Dissipative relativistic standard map: Periodic attractors and basins of attraction

The dissipative relativistic standard map, introduced by Ciubotariu et al. [Ciubotariu C, Badelita L, Stancu V. Chaos in dissipative relativistic standard maps. Chaos, Solitons & Fractals 2002;13:1253–67.], is further studied numerically for small damping in the resonant case. We find that the attractors are all periodic; their basins of attraction have fractal boundaries and are closely interwoven. The number of attractors increases with decreasing damping. For a very small damping, there are thousands of periodic attractors, comprising mostly of the lowest-period attractors of period one or two; the basin of attraction of these lowest-period attractors is significantly larger compared to the basins of the higher-period attractors.     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

Structurally stable but chaotic limit set of E-infinity Cantorian space–time

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E-infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.     

A modified variable-coefficient projective Riccati equation method and its application to (2 + 1)-dimensional simplified generalized Broer–Kaup system

A modified variable-coefficient projective Riccati equation method is proposed and applied to a (2 + 1)-dimensional simplified and generalized Broer–Kaup system. It is shown that the method presented by Huang and Zhang [Huang DJ, Zhang HQ. Chaos, Solitons & Fractals 2005; 23:601] is a special case of our method. The results obtained in the paper include many new formal solutions besides the all solutions found by Huang and Zhang.     

Interest rate option pricing and volatility forecasting: An application to Brazil

Assessing the markets perception of future interest and inflation rate volatility is of crucial importance to assess the evolution of expectations in an inflation targeting framework. This article aims to evaluate the information content of implied volatilities extracted from a Brazilian interest-rate call option. We compared the predictive performance of three different approaches: one using the traditional [Black F. The pricing of commodity contracts. J Financ Econ 1976;3:167–79] method, another one using the extended-Vasicek model, and in the third approach, we use a GARCH(2, 1) model. The empirical evidence was more favorable to the extended-Vasicek method. Moreover, extended-Vasicek’s implied volatilities could predict around 33% (adjusted R²) of the variations in realized volatility. Further research could test for the predictive content of long memory options such as those suggested in Wang et al. [Wang X-T, Qiu W-Y, Ren F-Y. Option pricing of fractional version of the Black–Scholes model with Hurst exponent H being in . Chaos, Solitons & Fractals 2001;12:599–608; Wang X-T, Ren F-Y, Liang X-Q. A fractional version of the Merton model. Chaos, Solitons & Fractals 2003;15:455–63].     

Nonlinear dynamics of regenerative cutting processes—Comparison of two models

Understanding the nonlinear dynamics of cutting processes is essential for the improvement of machining technology. We study machine cutting processes by two different models, one has been recently introduced by Litak [Litak G. Chaotic vibrations in a regenerative cutting process. Chaos, Solitons & Fractals 2002;13:1531–5] and the other is the classic delay differential equation model. Although chaotic solutions have been found in both models, well known routes to chaos, such as period-doubling or quasi-periodic motion to chaos are not observed in either model. Careful analysis shows that the chaotic motion from the Litak’s model has sharper spectral peaks, a smaller correlation dimension and a smaller value for the largest positive Lyapunov exponent. Implications to the control of chaos in cutting processes are discussed.     

Statistical convergence of double sequences in intuitionistic fuzzy normed spaces

Recently, the concept of intuitionistic fuzzy normed spaces was introduced by Saadati and Park [Saadati R, Park JH. Chaos, Solitons & Fractals 2006;27:331–44]. Karakus et al. [Karakus S, Demirci K, Duman O. Chaos, Solitons & Fractals 2008;35:763–69] have quite recently studied the notion of statistical convergence for single sequences in intuitionistic fuzzy normed spaces. In this paper, we study the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces. Furthermore, we construct an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and our method of convergence even for double sequences is stronger than the usual convergence in intuitionistic fuzzy normed space.     

A note on a fixed point theorem in Menger probabilistic quasi-metric spaces

The main result in Rezaiyan et al. [Rezaiyan R, Cho YJ, Saadati R. A common fixed point theorem in Menger probabilistic quasi-metric spaces, Chaos, Solitons & Fractals 2008;37:1153–7] is proved under two (necessary) additional conditions.     

