Chaos Soliton Fract Research Articles

Comment on “Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system”

• We work on a generalized ( 2 + 1 )-dimensional dispersive long-wave system. • With the help of symbolic computation, we give rise to four sets of the similarity reductions, each of which leads to a known ordinary differential equation . • Our similarity reductions are in respect of the horizontal velocity and the wave elevation above the undisturbed water surface. • All of our results depend on the constant coefficients in the system. Active researches on the oceanic water waves have been done. As for the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth, the paper commented [i.e., Chaos Solitons Fract. 138, 109950 (2020)] has investigated a generalized ( 2 + 1 )-dimensional dispersive long-wave system. In respect of the horizontal velocity and the wave elevation above the undisturbed water surface, with the help of symbolic computation, we give rise to four sets of the similarity reductions, each of which leads to a known ordinary differential equation. All of our results depend on the constant coefficients in the original system.

A coupled system of Langevin differential equations of fractional order and associated to antiperiodic boundary conditions

In this paper, we provide an extension result for the existence and uniqueness of solutions for a coupled system of fractional Langevin differential equations with antiperiodic boundary conditions. The system involves two different Caputo fractional derivatives defined on different intervals and associated with boundary conditions described by sequential fractional derivatives. As a conclusion for our main result, we deduce the results of H. Fazli, J.J. Nieto, Fractional Langevin equation with anti‐periodic boundary conditions, Chaos Soliton Fract, 114 (2018),332–337 under less restrictive conditions. The consistency of the main results is demonstrated by two numerical examples. For the sake of completeness, we end the paper by a concluding discussion.

Criteria of the electron pumping in electron-hole quantum plasma

The degradation of quantum semiconductor plasma during electron pumping is an important problem since it has several applications in the technological industry. Moreover, the identification of such a problem cannot be accomplished by experiment alone and a further theoretical investigation is required. Owing to the energy gap of semiconductors, electrons should acquire energy in order to move from the valence band to the conduction band. Therefore, to employ the quantum fluid model that includes the exchange-correlation potentials, Bohm potential, and the degenerate pressure, the electron-hole lifetime should be greater than the plasma oscillation time. If the latter condition is not satisfied, the quantum fluid model failed to describe the system and it is convenient to use the density functional theory. As a result of that, we apply the quantum fluid model to GaN where the plasma oscillation has a time of 0.002 ps while the electron-hole lifetime is 1 ps. The modified Kadomtsev-Petviashvili (mKP) equation had been obtained using a new set of stretched coordinates as the Kadomtsev-Petviashvili (KP) equation had been derived in (Afify et al 2019 Chaos Soliton Fract 124, 18–25). Our findings determine whether the quantum fluid model is valid or not in the case of semiconductor plasma. Studying exchange effects in a dynamic context has certain limitations. Thus, we stressed that the more accurate approaches introduced in the literature are hard to apply for nonlinear issues. Further, the applicability of our results is discussed, but not this much. Moreover, the numerical results showed that the electron beam parameters and quantum effects have significant effects on the growth of the solitary waves.

A characterisation of duopoly dynamics with frictions in production adjustments

This article revisits the classical work of Puu (Chaos Soliton Fract 1(6):573–581, 1991) on duopoly dynamics by gathering two distinct aspects of the functioning of markets: production of goods requires time and is subject to some gestation lags, but trading takes place continuously. Dynamics are characterized by a two-dimensional system of delay differential equations. The main aim of this work is to show that regular and non-regular fluctuations may emerge endogenously because of the existence of heterogeneous interacting agents that choose production over time in a myopic way. Chaotic dynamics in the discrete-time model of Puu (Chaos Soliton Fract 1(6):573–581, 1991) appear to be close enough to the origin of axes (implying that quantities produced by both firms are close to zero). In contrast, in our continuous-time version of the model with discrete delays, the dynamic system is more suitable of generating complex dynamics far enough from the origin when marginal costs vary. This is because of the role played by time delays and inertia. From a mathematical point of view, we show the existence of Hopf bifurcations and detect how time delays and inertia affect the stability of the system by using the recent techniques of stability crossing curves introduced by Gu et al. (J Math Anal Appl 311(1):231–253, 2005) and generalized by Lin and Wang (Can Appl Math Quart 20(4):519–533, 2012). The article also provides some findings about global bifurcations and chaotic dynamics by combining analytical studies and simulation exercises.

Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that from the time series of two identical systems, one can simply extract value of the stability parameter which value can be treated as largest LLE. Unlike the method presented in part, one developed method (LLEDPT) can be applied to the dynamical systems of any type, continuous, with discontinuities, with time delay and others. The theoretical improvement shows simplicity of the method and its obvious physical background. The proofs for the method effectiveness are based on results of the simulations of the experiments for Duffing and Van der Pole oscillators. These results were compared with ones obtained with use of the Stefanski method (Stefanski in Chaos Soliton Fract 11(15):2443–2451, 2000; Chaos Soliton Fract 15:233–244, 2003; Chaos Soliton Fract 23:1651–1659, 2005; J Theor Appl Mech 46(3):665–678, 2008) and LLEDP method. LLEDPT can be used also as the criterion of stability of the control system, where desired behavior of controlled system is explicitly known (Balcerzak et al. in Mech Mech Eng 17(4):325–339, 2013). The next step of development of the method can be considered in direction that allows estimation of LLE from the real time series, systems with discontinuities, with time delay and others.

Some fixed point results for a generalized ψ-weak contraction mappings in orbitally metric spaces

Samet and Vetro [Samet B, Vetro C. Berinde mappings in orbitally complete metric spaces. Chaos Solitons Fract 2011;44:1075–9.] studied a fixed point theorem for a self-mapping satisfying a general contractive condition of integral type in orbitally complete metric spaces. In this paper, we introduce the notion of a generalized ψ-weak contraction mapping and establish some results in orbitally complete metric spaces. Our results generalize several well-known comparable results in the literature. As an application of our results we deduce the result of Samet and Vetro. Some examples are given to illustrate the useability of our results.

A novel image encryption algorithm based on a 3D chaotic map

Recently [Solak E, Çokal C, Yildiz OT Biyikogˇlu T. Cryptanalysis of Fridrich’s chaotic image encryption. Int J Bifur Chaos 2010;20:1405–1413] cryptanalyzed the chaotic image encryption algorithm of [Fridrich J. Symmetric ciphers based on two-dimensional chaotic maps. Int J Bifur Chaos 1998;8(6):1259–1284], which was considered a benchmark for measuring security of many image encryption algorithms. This attack can also be applied to other encryption algorithms that have a structure similar to Fridrich’s algorithm, such as that of [Chen G, Mao Y, Chui, C. A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos Soliton Fract 2004;21:749–761]. In this paper, we suggest a novel image encryption algorithm based on a three dimensional (3D) chaotic map that can defeat the aforementioned attack among other existing attacks. The design of the proposed algorithm is simple and efficient, and based on three phases which provide the necessary properties for a secure image encryption algorithm including the confusion and diffusion properties. In phase I, the image pixels are shuffled according to a search rule based on the 3D chaotic map. In phases II and III, 3D chaotic maps are used to scramble shuffled pixels through mixing and masking rules, respectively. Simulation results show that the suggested algorithm satisfies the required performance tests such as high level security, large key space and acceptable encryption speed. These characteristics make it a suitable candidate for use in cryptographic applications.

Novel soliton-like and multi-solitary wave solutions of (3 + 1)-dimensional Burgers equation

In this paper, the (3 + 1)-dimensional Burgers equation is studied by means of Painlevé truncation expansion, extended Riccati equation method and Wang’s multiple Riccati equations method in Wang and Li [D.S. Wang, H. Li, Symbolic computation and non-travelling wave solutions of (2 + 1)-dimensional non-linear evolution equations, Chaos Soliton Fract. 38 (2008) 383–390]. As a result, some novel soliton-like, complexiton solutions and multi-solitary wave solutions are derived. It is shown that the process to derive them is very direct. Subsequently, an ( n + 1 ) -dimensional Burgers type equation is presented and its solutions are given.

WITHDRAWN: Charged superstring attached two different D-branes

The publisher regrets that this article is an accidental duplication of an article that has already been published in Chaos Soliton Fract. , 38 (2008) 615–619, doi: 10.1016/j.chaos.2008.01.025 . The duplicate article has therefore been withdrawn.

Stabilization of parameters perturbation chaotic system via adaptive backstepping technique

The work of Yassen [M.T. Yassen, Chaos control of chaotic dynamical systems using backstepping design, Chaos Soliton Fract. 27 (2006) 537–548] which mainly investigated the stabilization problem for a class of chaotic systems without the parameters perturbation. This paper is concerned with stabilization problem for a class of parameters perturbation chaotic systems via both backstepping design method and adaptive technique. The proposed controllers can guarantee that the parameters perturbation systems will be stabilized at a fixed bounded point. Furthermore, the paper also proposes controllers to stabilize the uncertain chaotic system at equilibrium point with only backstepping design method. Finally, numerical simulations are given to illustrate the effectiveness of the proposed controllers.

