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This paper gives the controllability analysis of continuous-time dynamic fuzzy control system from the aspect of fuzzy differential equations. The fuzzy state is different from the crisp state, as the counterpart of the controllability concept in the classical control theory, the controllable target state must be restricted within some limits. Hence, the concepts of admissible controllable state subset and quasi-controllability are introduced to describe the controllability property for fuzzy control system. The sufficient and necessary conditions for the fuzzy control system to be quasi-controllable are obtained and some examples are given to demonstrate the problems discussed in this paper.

In this paper, we investigate the nonautonomous recurrent neural networks systems with distributed delay. By applying Lyapunov functional method and using some analytic technique, several sufficient conditions to ensure the ultimate boundedness, global asymptotic stability and global exponential stability are established. The results obtained in this paper are new and useful.

A nonlinear differential kinetic model describing dynamical behaviours of an atom at a fatigue crack tip is developed in this paper. It is assumed that the forces acted on this atom by its surrounding atoms consist of the following three components: (1) an elastic restoring force governed by Leonard-Jones potential, which describes the elastic interaction between atoms; (2) a nonlinear damping force proportional to its velocity through a linear function of its displacement as a coefficient that empirically simulates the energy loss from the crack tip to its surroundings; (3) an external remote driving force to represent thermally activated energy supplied to the crack tip from the surroundings. Based on these assumptions of the interaction forces between the atoms around the crack tip, a nonlinear dynamic equation describing the motion of the atom at a crack tip using the Newton’s second principle is derived. For a periodic external force and a random one influenced by parameters omitted, deterministic and a stochastic analyses on the dynamic equation obtained above are completed. Based on the theories of the Hopf bifurcation, global bifurcation and stochastic bifurcation, the extent and some possible implications of the existence of atomic-scale chaotic and stochastic bifurcative motions involving the fracture behaviour of actual materials are systematically and qualitatively discussed and the extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. As demonstrated in the paper, chaotic behaviour may be observed in the case of a larger amplitude of the driving force and a smaller damping constant. The white noise introduced in the atomistic motion process may leads to a drift of the divergence point of the nonlinear stochastic differential kinetic system in contrast to the homoclinic divergence of the nonlinear deterministic differential kinetic system.

In this paper we study some global bifurcations arising in the Puu’s oligopoly model when we assume that the producers do not adjust to the best reply but use an adaptive process to obtain at each step the new production. Such bifurcations cause the appearance of a pair of closed invariant curves, one attracting and one repelling, this latter being involved in the subcritical Neimark bifurcation of the Cournot equilibrium point. The aim of the paper is to highlight the relationship between the global bifurcations causing the appearance/disappearance of two invariant closed curves and the homoclinic connections of some saddle cycle, already conjectured in [Agliari A, Gardini L, Puu T. Some global bifurcations related to the appearance of closed invariant curves. Comput Math Simul 2005;68:201–19]. We refine the results obtained in such a paper, showing that the appearance/disappearance of closed invariant curves is not necessarily related to the existence of an attracting cycle. The characterization of the periodicity tongues (i.e. a region of the parameter space in which an attracting cycle exists) associated with a subcritical Neimark bifurcation is also discussed.

This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.

In this paper, a traveling wave reduction combined with the transformation method in terms of Weierstras elliptic functions is used to find a class of new exact solutions for a non-linear partial differential equation (nPDE) of third order, the so called combined KdV–mKdV equation. The usual starting point is a special transformation (the traveling wave “ansatz”) converting the nPDG in its two variables x and t to the belonging non-linear ordinary differential equation (nODE) in the single variable ξ. Using the Weierstras elliptic-function method, new exact class of solutions in terms of the function ℘(ξ; g2, g3) are obtained. Moreover, class of solutions showing typical solitary behavior results as a special case. The important aspect of this paper however is the fact, that we are able to calculate distinct class of solutions which cannot be found in current literature. In other words, using this method, the solution manifold is augmented to new class of solution functions. In the same time we would like to stress the necessity of such sophisticated methods since a general theory of nPDEs does not exist at present.