A zero-crossing approach to uncover the mask by chaotic encryption with periodic modulation

We present a so-called zero-crossing identification method that can crack the security shell of the chaotic encryption method [Chaos, Solitons & Fractals 19 (2004) 919] with periodic modulation. By collecting a special set of truncated data from the zero-crossing incidents of the modulated signal, we can detect the modulating function from chaotic signal. Furthermore we extend the technique to extract modulating function from noise and discuss the potential applications of this method in engineering.     

An oscillatory population model

We consider a simple population model which includes time-dependent parameters prompted by the recent work of Lakshmi [Chaos, Solitons & Fractals 16 (2003) 183]. Time-dependent parameters introduce the possibility of chaos into the dynamics of even simple models. We provide some solutions of the model, compare them with the ones obtained by Lakshmi and discuss their behaviour and properties.     

A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation

In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons & Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained.     

Some bitopological concepts based on the alternative effects of closure and interior operator

In this paper we give an approach for constructing classes of near open and near closed sets which have unusual implication relations. These new classes of subsets are based on the alternative effect of closure and interior operators with respect to two topologies. Also these classes of subsets are applied for constructing several classes of near continuous functions and some types of separation axioms called mildly binormal, almost ij-normal, almost ij-regular, quasi ij-regular and strongly S-ij-regular. Using the introduced functions, we generalize several preservation theorems of normality and regularity to bitopological spaces. Implications between notions are given and counter examples for some reverse directions are obtained. It should be noted that considering the space time as the product of two topologies, the topology of space and that of the space time will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie and many others (cf. [Chaos, Solitons & Fractals 17 (2003) 989; Chaos, Solitons & Fractals 7 (1996) 499; Int. J. Theor. Phys. 37 (1998) 2935; Phys. Today (1996) 24]).     

Fractals via iterated functions and multifunctions

Fractals have wide applications in biology, computer graphics, quantum physics and several other areas of applied sciences (see, for instance [Daya Sagar BS, Rangarajan Govindan, Veneziano Daniele. Preface – fractals in geophysics. Chaos, Solitons & Fractals 2004;19:237–39; El Naschie MS. Young double-split experiment Heisenberg uncertainty principles and cantorian space-time. Chaos, Solitons & Fractals 1994;4(3):403–09; El Naschie MS. Quantum measurement, information, diffusion and cantorian geodesics. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995. p. 191–205; El Naschie MS. Iterated function systems, information and the two-slit experiment of quantum mechanics. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995. p. 185–9; El Naschie MS, Rossler OE, Prigogine I. Forward. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995; El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36; El Naschie MS. Fractal black holes and information. Chaos, Solitons & Fractals 2006;29:23–35; El Naschie MS. Superstring theory: what it cannot do but E-infinity could. Chaos, Solitons & Fractals 2006;29:65–8). Especially, the study of iterated functions has been found very useful in the theory of black holes, two-slit experiment in quantum mechanics (cf. El Naschie, as mentioned above). The intent of this paper is to give a brief account of recent developments of fractals arising from IFS. We also discuss iterated multifunctions.     

On a family of models of cell division cycle

The aim of this work is to generalize and study a model of cell division cycle proposed recently by Zheng et al. [Zheng Z, Zhou T, Zhang S. Dynamical behavior in the modeling of cell division cycle. Chaos, Solitons & Fractals 2000;11:2371–8]. Here we study the qualitative properties of a general family to which the above model belongs. The global asymptotic stability (GAS) of the unique equilibrium point E (idest of the arrest of cell cycling) is investigated and some conditions are given. Hopf’s bifurcation is showed to happen. In the second part of the work, the theorems given in the first part are used to analyze the GAS of E and improved conditions are given. Theorem on uniqueness of limit cycle in Lienard’s systems are used to show that, for some combination of parameters, the model has GAS limit cycles.     

