sufficient conditions for set stability and set stabilizability are established and formulae are provided to calculate the shortest transient periods for respective initial states a design procedure is proposed for finding all the time optimal set stabilizers finally an example is used to show the application of the proposed results to the synchronization problem of bns

nonchaotic stagnant motion in a marginal quasiperiodic gradient system a one dimensional dynamical system with a marginal quasiperiodic gradient is presented as a mathematical extension of a nonuniform oscillator the system exhibits a nonchaotic stagnant motion which is reminiscent of intermittent chaos in fact the density function of residence times near stagnation points obeys an inverse square law due to a mechanism similar to type i intermittency however unlike intermittent chaos in which the alternation between long stagnant phases and rapid moving phases occurs in a random manner here the alternation occurs in a quasiperiodic manner in particular in the case of a gradient with the golden ratio the renewal of the largest residence time occurs at positions corresponding to the fibonacci sequence finally the asymptotic long time behavior in the form of a nested logarithm is theoretically derived compared with the pomeau manneville intermittency a significant difference in the relaxation property of the long time average of the dynamical variable is found