averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time varying within the liapunov framework a sufficient condition for uniform asymptotic stability of ordinary differential equations is proposed unlike with classical liapunov theory the time derivative of the v function taken along solutions of the system may have positive and negative values it is shown that the proposed condition is useful for the study of uniform asymptotic stability of homogeneous systems with order tau 0 in particular it is established that asymptotic stability of the averaged homogeneous system implies local uniform asymptotic stability of the original time varying homogeneous system this shows that averaging techniques play a prominent role in the study of homogeneous not necessarily fast time varying systems

aubry mather sets for finite twist maps of a cylinder and semilinear duffing equations in this paper based on a generalized version of the aubry mather theorem about twist maps of an infinite cylinder without the usual infinite twist assumption we establish the existence of certain quasiperiodic solutions and unlinked periodic solutions in some semilinear duffing equations x g x p t where p t is a continuous periodic function of period 1 and the semilinearity means that 0 g x x k

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

fuzzy normal forms abstract linguistic expressions of a natural language are transformed at least three times before they are expressed in computational formulas between the linguistic and the computational expressions there are the meta linguistic and propositional expressions under various interpretations and assumptions fuzzy normal forms are canonical expressions derived at the propositional level with particular assumptions the fuzzy normal forms can be generated from fuzzy truth tables with a normal form derivation algorithm and an orthodox interpretation of conjunction disjunction and complementation operations it is shown that these fuzzy normal forms display a unique structure for certain general classes of crisp operators over fuzzy sets the fuzzy normal forms generated by crisp operators over fuzzy sets that correspond to max min standard complement de morgan triple are shown to be equivalent to the classical normal forms of two valued logic i e disjunctive and conjunctive normal forms known as dnf and cnf with a particular property i e the crisp containment fdnf 3 fcnf 3 whereas the fuzzy normal forms generated by crisp operators that correspond to non idempotent t norms t conorms and standard complement de morgan triples do not have in general a similar crisp containment property i e fdnf 2 fcnf 2 neither do the fuzzy normal forms generated by crisp operators over fuzzy sets that correspond to non idempotent and non commutative and non associative conjunction disjunction and standard complement de morgan triples have a crisp containment property i e fdnf 1 fcnf 1 however they still provide bounds on the combination of concepts

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

global stability results for switched systems based on weak lyapunov functions in this paper we study the stability of nonlinear and time varying switched systems under restricted switching we approach the problem by decomposing the system dynamics into a nominal like part and a perturbation like one most stability results for perturbed systems are based on the use of strong lyapunov functions i e functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state however switched systems under restricted switching may not admit strong lyapunov functions even when asymptotic stability is uniform over the set of switching signals considered the main contribution of the current paper consists in providing stability results that are based on the stability of the nominal like part of the system and require only a weak lyapunov function these results may have wider applicability than results based on strong lyapunov functions the results provided follow two lines first we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal like and the perturbation like parts of the system second we provide input to state stability iss results for the case when the nominal like part is switched linear time varying we provide two types of iss results standard iss that involves the essential supremum norm of the input and a modified iss that involves a power type norm

subharmonic solutions of second order subquadratic hamiltonian systems with potential changing sign abstract with the aid of some symplectic transformations the existence of subharmonic solutions of the second order hamiltonian system v x t x is considered where the potential v is subquadratic in x as x goes to infinity and can change sign with respect to t

a general invariance principle for nonlinear time varying systems and its applications a general invariance principle from the output to state point of view is proposed for the dynamical analysis of nonlinear time varying systems this is achieved by the construction of a simple and intuitive criterion using integral inequality of the output function and modified detectability conditions the proposed scheme can be viewed as an extension of the integral invariance principle c i byrnes and c f martin 1995 for time invariant systems to time varying systems such extension is nontrivial and can be used in various research areas such as adaptive control tracking control and the control of driftless systems an application to global tracking control of four wheeled mobile robots is given to demonstrate the feasibility and validity of the proposed approach

the stability of the equilibrium of a conservative system abstract in this paper we consider the conservative system formula where a t is a continuous and 1 periodic function in the time t e t x is also 1 periodic in the time t and dominated by the power x 2 n 2 in a neighborhood of x 0 a sufficient and necessary condition for the stability in the liapunov sense of the trivial solution x 0 is given

numerical determination of the basin of attraction for asymptotically autonomous dynamical systems we develop a method to numerically analyse asymptotically autonomous systems of the form dot x f t x where f t x tends to g x as t the rate of convergence is not limited to exponential but may be polynomial logarithmic or any other rate for these systems we propose a transformation of the infinite time interval to a finite compact one which reflects the rate of convergence of f to g in the transformed system the origin is an asymptotically stable equilibrium which is exponentially stable in x direction weconsider a lyapunov function in this transformed system as a solution of a suitable linear first order partial differential equation and approximate it using radial basis functions

growth conditions for exponential stability of time varying perturbed systems in this paper we derive some sufficient conditions for practical uniform exponential stability of time varying perturbed systems based on lyapunov techniques whose dynamics are in general unbounded in time in the sense that the solutions are uniform stable and converge to a small neighbourhood of the origin under quite general assumptions we first present a new converse stability theorem for a large class of time varying systems which will be used to prove certain stability properties of nonlinear systems with perturbation therefore a new lyapunov function is presented that guarantees practical uniform exponential stability of perturbed systems furthermore some illustrative examples are presented