This paper presents an approach to the well-known traveling salesman problem (TSP) using self-organizing maps (SOM). There are many types of SOM algorithms to solve the TSP found in the literature, whereas the purpose of this paper is to look for the incorporation of an efficient initialization methods and the definition of a parameters adaptation law to achieve better results and a faster convergence. Aspects of parameters adaptation, selecting the number of nodes of neurons, index of winner neurons and effect of the initial ordering of the cities, as well as the initial synaptic weights of the modified SOM algorithm are discussed. The complexity of the modified SOM algorithm is analyzed. The simulated results show an average deviation of 2.32% from the optimal tour length for a set of 12 TSP instances.

This paper investigates the stability for a delayed recurrent neural network. Sufficient conditions are obtained for ascertaining the global asymptotic stability of the unique equilibrium of the network based on LMI technique. The results are computationally efficient, since they are in the form of linear matrix inequality (LMI), which can be checked easily by various recently developed convex optimization algorithms. Besides, the analysis approach allows one to consider different types of activation functions, such as piecewise linear, sigmoids with bounded activations as well as C1-smooth sigmoids. In the end of this paper two illustrative examples are also provided to show the effectiveness of our results.

It is well known that a stable torus is observed as a result after the system meets the super-critical Neimark–Sacker bifurcation for a limit cycle. Although tori are easily observed in two-dimensional and periodically forced dynamical systems, there is a few papers about stable tori in three-dimensional autonomous systems. Besides, as physical circuit implementations, such circuits contain very special active elements, or have difficulty in realizing. In this paper, we show a very simple circuit of three-dimensional autonomous system, an extended Bonhoffer–van der Pol (BVP) oscillator, which is demonstrating a stable torus. Firstly we explain a discovery of the torus in a computer simulation of the model equation. To implement it as a circuitry, we design a new nonlinear resistor. Although this contains an FET and an op-amp, it is simpler than any other nonlinear resistors proposed in previous papers. We confirm that this BVP oscillator can generate a stable torus in a real circuitry. We thoroughly investigate the bifurcation phenomena of various limit cycles and tori in this circuit, i.e., a super-critical Neimark–Sacker and tangent bifurcations of limit cycles are concretely obtained, furthermore, phase locking and chaos regions are clarified in a bifurcation diagram.

In this paper, we mainly research a multifractal formalism in a probability space.The first objective of this paper is to define a multifractal generalisation of the Hausdorff measure and the packing measure. We explore the properties of the multifractal Hausdorff measure and the multifractal packing measure in a probability space, and invesigate the relative results of multifractal spectra functions in a probability space. We also describe a sufficient condion and a necessary one of validity for the multifractal formalism in a probability space.

In this paper, we investigate the dynamical behaviors of neural excitable system without periodic external current (proposed by Chialvo [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons & Fractals 1995;5(3–4):461–79] and with periodic external current as system’s parameters vary. The existence and stability of three fixed points, bifurcation of fixed points, the conditions of existences of fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using bifurcation theory and center manifold theorem. The chaotic existence in the sense of Marotto’s definition of chaos is proved. We then give the numerical simulated results (using bifurcation diagrams, computations of Maximum Lyapunov exponent and phase portraits), which not only show the consistence with the analytic results but also display new and interesting dynamical behaviors, including the complete period-doubling and inverse period-doubling bifurcation, symmetry period-doubling bifurcations of period-3 orbit, simultaneous occurrence of two different routes (invariant cycle and period-doubling bifurcations) to chaos for a given bifurcation parameter, sudden disappearance of chaos at one critical point, a great abundance of period windows (period 2 to 10, 12, 19, 20 orbits, and so on) in transient chaotic regions with interior crises, strange chaotic attractors and strange non-chaotic attractor. In particular, the parameter k plays a important role in the system, which can leave the chaotic behavior or the quasi-periodic behavior to period-1 orbit as k varies, and it can be considered as an control strategy of chaos by adjusting the parameter k . Combining the existing results in [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons & Fractals 1995;5(3–4):461–79] with the new results reported in this paper, a more complete description of the system is now obtained.

