A study of a modified nonlinear dynamical system with fractal-fractional derivative

Abstract

Purpose This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control. Design/methodology/approach The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems. Findings Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved. Originality/value This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised Pochhammer-Chree equation in mathematical physics

“…Some fundamental definitions of the Riemann-Liouville (R-L) fractional differentiation, Laplace transform (LT) and FCD are presented [8,18,20,21,22,41,44]. Definition 2.1 For α > 0 left (R-L) order fractional integral of α is defined as below [6][7][9][10]…”

“…First,ourself receive the solutions for existence and uniqueness. And other research solved timefractional generalised analytical-approximate solutions PC equations for waves publicity of an elastic rod using the q-homotopy analysis of the transform method [13][14][15], Modulation instability analysis [16], Hydro-magnetic [17], solitary wave [18], Carreau fluid [19], kink wave [20], Existence and Uniqueness [21,24], Hilbert space [22], natural reduced differential transform method [23], iterative Laplace transform method [25], tanh-coth and the sine-cosine methods [26], explicit fourthorder time stepping methods [27], decomposition method [28], weierstrass elliptic function method [30], modified F-expansion methods [31], global existence [32], generalized exponential rational function (GERF) technique [33][34], bernoulli sub-equation function method [35], lie group method [36], fractional natural decomposition method [37], modified exponential method [39], residual power series method [40], adams-bashforth scheme [41], laplace transform [42][43], conformable derivative [44,47], Mittag-Leffler function [45], caputo derivatives [46], Sumudu transform [48],and so on [49][50][51][52][53]…”

Dynamical analysis of a rotating rigid body containing a viscous incompressible fluid

“…The problem of the rotatory charged rigid body (RB), which contains a hollow filled with a viscous fluid, around a fixed point has piqued the enthusiasm of scientists and engineers since 1885 when Russian scientist Zhukovskii, who is best known for the so-called Joukowski aerofoil, proposed the concept (Zhukovskii, 1885; Sobolev, 1960; Moiseyev and Rumyantsev, 1968; Kostyuchenko et al , 1998; Kopachevsky and Krein, 2000; Chernous’ko, 1968; Chernousko, 1972; Smirnova, 1974; Vil’ke, 1993; Baranova and Vil’ke, 2013), and it has now wide applications in submarines, spacecraft and airplanes. Such a problem is too difficult to deal with owing to it being controlled by complicated differential equations of a nonlinear system, and a numerical method must be adopted to study its dynamical properties (Kumar et al , 2022; Sastre et al , 2022; He, 2023). Tian and her colleagues found that a nano/microelectromechanical system filled with plasma or fluids can control its dynamical property (Tian et al , 2021; Tian and He, 2021, He, 2023), and the idea has led to a new discipline, which is the fractal fluid mechanics (Wang, 2021; Wang, 2022; Khan, 2022; Wang, 2023; Wang and Wei, 2023; Wang and He, 2019; He and Liu, 2023).…”

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

“…First,ourself receive the solutions for existence and uniqueness. And other research solved timefractional generalised analytical-approximate solutions PC equations for waves publicity of an elastic rod using the q-homotopy analysis of the transform method [13][14][15], Modulation instability analysis [16], Hydro-magnetic [17], solitary wave [18], Carreau fluid [19], kink wave [20], Existence and Uniqueness [21,24], Hilbert space [22], natural reduced differential transform method [23], iterative Laplace transform method [25], tanh-coth and the sine-cosine methods [26], explicit fourthorder time stepping methods [27], decomposition method [28], weierstrass elliptic function method [30], modified F-expansion methods [31], global existence [32], generalized exponential rational function (GERF) technique [33][34], bernoulli sub-equation function method [35], lie group method [36], fractional natural decomposition method [37], modified exponential method [39], residual power series method [40], adams-bashforth scheme [41], laplace transform [42][43], conformable derivative [44,47], Mittag-Leffler function [45], caputo derivatives [46], Sumudu transform [48],and so on [49][50][51][52][53]…”