regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

nonlinear measures a new approach to exponential stability analysis for hopfield type neural networks in this paper a new concept called nonlinear measure is introduced to quantify stability of nonlinear systems in the way similar to the matrix measure for stability of linear systems based on the new concept a novel approach for stability analysis of neural networks is developed with this approach a series of new sufficient conditions for global and local exponential stability of hopfield type neural networks is presented which generalizes those existing results by means of the introduced nonlinear measure the exponential convergence rate of the neural networks to stable equilibrium point is estimated and for local stability the attraction region of the stable equilibrium point is characterized the developed approach can be generalized to stability analysis of other general nonlinear systems

complex bounds for real maps in this paper we prove complex bounds also referred to as a priori bounds for c 3 and in particular for analytic maps of the interval any c 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval we associate to such a map a complex box mapping which provides a kind of markov structure for the dynamics moreover we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain such bounds show that the first return maps to these domains are well controlled and consequently such bounds form one of the corner stones in many recent results in one dimensional dynamics for example renormalization theory rigidity density of hyperbolicity and local connectivity of julia sets

global exponential stability of impulsive discrete time neural networks with time varying delays this paper studies the problem of global exponential stability and exponential convergence rate for a class of impulsive discrete time neural networks with time varying delays firstly by means of the lyapunov stability theory some inequality analysis techniques and a discrete time halanay type inequality technique sufficient conditions for ensuring global exponential stability of discrete time neural networks are derived and the estimated exponential convergence rate is provided as well the obtained results are then applied to derive global exponential stability criteria and exponential convergence rate of impulsive discrete time neural networks with time varying delays finally numerical examples are provided to illustrate the effectiveness and usefulness of the obtained criteria

a new development for the tikhonov theorem in nonlinear singular perturbation systems abstract this paper deals with the exponential stability of nonlinear perturbation systems under a new condition a novel criterion of exponential stability of nonlinear systems is firstly given in a general form in this criterion a new kind of characteristic value is introduced which makes the exponential stability measurable based on this criterion a new development for the tikhonov theorem in nonlinear singular perturbation systems is presented

stability analysis of dynamical neural networks in this paper we use the matrix measure technique to study the stability of dynamical neural networks testable conditions for global exponential stability of nonlinear dynamical systems and dynamical neural networks are given it shows how a few well known results can be unified and generalized in a straightforward way local exponential stability of a class of dynamical neural networks is also studied we point out that the local exponential stability of any equilibrium point of dynamical neural networks is equivalent to the stability of the linearized system around that equilibrium point from this some well known and new sufficient conditions for local exponential stability of neural networks are obtained

erratum to real bounds ergodicity and negative schwarzian for multimodal maps here as before we define an open interval k to be nice if no iterate of k enters k this implies that if k1 and k2 are pullbacks of k then they are either disjoint or nested in lemma 9 page 762 it was implicitly assumed that v is disjoint from jn it is for this reason that the proof of theorem c 1 does not work unless we assume v is nice or something similar the proof of theorem c 1 as stated above is essentially the same as before using lemma 6 below instead of lemma 6 then in lemma 9 page 762 we do not need to require that kn 1 is a jump time provided we assume that v is nice making the additional assumption that v is nice proposition 1 and its proof and the rest of the paper go through unchanged lemma 6 for each 0 sufficiently small there exists 3 0 such that if i is a scaled neighbourhood of a nice interval v i then j is a 3 scaled neighbourhood of any component a of k j v where k 1 is arbitrary

novel delay dependent global asymptotic stability condition of hopfield neural networks with delays in this paper the global asymptotic stability of hopfield neural networks with delays is investigated distinct differences from other analytical approaches lie in transforming to an equivalent system by using a parameterized transformation which allows free variables in an operator a novel less conservative and restrictive criterion than those established in the earlier references is given in terms of several matrix inequalities by utilizing the lyapunov theory and matrix inequality framework the results are related to the size of delays numerical examples are given to show the effectiveness of our proposed method

nonfragile h state estimation for recurrent neural networks with time varying delays on proportional integral observer design in this article a novel proportional integral observer pio design approach is proposed for the nonfragile h state estimation problem for a class of discrete time recurrent neural networks with time varying delays the developed pio is equipped with more design freedom leading to better steady state accuracy compared with the conventional luenberger observer the phenomena of randomly occurring gain variations which are characterized by the bernoulli distributed random variables with certain probabilities are taken into consideration in the implementation of the addressed pio attention is focused on the design of a nonfragile pio such that the error dynamics of the state estimation is exponentially stable in a mean square sense and the prescribed h performance index is also achieved sufficient conditions for the existence of the desired pio are established by virtue of the lyapunov krasovskii functional approach and the matrix inequality technique finally a simulation example is provided to demonstrate the effectiveness of the proposed pio design scheme