This paper refines existing techniques into an algorithmic method for deriving the generalization of a Lax Pair directly from a general integrable nonlinear evolution equation via the use of truncated Painleve expansions. The resulting algorithm is also applicable to multicomponent integrable systems, and is thus expected to be of great value for complicated variants of such systems in various applications areas. Although a related method has existed for simple scalar integrable evolution equations for many years now, nevertheless no systematic procedure has been given that would work in general for scalar as well as for multicomponent systems. The method presented here largely systematizes the necessary operations in applying the Painleve method to a general integrable evolution equation or system of equations. We demonstrate that by following the concept of enforcing integrability at each step (referred to here as the Principle of Integrability), one is led to an appropriate generalization of a Lax Pair, although perhaps in nonlinear form, called a “Lax Complex”. One new feature of this procedure is that it utilizes, as needed, a technique from the well-known Estabrook–Wahlquist method for determining necessary integrating factors. The end result of this procedure is to obtain a Lax Complex, whose integrability condition will contain the original evolution equation as a necessary condition. This in itself is sufficient to ensure that the Lax Complex may be used to construct Backlund solutions of the evolution equation, to obtain Darboux Transformations, and also to obtain Hirota’s tau functions, in a manner analogous to the procedure for single component systems. The additional problem of finding a general procedure for the linearization of any Lax Complex is not treated in this paper. However, we do demonstrate that a particular technique, which can be derived self-consistently from the Painleve–Backlund equations, has proven to be sufficient so far. The Nonlinear Schrodinger equation is used to illustrate the method, and then the method is applied to obtain, for the first time via the Painleve method, a Lax Complex for the vector Manakov system. Limitations in the algorithm remain, especially for cases with more than one principal branch, and these are briefly mentioned as directions for future work.

In this paper, the H ∞ analysis problem is studied for a general class of nonlinear stochastic systems with time-delay. The stochastic systems are described in terms of stochastic functional differential equations. The Razumikhin-type lemma is employed to establish sufficient conditions for the time-delay stochastic systems to be internally stable, and the H ∞ analysis problem is studied in order to quantify the disturbance rejection attenuation level of the nonlinear stochastic time-delay system. In particular, the paper obtains the general conditions under which the L 2 gain of the system is less than or equal to a given constant. Some easy-to-test criteria are also given so as to determine whether the nonlinear stochastic time-delay system under investigation is internally stable and whether it achieves certain H ∞ performance index. Finally, illustrative examples are provided to show the usefulness of the proposed theory.

In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.

In this paper, the BBM-like equations with fully nonlinear dispersion, B ( m , n ) equations: u t + ( u m ) x − ( u n ) xxt = 0 which exhibit solutions with compact support and with solitary patterns, are studied. The exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of the equations are found by a new method. The special cases, B (2, 2) and B (3, 3), are used to illustrate the concrete scheme of our approach presented by this paper in B ( m , n ) equations. The nonlinear equations B ( m , n ) are addressed for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of B ( m , n ) equations are established.

This paper proposes a general chaotification algorithm for a class of nonlinear discrete-time dynamical systems. Based on a given nonlinear discrete-time system, the new chaotification algorithm uses the decentralized linear state feedback control and a class of nonlinear functions that are only required to satisfy some mild assumptions to construct a chaotic nonlinear system. The sine and sawtooth function used in the existing literature on chaotification only are the special cases of the proposed nonlinear function. Based on the corrected version of the Marotto theorem, we mathematically prove that the constructed nonlinear system is indeed chaotic in the sense of Li and Yorke. In particular, an explicit formula for the computation of chaotification parameters is also obtained. The theoretical results in this paper can be used to supervise one to construct different chaotification algorithms.

