cyclic differential equations and period three solutions of differential delay equations where g c is a given function ni are integers and a is a real parameter such equations have been considered in several papers for instance 2 5 6 it has become apparent that it is important to be able to show that 0 1 has no nontrivial periodic solutions of certain periods for example the global hopf bifurcation theorem as developed by alexander and yorke for ordinary differential equations and later extended to functional differential equations 12 6 1 gives relatively little information about 0 i if one does not know that 0 1 has no nontrivial solutions of certain periods a well known result see lemma 4 1 in 21 asserts that eq 0 i can have no nonconstant periodic solutions of period 2 in this paper we show that for certain classes of g 0 1 can have no nonconstant periodic solutions of period 3 the general idea that periodic solutions of 0 1 of period n n an integer satisfy an associated system of ordinary differential equations has been used before see 141 and lemma 4 1 of 121 however in our case we deal with a system of three first order odes instead of two odes as in lemma4 1 of 121

adaptive neural control for a class of switched nonlinear systems in this paper adaptive neural control is presented for a class of switched nonlinear systems with switching jumps and uncertainties in both system models and switching signals condition on limiting variation of the lyapunov function is given for input to state asymptotic stability of switched systems with switching jumps the control objective is achieved uniformly with respect to a class of switching signals the coupled difficulties from the discrepancy between control gains and switching jumps are overcome by discontinuous adaptive neural control combined with the classical adaptive control smooth approximations of the discontinuous controls are then presented for a systematic design procedure

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

uniform stabilization of nonlinear systems with arbitrary switchings and dynamic uncertainties we solve the problem of global uniform input to state stabilization of nonlinear switched systems with time varying and periodic dynamics with dynamic uncertainties and with external disturbances the switching signal is assumed to be unknown and the dynamics of the known components of the state vector is equivalent to the general triangular form gtf with non invertible input output maps in our first and most general result we prove that if the dynamic uncertainty is treated as external disturbance then the general triangular form system can be stabilized with arbitrarily small gain w r t the dynamic uncertainty by means of a switching independent smooth and periodic feedback hence using a suitable extension of the well known small gain theorem to our case of switched systems with arbitrary switchings we obtain the uniform input to state stabilization of the entire interconnected system the second part of the paper addresses a more special case of triangular form tf switched systems with right invertible input output i o maps with unknown switchings and with dynamic uncertainties we show that the design becomes simpler and more constructive and the controllers become time invariant if the dynamics is autonomous in this special case finally we consider an example with explicit design of the stabilizing controllers

multiplicity results of periodic solutions for a class of second order delay differential systems abstract this paper is concerned with the periodic boundary value problem 1 u t u t f t u t a e t 0 2 u 0 u 2 u 0 u 2 0 where 0 is a given constant r is a parameter the variational principle is given and some multiplicity results of periodic solutions of 1 are obtained directly via variational methods

an explicit formula of bounded feedback stabilizers for feedforward systems abstract this work constitutes continuation of a recent result of the same authors which provides an explicit formula for bounded feedback stabilizers for a wide class of triangular nonlinear systems in the present paper the previous result is extended for nonlinear systems having feedforward structure and whose linearization at the equilibrium is in general uncontrollable

nontrivial periodic solutions for delay differential systems via morse theory abstract the existence of the nontrivial periodic solutions to the system of delay differential equations 1 1 x t f x t is investigated where x r n f c r n r n we prove several new existence results via morse theory and the galerkin methods

multiplicity results of periodic solutions for systems of first order delay differential equation abstract this paper is concerned with the periodic solutions of the following delay non autonomous systems 1 u t f t u t r where r 0 f c r 1 r n r n satisfies f t r z f t z for all z r n some multiplicity results of periodic solutions of 1 are obtained via variational methods

equivalence of nonlinear systems to triangular form the singular case the problem of state equivalence of a given nonlinear system to a triangular form is considered here the solution of this problem has been known for the regular case i e when there exists a certain nested sequence of regular and involutive distributions it is also known that in this case the corresponding system is linearizable using a smooth coordinate change and static state feedback this paper deals with the singular case i e when the nested sequence of involutive distributions of the system contains singular distributions special attention is paid to the so called bijective triangular form geometric coordinates free criteria for the solution of the above problem as well as constructive verifiable procedures are given these results are used to obtain a result in the nonsmooth stabilization problem

