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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.

This paper is concerned with the convergence rates to travelling waves for a relaxation model with general flux functions. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen large in suitable norm. Our analysis is based on the L 1 -stability results obtained by C. Mascia and R. Natalini in [12].

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.

This paper is a further continuation of the paper [1]. In the present paper Mellin transform and Munts formula of weak functions in complex domain will be treated.

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.

In this paper, the new subclass Bn(λ, α, A, B) of analytic functions is introduced. Its subordination relations, inclusion relations, the inequality properties and convering theorem are discussed. The results obtained in this paper are sharp, which generalize the related works of some authors and some other new results are obtained.

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.

Based on the Cass-Koopmans Model with endogenous fertility given in the paper [1-4], the fertility dynamics is discussed in this paper. It is proved that the first quadrant of the k,c plane is separated into two parts by a increasingly differentiable curve. The fertility increases with time on the upper half and decreases with time on the lower half. The fertility decreases along one arm of the economic growth path on which the per capita capital and per consumption increase and increases along the other arm of the economic growth path on which the per capita capital and per person consumption decrease under the condition 0

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 deals with the existence of positive periodic solutions for a kind of nonautonomous Volterra intergo-differential equations by employing the Krasnoselskii fixed point theorem. Applying the general theorems established to several biomathematical models, the paper improves some previous results and obtains some new results.

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.

This paper is concerned with the existence and uniqueness of solution for a class of stochastic functional equation: X = ϕ ˜ ( X ) , where ϕ ˜ : B → B and B is a Banach space consisted of all left-continuous, (Ft)-adapted processes. Also, the main result is applied to some S.D.E (or S.I.E.). And the authors adopted some of the results in current research in the models of stochastic control recently. This paper proves the existence and uniquence and uniqueness of solution for stochastic functional equation. A series of corollaries are deduced from the special examples of the theorems in this paper.

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.

In this paper three sets of boundary conditions are considered for reconstructing the stiffness of the inhomogeneous Bernoulli–Euler beams. The essence of the paper consists in postulating the mode shape of the vibrating beam as a static deflection of associated uniform, homogeneous beam. This unconventional way of problem formulation turns out to lead to series of new closed-form solutions. For each combination of the boundary conditions several cases of the inertial coefficients are considered. All exact solutions for natural frequencies are represented as rational expressions of the involved coefficients. Solutions are written in terms of two positive integers: ` m ' representing the degree of the polynomial in the inertial term and `n' indicating power in the postulated mode shape. A remarkable conclusion is reached: For specified ` m ' and ` n ', the natural frequencies of the inhomogeneous beams with different boundary conditions coalesce. This remarkable nature does not imply that these beams share the same frequencies. In fact, these are different beams for each set of boundary conditions the expression for the stiffness is different. The paper should be considered as a first step towards analysis of uncertainty, inherently present in structures.

Creation of the programs for determination of the Minkowski dimension and other corresponding fractal characteristics are the main result of this paper. The programs were created also for the one-side fractal dimension. The stable numerical algorithm of multiplicative differentiate is given in this paper. One confirms that the critical effect occurs and it is corresponding to creation of the macroscopic crack through the breaking of the early formed microdefects. The singularity occurs at the moment at which the percolation is simultaneously observed. A character of the singularity is well visible – the jump for tens of orders occurs. The critical effect was observed in the computer simulations of the fatigue process.

This paper studies the bulk-arrival Mx/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient queue-length distribution is derived. Furthermore, the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are obtained. Especially some results for the single-arrival M/G/1 queue with single server vacation and bulk-arrival Mx/G/1 queue but with no server vacation can be derived directly by the results obtained in this paper.

A model for firms’ financial conditions is proposed, which ultimately reduces to a two-dimensional non-invertible map in the variables mean and variance of the equity ratio. The possible dynamics of the model and the global behaviour are investigated. We describe the mechanism of bifurcations leading to fractalization of the basins and/or fractalization of their boundaries, showing how a locally stable attractor may be almost globally unstable. Multistability is also investigated. Two, three or four co-existing attractors have been found and we describe the mechanism of bifurcations leading their basins to become chaotically intermingled, and thus to unpredictability of the asymptotic state in a wide region. The knowledge of such regimes, besides those associated with simple dynamics, may be of help for the operators. While the use of the technical tools we propose to study the global dynamics and bifurcations may be of help for further investigations. ” 2000 Elsevier Science Ltd. All rights reserved. In this paper we present a model of fluctuating growth in which firms’ financial conditions play a crucial role. Our analysis starts from the distribution of firms according to their equity ratio, that is the ratio of the equity base or net worth to the capital stock, a proxy of financial robustness. We identify two dynamic laws for the mean and the variance of this distribution. The motion over time of the average equity ratio is the engine of growth and fluctuations. The dynamic pattern of the dispersion of the distribution, captured by the variance, however, interacts with evolution of the average equity ratio. Given the non-linear nature of the map which describes the laws of motion of the mean and the variance of the equity ratio, a wide range of dynamic patterns are possible. Fixed points or periodic orbits, attracting closed invariant curves and thin annular chaotic areas wide chaotic areas or explosions may occur. For quite plausible values of the parameters which characterize the map, the dynamics of the equity ratio can be regular or chaotic, and the motion of capital and output can be characterized as a process of fluctuating growth, although, as we shall see, often very sensitive to small perturbations. The goal of the present paper is to show how the global properties (deriving from the structure of the basins, their boundaries and the critical curves of non-invertible two-dimensional maps) may be used to understand the dynamic behaviour of the model, especially when analytical results are not accessible, as in our case, where not only the equilibrium values, but also the number of existing fixed points, cannot be explicitly known.