A planar pendulum is considered which is parametrically excited by a periodic vertical force. The amplitude and frequency of the excitation are used as control parameters. The downward, hanging and the upward, inverted positions correspond to equilibrium positions if we only consider the variation in angle measured from the downward position. For moderate levels of forcing, there are zones that exist in the space of control parameters, where the downward hanging position is unstable and initial conditions that are close to the hanging position lead to steady state oscillations of period-2. To review this situation, this paper describes the development of these oscillations as the amplitude of forcing is varied. In the largest zone, a symmetry-breaking occurs which brings about a pair of asymmetric oscillations. This break in symmetry of the period-2 solution can lead to either an increase or decrease in the amplitude of the forthcoming swing and reference to the experimental significance of this angle change is noted in this paper. Typically, further increases of the parameter produce a cascade of period doubling bifurcations, before most oscillating solutions eventually lose their stability so that the system must experience a rotation. As a result, symmetry-breaking becomes an effective precursor to escape from the local potential well around the hanging position. Here we compare this behaviour with that in other resonance zones. The change of geometric structure when the symmetry-breaking bifurcation occurs is examined and graphically represented as a ‘pinched’ cylinder-like shape, compared with the Mobius strip that has been associated with the period-doubling bifurcation. The paper also refers to practical problems, where the introduction of nonlinearity means that potentially all frequencies below the main zone of the control space lead to dangerous effects and in some scenarios disastrous outcomes.

In this paper, a nonautonomous two-dimensional competitive Lotka–Volterra system with impulsive is considered. we study the persistence and extinction, giving two inequalities involving averages of the growth rates and impulsive value, which guarantees persistence of the system. An extension of the principle of competition exclusion is obtained in this paper. Moreover, several examples are also workedout, they show that the impulsive can change the persistence of the system. � 2004 Elsevier Ltd. All rights reserved.

For bi-directional associative memory (BAM) neural networks (NNs) with different constant or time-varying delays, the problems of determining the exponential stability and estimating the exponential convergence rate are investigated in this paper. An approach combining the Lyapunov–Krasovskii functional with the linear matrix inequality (LMI) is taken to study the problems, which provide bounds on the interconnection matrix and the activation functions, so as to guarantee the system’s exponential stability. Some criteria for the exponential stability, which give information on the delay-dependent property, are derived. The results obtained in this paper provide one more set of easily verified guidelines for determining the exponential stability of delayed BAM (DBAM) neural networks, which are less conservative and less restrictive than the ones reported so far in the literature. Some typical examples are presented to show the application of the criteria obtained in this paper.

This paper deals with the problem of identification of the unknown parameters of a dynamical system. Our work in this paper largely simulation related, follows the approach suggested by Kim and Lee [IEEE Int. Conf. Neural Networks (1993) 438–443] and it depends on the introduction of auxiliary polynomial. The merit of our work lies on the facts that no assumptions are placed on the stability of auxiliary polynomial and also that it does not require the a priori knowledge of the stability of the dynamical system, in question. Our method applies to both linear and quasi-linear dynamical systems. The results obtained involve the construction of appropriate neural networks. A variety of configurations of the dynamical systems are considered for simulation.

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads f =2.5 , η1=1.1, β=0.5, δ =0.01 and e∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection.

Based on a more general transformation presented in this paper, a generalized method for finding more types travelling wave solutions of nonlinear evolution equations (NLEEs) is presented and implemented in a computer algebraic system. As an application of the method, Whitham–Broer–Kaup (WBK) equation is studied to illustrate the method. As a result, We cannot only successfully recover the previously known travelling wave solutions found by Fan's method [J. Phys. A 35 (2002) 6853; Comput. Phys. Commun. 53 (2003) 17], but also obtain some new formal solutions. The solutions obtained in this paper include polynomial, exponential, solitary wave, rational, triangular periodic, Jacobi and Weierstrass doubly periodic solutions.

This paper characterizes the complex dynamics of the permanent-magnet synchronous motor (PMSM) model with a non-smooth-air-gap, extending the work on the smooth case studied elsewhere. The stability, the number of equilibrium points, and the pitchfork and Hopf bifurcations are analyzed by using bifurcation theory and the center manifold theorem. Numerical simulations not only confirm the theoretical analysis results but also show some more new results including the period-doubling bifurcation, cyclic fold bifurcation, single-scroll and double-scroll chaotic attractors, ribbon-chaotic attractor, as well as intermittent chaos that are different from those reported in the literature before. Moreover, analytical expressions of an approximate stability boundary are given, by computing the local quadratic approximation of the two-dimensional stable manifold at an order-2 saddle point. Combining the existing results with the new results reported in this paper, a fairly complete description of the complex dynamics of the PMSM model is now obtained.