Different Higgs models and the number of Higgs particles

In this short paper we discuss some interesting Higgs models. It is concluded that the most likely scheme for the Higgs particles consists of five physical Higgs particles. These are two charged H⁺, H⁻ and three neutrals h⁰, H⁰, A⁰. Further more the most probably total number of elementary particles for each model is calculated [El Naschie MS. Experimental and theoretical arguments for the number of the mass of the Higgs particles. Chaos, Solitons & Fractals 2005;23:1091–8; El Naschie MS. Determining the mass of the Higgs and the electroweak bosons. Chaos, Solitons & Fractals 2005;24:899–905; El Naschie MS. On 366 kissing spheres in 10 dimensions, 528 P-Brane states in 11 dimensions and the 60 elementary particles of the standard model. Chaos, Solitons & Fractals 2005;24:447–57].     

Comment on “Reconnection scenarios…” Chaos Solitons and Fractals 2002;14(1):117–127

We point out that Proposition 3.1 in [E. Petrisor. Reconnection scenarios and the threshold of reconnection in the dynamics of non-twist maps. Chaos Solitons Fractals 2002;14(1):117–27] is, strictly speaking, false. On the other hand, we suggest that for near integrable mappings, the results of [E. Petrisor. Reconnection scenarios and the threshold of reconnection in the dynamics of non-twist maps. Chaos Solitons Fractals 2002;14(1):117–27] are qualitatively correct and quantitatively very approximate.     

Comments on “Interactions among different types of solitary waves in (2 + 1)-dimensional system” by Cheng-lin Bai, Cheng-jie Bai, Hong Zhao [Chaos, Solitons and Fractals 25 (2005) 481]

Starting from a quite universal formula, which is valid for some quite universal (2 + 1)-dimensional physical models, Bai et al. [Bai CL, Bai CJ, Zhao H. Chaos, Solitons & Fractals 2005;25:481] have claimed that the interactions among different types of solitary waves like peakons, dromions, and compactons are investigated both analytically and graphically. We show that all of the claims by Bai et al. are wrong since the adopted q-functions do not satisfy the differential equation. From this point of view, the interactions for peakon–antidromion, compacton–antidromion, and antipeakon–compacton say nothing of completely inelastic or not completely elastic.     

Crisis-induced chaos in the Rose-Hindmarsh model for neuronal activity

The bifurcation diagrams for the Rose-Hindmarsh model are obtained from the Poincaré maps which govern the dynamics of this differential system. The Lyapunov spectra for this model are estimated from time series. The transition from periodicity to crisis-induced chaos. and back to periodicity is presented for I [2.5, 2.69]. and is qualitatively different from the transitions described for different parameter regions [A. V. Holden and Yinshui Fan, Chaos, Solitons & Fractals 2, 221–236 (1992); Chaos, Solitons & Fractals 2, 349–369 (1992)]. A piecewise smooth, one-dimensional map is constructed to simulate the dynamics of the model and to reproduce the process of crisis-induced chaos.     

Families of non-travelling wave solutions to a generalized variable coefficient two-dimensional KdV equation using symbolic computation

Making use of symbolic computation and the generalized Riccati equation expansion method, some exact non-travelling wave solutions for a generalized variable coefficients two-dimensional KdV equation are obtained. By means of some suitable selections of the arbitrary functions including in the obtained solutions, the results obtained by Elwakil et al. [see: Chaos, Solitons & Fractals 19 (2004) 1083] can be recovered. From our results, some exact solutions for the cylindrical Kadomatsev–Petviashvilli equation can be also derived.     

Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator

We investigate the interaction effect of fast vertical parametric excitation and time delay on self-oscillation in a van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic and then we apply the averaging technique on this slow dynamic to derive a slow flow. In particular we analyze the slow flow to analytically approximate regions where self-excited vibrations can be eliminated. Numerical integration is performed and compared to the analytical results showing a good agreement for small time delay. It was shown that vertical parametric excitation, in the presence of delay, can suppress self-excited vibrations. These vibrations, however, persist for all values of the excitation frequency in the case of a fast vertical parametric excitation without delay [Bourkha R, Belhaq M. Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos, Solitons & Fractals, 2007;34(2):621–7.].