In a recent paper Bahar [Chaos, Solitons + Fractals, 1996, 7(1), 41] described bifurcation from a fixed point generated by iterated function systems. An analytical study of it, by using Banach theorem, was proposed by us in Chaos, Solitons + Fractals, 1998, 9(3), 449. In this paper we present an extension of our previous study and we prove that by a special transformation, the considered two-dimensional map can be reduced to two distinctive one-dimensional maps, such that each one determines the behavior of the entire system.© 1999 Elsevier Science Ltd. All rights reserved.

Complexity studies are burgeoning into an ever-increasing number of fields. This paper seeks to abstract from a large number of non-biological complexity areas some exemplar common thread representative of many complexity areas. One such common thread is identified, a thread leading immediately to the notion of Universal Library, as an additional complexity area, containing all possible texts and, correspondingly, encompassing all possible knowledge (amidst, naturally, a welter of partially and wholly senseless texts). This notion is then wedded to an elementary portion of number theory, to indicate where replicas of universal libraries exist and with what certain attributes. Next, a representative example of a biological complexity area—DNA sequences—is introduced. While DNA is often casually termed a library specifying the heritability machinery of individuals and species, this paper briefly explores whether explicit relations exist between DNA sequences and universal libraries, to help test, inter alia, the strength of the initially identified complexity thread to link non-biological and biological complexity areas. This theme of common linking threads has many interesting open research issues.

This paper deals with optimal control problem of parabolic differential equation with two point boundary conditions (in the time variable). The results here extend those in [3] on optimal control of the heat equations. Moreover, the technique used in this paper is based on some smooth approximations of “tangent cones” in the sense of Clarke and some maximal monotone operators.

This paper presents a knot theoretic approach to the study of molecular graphs. It provides examples of molecular knots and links as well as other topologically complex molecules. The paper discusses various topological problems and theorems which were motivated by questions in chemistry. Some of these results have direct applications to the analysis of molecular structures, while other results, though arising out of questions about chemistry, are purely topological in nature.

In this paper, the transient solutions for M/G/1 queues with single server vacation and multiple server vacations are firstly studied, and the recursion expressions of their Laplace transform are given. Further the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are directly obtained from the transient solution. As will be seen this paper provides a intuitive and elegant method for studying transient solutions for M/G/l queues with single server.

Buckling mode localization in nonhomogeneous beams on elastic foundations is studied in this paper. The classical theory of localization for randomly disordered nearly periodic structures is extended to nonhomogeneous continuous structures. Lyapunov exponents, which characterize the average exponential rates of growth or decay of amplitudes of deformation, are determined using the method of transfer matrix. The smallest positive Lyapunov exponent is the localization factor of interest. As an alternative approach, Green's function method coupled with a finite element formulation is applied to evaluate the localization factors. The results obtained in this paper provide an explanation on localized buckling observed in experiments.

Experimental results indicate that the fracture in many structures is rough and irregular and depends strongly on their microstructures and loading situations. In many cases the problem is to repair structures involving such irregular cracks and interfaces. The aim of this paper is to contribute to the analysis of these problems by using a new geometry, fractal geometry which describes with great accuracy irregular phenomena in nature. According to fractal geometry the interface of a crack can be considered to be the unique ‘fixed point’ of a given iterative function system (IFS) or the graph of a fractal interpolation function. On these interfaces, conditions describing the mechanical behaviour of the bonding material will be assumed to hold. The general case of a mechanical behaviour described by a non-monotone possibly multivalued stress-strain law will be treated. The method that will be developed in the framework of this paper will be an extension of the classical FEM to the case of fractal interfaces.

In recent papers, Babolian & Delves[2] and Belward[3] described a Chebyshev series method for the solution of first kind integral equations. The expansion coefficients of the solution are determined as the solution of a mathematical programming problem. The method involves two regularization parameters, Cf and r, but values assigned to these parameters are heuristic in nature. Essah & Delves[7] described an algorithm for setting these parameters automatically, but it has some difficulties. In this paper we describe three iterative algorithms for computing these parameters for singular and non-singular first kind integral equations. We give also error estimates which are cheap to compute. Finally, we give a number of numerical examples showing that these algorithms work well in practice.

In this paper we consider a simple family of nonlinear dynamical systems generated by smooth functions. Some theorems for the existence and the uniqueness of the limit cycles of the systems are presented. If f and g are generating functions with unique limit cycles and xf ( x ) xg ( x ), for all x ≠ 0, then according to the ‘bounding theorem’ proved in the paper, the limit cycle of the system generated by f is bounded by the limit cycle of the system generated by g . This gives us a method to estimate the amplitude of the oscillations also for systems for which we do not know the generating function exactly. As an application we extend the nonlinear business cycle model proposed by Tonu Puu (1989).

This paper deals with the global stability of the long term dynamics of nonlinear mechanical systems under periodic excitation. Generally, the boundaries of the basins of attraction are formed by the stable manifolds of unstable periodic solutions. These stable manifolds are the set of initial conditions of trajectories which approach an unstable periodic saddle solution. Because these are the only trajectories which do not approach an attractor, in general the stable manifolds are the boundaries of the basins of attraction. In this paper manifolds are calculated of a beam system supported by a one-sided spring in order to identify the global stability of the coexisting attractors. The numerical results are compared with experimental results.