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smital [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φxy and Φ xy ∗ generated by F such that Φ xy ∗ ≡1 and Φxy(0+) Φ uv ∗ such that Φ uv ∗ ≡1 and Φuv(t)=0 whenever 0 0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.

Aim of the present paper is to show an application of the fractal integral, which was recently introduced in the framework of fractional calculus, to the mechanics of materials with disordered microstructure. Some rules of integration for simple functions performed on generalized Cantor sets have been developed. We believe that these results can be of some interest in the modeling of materials whose microstructure is fractal-like. Part of the paper is devoted, as an example, to show why concrete can be considered such a material. The fractal dimension of the stress carrying material ligament is found. Finally, by the fractal integration rules previously developed, the computation of the tensile and flexural strength for structures made by a concrete-like material is performed and consequent size effects are highlighted.

A subalgebra of loop algebra A 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given.

A novel adaptive complementary variable structure control is proposed in this paper for chaotic synchronization. The bounded parameters of the model approximation error and the external disturbance are all regarded as unknown constants in this paper. Based on Lyapunov’s stability theory and the Babalat’s lemma the proposed controller has been shown to render the synchronous error to zero. The Duffing–Holmes oscillator was used as an illustrative example. Simulation results validated that the proposed scheme in the application of secure communication.

This paper discussed the interference between external electric exposure and biological objects. Based on results of space-time characteristics of trans-membrane voltage, the variation of cell trans-membrane voltage exposed to extremely low frequency (ELF) electrical field was analyzed; the traditional Hodgkin–Huxley model was modified by importing new parameter denoting external electrical fields, and the bifurcation caused by new parameter war found. This paper employed the algebra criterion in high dimension equations to perform the analysis of single parameter dynamical bifurcation. The results were biological significant and suggested that the aberration of dynamics in bio-systems may be accounted for diseases caused by electric exposure.

In this paper, we proposed a scheme of the bidirectional secure communication hyperchaotic system, where the chaotic circuits are constituted by two pairs of Lorenz circuits with coupling characteristics between the transmitter and receiver. Because of the coupling relationship between the transmitter and the receiver, even if an intruder knows the coupling values of the system, he could not steal the message from the system from practical simulation results. On the other hand, to achieve the synchronization between transmitter and receiver, two methods are presented to design the feedback control gain. The design idea is to apply a suitable Lyapunov function and the property of positive definite of a matrix to satisfy the direct Lyapunov theorem. Therefore, the transmitting message masked by chaotic state from the transmitter can be guaranteed to perfectly recover in the receiver. To implement the proposed communication scheme, the pairs of Lorenz circuits designed by the operational amplifier, resistor and capacitor elements are also presented in the paper. Finally, the test results containing simulation and the circuit measurement are shown to demonstrate that the proposed method is correct and feasible.

In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding dimension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of necessary variables to model the dynamics. Therefore, the optimality of this dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor. The method of this paper relies on testing this property by locally fitting a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The corresponding algorithms are developed in uni/multivariate form and some probable advantages of using information from other time series are discussed. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic benchmark systems. Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension.

In this paper, we propose a new method to design a feedback controller for synchronizing the hyperchaotic circuits. The main idea of the method is to design a controller by applying a suitable Lyapunov function and Gerschgorin theorem. Therefore, the feedback controller can guarantee that the signal by chaotic masking from the transmitter can be recovered at the receiver. A scheme of the hyperchaotic communication systems is proposed, where the transmitter and receiver are constituted of the coupled Chua’s chaotic circuits in this paper. On the other hand, to enhance the degree of security in chaotic communication systems, the techniques of cipher are also considered in this paper. Therefore, the cryptosystems can be obtained by using the encrypter, and decrypter as the chaotic transmitter–receiver pairs. To the purpose of the synchronization, the concepts of identical and generalized synchronizations are used to achieve the aim that the plaintext signals from the encrypter can be retrieved at the decrypter. Finally, the simulation results are given to demonstrate the performance of the proposed cryptosystems.