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

on lasalle s invariance principle and its application to robust synchronization of general vector lie spl acute nard equations a functional version of lasalle s invariance principle is derived i e rather than the usual pointwise lyapunov like functions it uses specially constructed functionals along system trajectories this modification enables the principle to handle even nonautonomous systems to which the classical lasalle s principle is not directly applicable the new theoretical results are then used to study robust synchronization of general lie spl acute nard type of systems the developed technique is finally applied to chaotic oscillators synchronization numerical simulation is included to demonstrate the effectiveness of the proposed methodology

brief on exponential stability of nonlinear time varying differential equations within the liapunov framework a sufficient condition for exponential stability of ordinary differential equations is proposed unlike with classical liapunov theory the time derivative of the liapunov function taken along solutions of the system may have positive and negative values verification of the conditions of the main theorem may be harder than in the classical case it is shown that the proposed conditions are useful for the investigation of the exponential stability of fast time varying systems this sets the stability study by means of averaging in a liapunov context in particular it is established that exponential stability of the averaged system implies exponential stability of the original fast time varying system a comparison of our work with results taken from the literature is included

periodic solutions of second order superquadratic hamiltonian systems with potential changing sign ii abstract we consider the periodic solutions of the second order hamiltonian system x x h t v x with v being positive and superquadratic at infinity h being c 1 2 periodic sign changing and all zeros being simple some existence and multiplicity results of periodic solutions are given

subharmonic solutions for nonautonomous sublinear second order hamiltonian systems some existence theorems are obtained for subharmonic solutions of nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

on the equivalence relations of detectability and pe conditions this note investigates equivalent relations of detectability and persistency excitation pe for general nonlinear time varying systems three new definitions relating to the detectability are proposed and discussed as a preliminary result we show that these detectability conditions are all equivalent under a mild assumption furthermore two pe conditions that are a nonlinear version extension of the pe proposed in present literature are defined very interestingly it can be shown that these nonlinear version pe conditions are both equivalent to the proposed detectability conditions under the same assumption to the best of our knowledge this is a first result about the connection between detectability and pe in the area of nonlinear time varying systems moreover by combining a newly developed stability criterion with the proposed results an interesting example is given to illustrate how the classic pe conditions for linear systems can be extended and used in nonlinear systems from this example it can also be seen that the uniformly asymptotic stability can be guaranteed either using the detectability conditions or by the pe conditions based on our approach

symplectic transformations and periodic solutions of hamiltonian systems abstract symplectic transformations with a kind of homogeneity are introduced which enable us to give a unified approach to some existence problems on the periodic solutions for the first and second order hamiltonian systems

periodic and subharmonic solutions of a class of subquadratic second order hamiltonian systems some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

asymptotic stability of nonautonomous systems by liapunov s direct method abstract this paper discusses the asymptotic stability of nonautonomous systems by means of the direct method of liapunov we assume existence of a positive time invariant liapunov function with negative semi definite derivative we focus on the extra conditions needed in order to guarantee asymptotic stability and present a theorem which offers a new criterion for asymptotic stability the theorem is illustrated first by recovering barbashin s principle for autonomous and periodic systems and second by studying the nonlinear pendulum with time varying damping for which we derive a new asymptotic stability result

a new asymptotic stability criterion for nonlinear time variant differential equations a new sufficient condition for asymptotic stability of ordinary differential equations is proposed unlike classical lyapunov theory the time derivative along solutions of the lyapunov function may have positive and negative values the classical lyapunov approach may be regarded as an infinitesimal version of the present theorem verification in practical problems is harder than in the classical case an example is included in order to indicate how the present theorem may be applied

synchronization with partial state coupling on so n this paper studies autonomous synchronization of k agent orientations in so n by using links that each convey a priori different partial state information between two agents the latter compare the action of their states on a common link dependent vector of mathbb r n such setting appears when each pair of agents compares the expression in their respective body frames of a common external direction force field heading of moving object etc to synchronize their states agents must combine this information over time or throughout the network a gradient coupling law for synchronization is proposed extensive convergence analysis of the coupled agents is provided both for fixed and time varying reference vectors the results are compared to a similar setting with states in mathbb r n and agents coupled through scalar product of their state with a reference vector

a sufficient condition for asymptotic stability for a class of time varying parameterized systems abstract we derive sufficient conditions for local asymptotic stability for a class of parameterized time varying systems whose dynamics are in general unbounded in time our main result generalizes a well known limiting dynamics based approach due to peuteman and aeyels 1999 in 9 for fast time varying systems whose dynamics are bounded in time its proof is based on a new result providing a lyapunov characterization for exponential stability