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

partial state global stabilization for general triangular systems abstract we deal with the partial state global stabilization problem for a wide class of nonlinear systems which contains those having triangular structure the sufficient conditions we propose guarantee global stabilization by means of an output smooth feedback integrator

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

smooth bifurcation of symmetric periodic solutions of functional differential equations with odd f c r in contrast to the one dimensional ordinary differential equation this equation can have periodic solutions as the example i t 7r 2 x t 1 with solutions x t z sin rc 2 t z 0 shows numerical bifurcation diagrams e g 6 143 indicate a complicated structure of the period solutions depending on the delay parameter u one finds various intersecting smooth curves one task of mathematics is to prove the existence of periodic solutions in this paper we are mainly interested in symmetric periodic solutions i e solutions x of c which fultil the symmetry property

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

robust stabilization of the generalized triangular form nonlinear systems with disturbances we investigate the problem of global uniform input to state stabilization of nonlinear generalized triangular form gtf control systems of ode with time varying and periodic dynamics and with external essentially bounded disturbances our first main result is a statement which is a kind of extension of theorems on adding an integrator to the case of input to state stabilization of gtf systems as a corollary we immediately obtain our second main result on global uniform input to state stabilzation of gtf systems w r t the external disturbances the latter is a generalization of the recent result devoted to the global asymptotic stabilization of the gtf systems without disturbances it is essential that the proof of the main result is based on the converse iss lyapunov theorems for the time varying systems which allows to simplify construction proposed our recent work

brief paper backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings this paper is concerned with the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings two classes of state feedback controllers and a common lyapunov function clf are simultaneously constructed by backstepping the first class uses the common state feedback controller which is independent of switching signals the other class utilizes individual state feedback controllers for the subsystems as an extension of the designed method the global stabilization problem under arbitrary switchings for switched nonlinear systems in nested lower triangular form is also studied an example is given to show the effectiveness of the proposed method

a modified prony algorithm for exponential function fitting a modification of the classical technique of prony for fitting sums of exponential functions to data is considered the method maximizes the likelihood for the problem unlike the usual implementation of prony s method which is not even consistent for transient signals proves to be remarkably effective in practice and is supported by an asymptotic stability result novel features include a discussion of the problem parametrization and its implications for consistency the asymptotic convergence proofs are made possible by an expression for the algorithm in terms of circulant divided difference operators

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization for a class of nonlinear systems having triangular structure we provide explicit formulas for feedback controllers that exhibit global stabilization the global part of our analysis is mainly based on a version of sontag s input to state stability property and constitutes a generalization of previous works of the second author

circulant matrices and differential delay equations on etudie les solutions periodiques non triviales de familles parametrisees d equations differentielles a retard

persistence of a class of periodic kolmogorov systems in this paper we study the persistence of a periodic kolmogorov system which models the growth of n interacting biological species the system is assumed to have a simple dynamic in the sense that each coordinate subsystem possesses a global attractor our research is mainly addressed to the case in which there exists at least a predator among the n species however we also obtain a result about persistence for a class of competitive systems

homoclinic orbits for nonlinear difference equations containing both advance and retardation in this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions moreover if the nonlinearity is an odd function the existence of an unbounded sequence of homoclinic orbits is obtained

global properties of the triangular systems in the singular case we introduce a new class of the triangular multi input and multi output control systems of o d e which are not feedback linearizable and investigate its global behavior the triangular form introduced is a generalization of the classes of triangular systems considered before for our class we solve the problem of global robust controllability combining our main result with that of f h clarke yu s ledyaev e d sontag a i subbotin asymptotic controllability implies feedback stabilization ieee trans automat control 42 1997 1394 1407 we obtain a corollary on the global discontinuous sampled stabilization an example showing that global smooth stabilization can be irrelevant to the singular case is considered to prove our main result we apply a certain back stepping algorithm and combine the technique proposed in v i korobov s s pavlichkov w h schmidt global robust controllability of the triangular integro differential volterra systems j math anal appl 309 2005 743 760 with solving a specific problem of global practical stabilization by means of a discontinuous time varying feedback law