The topic of this paper is as, the title shows, to introduce the formulation of fuzzy portfolio optimization problem as a convex quadratic programming approach and then give an acceptable solution to such problem. A numerical example included in the support of this paper for illustration.

In the first paper (I) we have shown that two-dimensional nonlinear dispersion Boussinesq equations, B(m,n,k,p), admitted abundant compact structures for different parameters m, n, k, p. In this paper we shall further investigate the solitary pattern solutions of B(m,n,k,p) equations. As a consequence, it is shown that these equations also possess many solitary pattern solutions expressed in terms of the hyperbolic sine and cosine functions.

In the recent paper [Int. J. Non-Linear Mech. 35 (2000) 309], periodic attractors were studied for a class of complex damped non-linear dynamical systems of the form: (*) z +ω 2 z+e z f(z, z , z , z )P(Ωt)=0, where z(t)=x(t)+iy(t), i = −1 , the bar denotes the complex conjugate and e is a small positive parameter. The object of this paper is to continue our investigation of this class by studying the stabilization of unstable periodic attractors of system (*) . A feedback control method which is suggested by Pyragas is used to stabilize these unstable periodic attractors. We construct Poincare plots before and after control to prove that the stabilization is achieved. Two examples are presented which served to illustrate this investigation. Maximal Lyapunov exponents and the power spectrum are calculated to show that our periodic attractors are not chaotic attractors.

In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.

This paper simulates the non-linear vibration of a gear pair including shaft flexibility. Backlash is included because of the clearance between the teeth and the periodic variation in the stiffness of the mesh produces a parametric excitation. This paper shows the different behaviour that may be observed when the additional degree of freedom is added to the single degree of freedom model previously analysed.

The present paper concerns with the investigation of the reflection of generalized magneto-thermo-viscoelastic waves at the boundary of a semi-infinite solid. The free surface of the solid is considered to be adjacent to vacuum. We assumed that the solid is subjected to a constant temperature and magnetic field. When a rotational wave is incident from within the solid on its boundary, the expressions for the reflection coefficients (and their approximate expressions) of the waves generated at the boundary are obtained. The assumptions that the body is a perfect conductor and the chosen that the viscoelastic model is of Kelvin–Voigt type are considered. The effect of the viscosity, the applied magnetic field and the thermo-elastic coupling factor on the reflection coefficients are studied. The numerical calculations of the absolute values of the reflection coefficients and their relative changes for the Lord–Shulman and Green–Lindsay in generalized thermoelastic models are presented graphically. Finally, some earlier studies may be deduced as special cases from our results presented in this paper.

It is known that the Hasimoto transformation of Heisenberg ferromagnetic system yields nonlinear Schrodinger equation. In this paper we investigate a series of moving space curve equations whose Hasimoto transformations are KdV type equations. In this paper we also propose their Lax pairs.

The model of a two-dimensional fluid flow past a cylinder is a relatively simple problem with a strong impact in many applied fields, such as aerodynamics or chemical sciences, although most of the involved physical mechanisms are not yet well known. This paper analyzes the fluid flow past a cylinder in a laminar regime with Reynolds number, Re , around 200, where two vortices appear behind the cylinder, by using an appropriate time-dependent stream function and applying non-linear dynamics techniques. The goal of the paper is to analyze under which circumstances the chaoticity in the wake of the cylinder might be modified, or even suppressed. And this has been achieved with the help of some indicators of the complexity of the trajectories for the cases of a rotating cylinder and an oscillating cylinder.