Comments on “Application of generalized differential transform method to multi-order fractional differential equations”, Vedat Suat Erturk, Shaher Momani, Zaid Odibat [Commun Nonlinear Sci Numer Simul 2008;13:1642–54

In this note, we would like to point some similarities between the study [Erturk VS, Momani S, Odibat Z. Application of generalized differential transform method to multi-order fractional differential equations. Commun Nonlinear Sci Numer Simul. doi:10.1016/j.cnsns.2007.02.006] with the already existing one [Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method. Chaos Soliton Fract. 10.1016/j.chaos.2006.09.004].

Elliptic equation’s new solutions and their applications to two nonlinear partial differential equations

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of first-order elliptic equation ϕ′ 2 = a 0 + a 1 ϕ + a 2 ϕ 2 + a 3 ϕ 3 + a 4 ϕ 4 (where ϕ ′ = d d x ϕ ) are obtained. To our knowledge, these nontrivial solutions can not be found in [Chaos Solitons Fract. 26 (2005) 785–794] and [Phys. Lett. A 336 (2005) 463–476] by Yomba and other existent papers until now. By using these nontrivial solutions, a direct algebraic method is described to construct several kinds of exact non-travelling wave solutions for the (2 + 1)-dimensional Breaking soliton equations and the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equation. By using this method, many other physically important nonlinear partial differential equations (NLPDEs) can be investigated and new non-travelling wave solutions can be explicitly obtained with the aid of symbolic computation system Maple.

Applications of a further extended tanh method

By using a further extended tanh method [Chaos Solitons Fract. 17 (2003) 669] and symbolic computation system Maple, we obtain abundant new explicit exact solutions of the (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional Boussinesq equation. The new solutions including soliton-like solutions, period form solutions and some other solutions.

Fractional calculus and some intermediate physical processes

One of the main applications of the fractional calculus (integration and differentiation of arbitrary (fractional) order) is the modeling of the intermediate processes [J. Frac. Calculus 7 (1995) 89; Int. J. Theo. Phys. 35(2) (1996) 311; R. Gorenflo, R. Rutman, Fachbereich Mathematik Reprint No. A-28/94 (1994); F. Mainardi, Proc. 14th IMACS World Congr., vol. 1, 1994, p. 329; Chaos Soliton Fract. 7(6) (1996) 1; J. Math. Phys. 30 (1989) 134]. In this work we use the fractional calculus to formulate the models which represent the two intermediate physical process; convection–diffusion and transport–diffusion processes. The existence and uniqueness of their solution will be proved and the continuation to the convection, diffusion and transport processes are studied.

The [formula omitted]-limit set and the limit set of subsets

In this paper, we discuss the properties of Ω-limit sets and limit sets of subsets in a metric space, and also we point out a mistake in [Chaos Soliton Fract. 11 (2000) 2319] that denies a Conley’s theorem.

Bifurcations in a Mathieu equation with cubic nonlinearities: Part II

In a previous paper [Chaos Solitons Fract. 14(2) (2002) 173], the authors investigated the dynamics of the equation: d 2x dt 2 +(δ+ϵ cost)x+ϵ Ax 3+Bx 2 dx dt +Cx dx dt 2+D dx dt 3 =0 We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented.

Design of an evaluation system for local blood pressure control programs: Bruce M. Brock, Kenneth M. Cummings, and Sanford B. Izenson; Michigan Department of Public Health, Lansing, Michigan

One of the main applications of the fractional calculus (integration and differentiation of arbitrary (fractional) order) is the modeling of the intermediate processes [J. Frac. Calculus 7 (1995) 89; Int. J. Theo. Phys. 35(2) (1996) 311; R. Gorenflo, R. Rutman, Fachbereich Mathematik Reprint No. A-28/94 (1994); F. Mainardi, Proc. 14th IMACS World Congr., vol. 1, 1994, p. 329; Chaos Soliton Fract. 7(6) (1996) 1; J. Math. Phys. 30 (1989) 134]. In this work we use the fractional calculus to formulate the models which represent the two intermediate physical process; convection–diffusion and transport–diffusion processes. The existence and uniqueness of their solution will be proved and the continuation to the convection, diffusion and transport processes are studied.