robust exponential stability of uncertain stochastic neural networks with distributed delays and reaction diffusions this paper considers the problem of stability analysis for uncertain stochastic neural networks with distributed delays and reaction diffusions two sufficient conditions for the robust exponential stability in the mean square of the given network are developed by using a lyapunov krasovskii functional an integral inequality and some analysis techniques the conditions which are expressed by linear matrix inequalities can be easily checked two simulation examples are given to demonstrate the reduced conservatism of the proposed conditions

diffusion limited aggregation and its response to anisotropy the stability of diffusion limited growths with n equivalent major fingers is investigated in two dimensions using a conformal mapping the results imply that square lattice but not hexagonal lattice bias are relevant in simple diffusion limited aggregation dla in general it is found that the maximum number of fingers stable with respect to finger loss by competition is given by nmax 2 2 d 1 where d is the apparent fractal dimension of the fingers at least when n is even computer simulations of dla with n fold symmetric growth rules are shown for n 5 8 12 finally it is argued that unbiased isotropic fractal dla should lie at the stability limit when combined with cone angle arguments independently relating d to the effective numbers of fingers this determines the value d 1 12 2 1 707 for dla in two dimensions this corresponds to a characteristic exterior half angle at the leading tips given by 2

absolute exponential stability of a class of continuous time recurrent neural networks this paper presents a new result on absolute exponential stability aest of a class of continuous time recurrent neural networks with locally lipschitz continuous and monotone nondecreasing activation functions the additively diagonally stable connection weight matrices are proven to be able to guarantee aest of the neural networks the aest result extends and improves the existing absolute stability and aest ones in the literature

adaptive boundary iterative learning control for an euler bernoulli beam system with input constraint this paper addresses the vibration control and the input constraint for an euler bernoulli beam system under aperiodic distributed disturbance and aperiodic boundary disturbance hyperbolic tangent functions and saturation functions are adopted to tackle the input constraint a restrained adaptive boundary iterative learning control abilc law is proposed based on a time weighted lyapunov krasovskii like composite energy function in order to deal with the uncertainty of a system parameter and reject the external disturbances three adaptive laws are designed and learned in the iteration domain all the system states of the closed loop system are proved to be bounded in each iteration along the iteration axis the displacements asymptotically converge toward zero simulation results are provided to illustrate the effectiveness of the proposed abilc scheme

on the global stability of delayed neural networks lyapunov functional methods combining with some inequality techniques are employed to study the global asymptotic stability of delayed neural networks without assuming lipschitz conditions on the activation functions a new sufficient condition is established such criteria allows us to include non lipschitzian activation functions in the design of delayed neural networks the result presented here is also discussed from the point of view of its relationship to some previous results

global attractivity in delayed hopfield neural network models two different approaches are employed to investigate the global attractivity of delayed hopfield neural network models without assuming the monotonicity and differentiability of the activation functions liapunov functionals and functions combined with the razumikhin technique are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays in the case of monotone and smooth activation functions the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model such criteria depend on the magnitude of delays and show that self inhibitory connections can contribute to the global convergence

global convergence and asymptotic stability of asymmetric hopfield neural networks abstract the global convergence and asymptotic stability of hopfield neural networks are known to be bases of successful applications of networks in various computing and recognition tasks however all previous studies on the networks assumed that the interconnection matrix is symmetric or antisymmetric in this paper the two fundamental properties of the networks are studied without a symmetry assumption it is proved that the networks will be globally convergent to a stable state if the interconnection matrix is weakly diagonally dominant in a sense to be defined furthermore under any one of conditions assuring global convergence of the network the maximal attraction radius of any one of stable states is found to be half of the distribution distance of the state to the network the obtained results not only generalize the existing results but also provide a theoretical foundation of performance analysis and new applications of the hopfield networks

event triggered state estimation for discrete time multidelayed neural networks with stochastic parameters and incomplete measurements in this paper the event triggered state estimation problem is investigated for a class of discrete time multidelayed neural networks with stochastic parameters and incomplete measurements in order to cater for more realistic transmission process of the neural signals we make the first attempt to introduce a set of stochastic variables to characterize the random fluctuations of system parameters in the addressed neural network model the delays among the interconnections are allowed to be different which are more general than those in the existing literature the incomplete information under consideration includes randomly occurring sensor saturations and quantizations for the purpose of energy saving an event triggered state estimator is constructed and a sufficient condition is given under which the estimation error dynamics is exponentially ultimately bounded in the mean square it is worth noting that the ultimate boundedness of the error dynamics is explicitly estimated the characterization of the desired estimator gain is designed in terms of the solution to a certain matrix inequality finally a numerical simulation example is presented to illustrate the effectiveness of the proposed event triggered state estimation scheme