In two recent papers [Phys. Lett. A 235 (1997) 35; J. Math. Phys. 39 (1998) 3271], some interesting conclusions were proposed that four pairs of different finite-dimensional integrable systems, which are the discrete Toda symplectic map and continuous constrained c-KdV system, AKNS and Dirac system, Harry–Dym and Heisenberg spin chain system, and Geng and Qiao system, respectively, possess the common non-dynamical r -matrix. A natural problem is whether there also exist any other pairs like them. In this paper, it is shown that the fifth pair of different finite-dimensional integrable systems of Kaup–Newell (KN) hierarchy and Yan hierarchy also share the same non-dynamical r -matrix being independent from the dynamical variables. The two finite-dimensional integrable systems possess Lax representation and can be shown to be Liouville integrable by using the r -matrix.

This review paper gives an extensive overview of the soliton solutions for some famous partial differential equations like KdV, mKdV, Sine–Gordon, and nonlinear Schrodinger equations. Different analytical methods of treatment as well as those of numerical methods are presented in this review paper. Finally relations between the soliton solution and fluid mechanics are shown.

In this paper we will introduce the new concept “There are no one- or two-dimensional Riemannian manifolds”. Another idea is the consideration of the geometric volume of the manifold as proportional to the energy. Some applications are given at the end of this paper.

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.

This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate of thermo-mechanical coupling by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of the thermo-mechanical coupling rectangular plate possesses many bifurcation points, chaotic motions and period-double phenomena under various lateral loads, bi-axial loads, thermo-mechanical coupling factors and aspect ratios. The modeling results of numerical simulation indicate that the chaotic motion may occur in the range of lateral loads Q =1.25 to 3.35 and near the bi-axial load η2=1.5. The dynamic motion of the thermal-couple plate is periodic if the aspect ratio is within a specific range or the thermo-mechanical coupling factor is within a specific range. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling plate in large deflection.

The purpose of this paper is to analyze the peak-to-peak dynamics (PPD) of a three-class model of the Dynastic Cycle in ancient China. While the asymptotic behavior generated by the prey–predator–superpredator structure of the model was thoroughly studied in an earlier paper [Feichtinger G, Forst CV, Piccardi C. A nonlinear dynamical model for the dynastic cycle. Chaos, Solitons & Fractals 1996;7:257–271], the analysis of the PPD provides new interesting insights into the properties of the system, by yielding a set of reduced models relating the peaks' amplitudes and their return times.

In this talk we present main points of some of Fock papers which were not properly followed when published and review Fock's interpretation of the energy–time uncertainty relation, as well as his ideas on generalization of the concept of physical space published in his last paper.

This paper presents an analysis of the dynamical behaviour of a non-symmetric oscillator with piecewise-linearity. The Chen–Langford (C–L) method is used to obtain the averaged system of the oscillator. Using this method, the local bifurcation and the stability of the steady-state solutions are studied. A Runge–Kutta method, Poincaré map and the largest Lyapunov’s exponent are used to detect the complex dynamical phenomena of the system. It is found that the system with piecewise-linearity exhibits periodic oscillations, period-doubling, period-3 solution and then chaos. When chaos is found, it is detected by examining the phase plane, bifurcation diagram and the largest Lyapunov’s exponent. The results obtained in this paper show that the vibration system with piecewise-linearity do exhibit quite similar dynamical behaviour to the discrete system given by the logistic map.

In this paper a discrete-time economic model is considered where the savings are proportional to income and the investment demand depends on the diAerence between the current income and its exogenously assumed equilibrium level, through a nonlinear S-shaped increasing function. The model can be ultimately reduced to a two-dimensional discrete dynamical system in income and capital, whose time evolution is ‘‘driven’’ by a family of two-dimensional maps of triangular type. These particular two-dimensional maps have the peculiarity that one of their components (the one driving the income evolution in the model at study) appears to be uncoupled from the other, i.e., an independent one-dimensional map. The structure of such maps allows one to completely understand the forward dynamics, i.e., the asymptotic dynamic behavior, starting from the properties of the associated one-dimensional map (a bimodal one in our model). The equilibrium points of the map are determined, and the influence of the main parameters (such as the propensity to save and the firms’ speed of adjustment to the excess demand) on the local stability of the equilibria is studied. More important, the paper analyzes how changes in the parameters’ values modify both the asymptotic dynamics of the system and the structure of the basins of the diAerent and often coexisting attractors in the phase-plane. Finally, a particular ‘‘global’’ (homoclinic) bifurcation is illustrated, occurring for suAciently high values of the firms’ adjustment parameter and causing the switching from a situation of bi-stability (coexistence of two stable equilibria, or attracting sets of diAerent nature) to a regime characterized by wide chaotic oscillations of

