Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays

We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].     

Time delay in a basic model of the immune response

The effects of time delay on the two-dimensional system of Mayer et al., which represents the basic model of the immune response, are analysed (cf. Mayer H, Zaenker KS, an der Heiden U. A basic mathematical model of the immune response. Chaos, Solitons and Fractals 1995;5:155–61). We studied variations of the stability of the fixed points due to the time delay and the possibility for the occurrence of the chaotic solutions.     

An Analytical Study of Bifurcations Generated by Some Iterated Function Systems

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.     

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].     

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.     

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.     

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.     

Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion

In this paper, we study a diffusive three-species ratio-dependent food chain model, using differential inequality, to obtain sufficient conditions that ensure the permanence of the system and the extinction of predator species. Our results reinforce the main result of Sun Wen, Shihua Chen and Huihai Mei [Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation, Chaos, Solitons and Fractals, in press].     

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 proof for a theorem on intertwining property of attraction basin boundaries in planar dynamical systems

In this paper we present a new rigorous proof to a theorem on intertwining property of attraction basin boundaries in planar systems discussed in the literature [Chaos, Solitons & Fractals 10 (1999) 1453].     

Exact solution of a thin film flow of an Oldroyd 6-constant fluid over a moving belt

We obtain exact solutions for thin film flow of an Oldroyd 6-constant fluid on a vertically moving belt. These are compared with the homotopy perturbation results of Siddiqui et al. [Siddiqui AM, Ahmed M, Ghori QK. Thin film flow of non-Newtonian fluids on a moving belt. Chaos, Solitons & Fractals 2007;33(3):1006–1016.].     

Application of the internal time operators for the Renyi map

Recently, the internal time operator for the Renyi map has been constructed (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals). It corresponds to a phase space given by the interval [0,1] and to the invariant Lebesgue measure. In this paper, following the idea of (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals), we construct the time operator for a dynamical system with an arbitrary invariant measure μ and an arbitrary phase space X=[a,b] with a and b finite or infinite. We illustrate also the action of such an operator on a fixed initial state.     

Structural Stability of Discontinuous Galerkin Schemes

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193?C2199, 2009; Chaos Solitons Fractals 41(2):655?C660, 2009).     

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.].     

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.     

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.     

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.     

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.     

Crisis-limited chaotic dynamics in ecological systems

We review our recent efforts to understand why chaotic dynamics is rarely observed in natural populations. The study of two-model ecosystems considered in this paper suggests that chaos exists in narrow parameter ranges. This dynamical behaviour is caused by the crisis-induced sudden death of chaotic attractors. The computed bifurcation diagrams and basin boundary calculations reinforce our earlier conclusion [Chaos, Solitons & Fractals 8 (12) (1997) 1933; Int J Bifurc Chaos 8 (6) (1998) 1325] that the reason why chaos is rarely observed in natural populations is hidden within the mathematical structure of the ecological interactions and not with the problem associated with the data (insufficient length, precision, noise, etc.) and its analysis. We also argue that crisis-limited chaotic dynamics can be commonly found in model terrestrial ecosystems.     

Blood flow simulation through fractal models of circulatory system

The blood flow in human arteries has been analytically calculated according to Poiseuille’s equation. Geometry of the fractal arterial trees has been described in previous article [Gabryś E, Rybaczuk M, Kędzia A. Fractal model of circulatory system. Symmetrical and asymmetrical approach comparison. Chaos, Solitons & Fractals, in press]. Blood vessel trees are consisted of straight, rigid cylindrical tubes. In each bifurcation two new children segments appears according to Murray law.Blood flow in circulatory system is driven by the pressure differences at the two ends of the blood vessel. A mathematical analysis shows the continuous dependence of the solution on vessel tree parameters and boundary condition.