stability and synchronization of discrete time markovian jumping neural networks with mixed mode dependent time delays in this paper we introduce a new class of discrete time neural networks dnns with markovian jumping parameters as well as mode dependent mixed time delays both discrete and distributed time delays specifically the parameters of the dnns are subject to the switching from one to another at different times according to a markov chain and the mixed time delays consist of both discrete and distributed delays that are dependent on the markovian jumping mode we first deal with the stability analysis problem of the addressed neural networks a special inequality is developed to account for the mixed time delays in the discrete time setting and a novel lyapunov krasovskii functional is put forward to reflect the mode dependent time delays sufficient conditions are established in terms of linear matrix inequalities lmis that guarantee the stochastic stability we then turn to the synchronization problem among an array of identical coupled markovian jumping neural networks with mixed mode dependent time delays by utilizing the lyapunov stability theory and the kronecker product it is shown that the addressed synchronization problem is solvable if several lmis are feasible hence different from the commonly used matrix norm theories such as the m matrix method a unified lmi approach is developed to solve the stability analysis and synchronization problems of the class of neural networks under investigation where the lmis can be easily solved by using the available matlab lmi toolbox two numerical examples are presented to illustrate the usefulness and effectiveness of the main results obtained

feigenbaum coullet tresser universality and milnor s hairiness conjecture we prove the feigenbaum coullet tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type this gives the first computer free proof of the original feigenbaum observation of the universal parameter scaling laws we use the hyperbolicity theorem to prove milnor s conjectures on self similarity and hairiness of the mandelbrot set near the corresponding parameter values we also conclude that the set of real infinitely renormalizable quadratics of type bounded by some n 1 has hausdorff dimension strictly between 0 and 1 in the course of getting these results we supply the space of quadratic like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension one foliation of the connectedness locus

some generalized sufficient convergence criteria for nonlinear continuous neural networks a reason for applying the direct method of lyapunov to artificial neural networks anns is to design dynamical neural networks so that they exhibit global asymptotic stability lyapunov functions that frequently appear in the ann literature include the quadratic function the persidskii function and the lure postnikov function this contribution revisits the quadratic function and shows that via krasovskii like stability criteria it is possible to have a very simple and systematic procedure to obtain not only new and generalized results but also well known sufficient conditions for convergence established recently by non lyapunov methods such as the matrix measure and nonlinear measure

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

on the metric properties of multimodal interval maps and c 2 density of axiom a in this paper we shall prove that axiom a maps are dense in the space of c 2 interval maps endowed with the c 2 topology as a step of the proof we shall prove real and complex a priori bounds for first return maps to certain small neighborhoods of the critical points of real analytic multimodal interval maps with non degenerate critical points we shall also discuss rigidity for interval maps without large bounds

exponential stability and oscillation of hopfield graded response neural network both exponential and stochastic stabilities of the hopfield neural network are analyzed the results are especially useful for analyzing the stabilities of asymmetric neural networks a constraint on the connection matrix has been found under which the neural network has a unique and exponentially stable equilibrium given any connection matrix this constraint can be satisfied through the adjustment of the gains of the amplifiers and the resistances in the neural net circuit a one to one and smooth map between input currents and the equilibria of the neural network can be set up the above results can be applied to the master slave net to prove that the master net can find the best connection matrix for the slave net for the neural network disturbed by some noise the stochastic stability of the network is also analyzed a special asymmetric neural network formed by a closed chain of formal neurons is also studied for its stability and oscillation both stable and oscillatory dynamics are obtained in the closed chain network through the adjustment of the gains and resistances of the amplifiers

frustration stability and delay induced oscillations in a neural network model the effect of time delays on the linear stability of equilibria in an artificial neural network of hopfield type is analyzed the possibility of delay induced oscillations occurring is characterized in terms of properties of the not necessarily symmetric connection matrix of the network such oscillations are possible exactly when the network is frustrated equivalently when the signed digraph of the matrix does not require the perron property nonlinear analysis centre manifold computation of a three unit frustrated network is presented giving the nature of the bifurcations taking place a supercritical hopf bifurcation is shown to occur and a codimension two bifurcation is unfolded

theory of laplacian fractals diffusion limited aggregation and dielectric breakdown model abstract we describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the laplace equation and a stochastic field this new theory provides a systematic method for the calculation of the fractal dimension d and of the multifractal spectrum of the growth probability a detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented our approach exploits the scale invariance of the laplace equation that implies that the structure is self similar both under growth and scale transformation this allows one to introduce a fixed scale transformation instead of coarse graining as in the renormalization group theory that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process for the calculation of the matrix elements of this transformation one has to consider an infinite but rapidly convergent number of processes that occurs outside a considered diagram

simplified diffusion limited aggregation models abstract computer simulations have been used to explore two dimensional diffusion limited aggregation models in which growth is allowed only at the tips of needle shaped arms that are directed away from the origin for the case of two adsorbing needles with growth velocities equal to the rate at which the needle tips capture random walkers the critical angle c between the two needles for stable growth is about 30 33 in good agreement with the theoretical value of 1 2 3 1 2 or about 33 03 obtained by derrida and hakim preprint 1992 for the growth of n symmetrically directed arms with the same boundary conditions all n arms grow in a stable pattern for n 6 for n 7 the growth of n needles is not stable quite different results are reported for other needle growth models with different boundary conditions