The present paper is a contribution within the E (∞) Cantorian space theory. After a brief review of the historical development of length and time measurements, we consider conventional definitions and methods of calculation of vector and fractal dimensions. A definition of emptiness as zero entropy is discussed. Treating the `fabric' of space-time as non-permanent, we consider other possible ways of filling emptiness. This leads to a definition of range dimension ( p ), the properties of which are then discussed. p =1/2 and p =1/4 range dimensions are linked to p =1 (conventional) range dimensions in fluid environments using differential velocities and accelerations. Applications of p =2 range dimensions to quantum events are also considered, and eclectic evidence for the existence of range dimensions is presented. The bijection relationship, defining space-filling fractal dimension ( d c , n ) is extended to cover the range 1 n l n / l 0 ) is developed. The n -dimensional term ( l n / l 0 ) n is then closely correlated to the ratio ( l 1 / l 0 ) d c , n . The bijection relationship is extended to other range dimensions. An approximate empirical relationship is now presented, which provides a basis for defining conventional ( p =1) range dimensional space, by introducing N n , the number of elements needed to set up 1–4 vector dimensions. Two other ways of obtaining approximately the same number of elements are also given, one of which leads to the concept of a p =2 range dimensional substrate for conventional ( p =1) range dimensions. It is found that a single track in p =1 range dimensions needs nearly p =2 range dimensions to support it. The required value then falls as n increases, equaling approximately unity for n =3 vector dimensions. A second empirical equation defining the same value of N n for 3 n p =2 range dimensions is also presented. A fractal form for conventional space is then defined, characterised by tiny elements of negative entropy (in log space). This definition is then readily amalgamated with dynamically generated space, and the latter is shown to correlate with the classical gas equation and Hooke's law for solids. Based on the new definition of space, a proof of both the well-known inverse square law and Hubble's astronomical law is given. A relationship between the number of events in p=1 and p=2 range dimensions is next established, and the effect of `information loss' corresponding to the reduction in the required value of range dimension as vector dimension ( n ) increases is tentatively linked to wave functions, for n =4 vector dimensions. The corresponding `useful' information is then shown to be contained in the exponent field, which is of the form {2 f ( e −1) pd c ,4 }, where f≈1, p=2 range dimensions, and e (within the exponent!) equals the numerical value of the exponent. This field is then sub-divided to various extents, and the corresponding entropies are used to determine sets of distinguishable arrangements N . These sets are successfully compared with `magic' numbers in atomic and nuclear physics, and with preferred sodium atom groupings. A prediction of the number of occupied energy levels in atomic electron shell theory is successfully carried out. Finally, combining `useful' and `hidden' information the p =2 range dimension substrate is re-created by allowing matter to collapse and `drain away' from its p =2 range dimensional space base. Light propagation through space having p =2 range dimensions is then discussed. The paper concludes with a discussion of various ideas implicit in the text, though not necessarily directly obvious. The interdependence of fractal, range and vector dimensions is first considered, followed by a review of the universal application of range dimensions. The paper ends with a study of the fractal nature of space and its justification.

Some definitions such as m-chain recurrent set, weakly gradient flow and generalized Morse decomposition for a flow defined on a topological space are introduced in this paper. Some conclusions, include the chain recurrent set contain the m-chain recurrent set; the m-chain recurrent set contain the non-wandering set, are proved. In some flows the non-wandering set is proper subset of the m-chain recurrent set; in the meantime the m-chain recurrent set is proper subset of the chain recurrent set. Moreover some criterions for the existence of trajectories joining singular points and a necessary and sufficient condition of the weakly gradient flow are also given here. At last the generalized Morse decomposition of the invariant set are discussed in the paper.