Simulation of Chaotic Oscillators of Fractional Order Simulación de Osciladores Caóticos de Orden Fraccional

In 1695 the theory of fractional calculus was introduced, but it only developed as a pure mathematical branch. Currently several research groups have focused on the control, the implementation of filters, PID controllers, synchronization, the implementation of circuits of chaotic systems of fractional order, etc. Currently, the number of applications of fractional calculus is increasing rapidly, these mathematical phenomena have allowed us to describe and model a real object more accurately than the classical "integer" methods. Along with the development of the fractional calculation, it was shown that many fractional-order nonlinear dynamic systems behave in a chaotic manner. This is the type of non-linear systems that are addressed in this research topic with the focus on derivatives of arbitrary order, where numerical simulations of chaotic behavior are presented in nonlinear, fractional-order autonomous models. The case studies are six chaotic oscillators of fractional order; The systems of Lorenz, Rӧssler, Financiero, Lui, Chen and Lü, whose attractors are obtained by applying the definitions of the Grünwald-Letnikov definitions and the predictive corrective method of Adams-Bashforth-Moulton. Caos, Sistemas de orden fraccional, Osciladores Resumen En 1695 se introdujo la teoría del cálculo fraccional, pero solo se desarrolló como una rama matemática pura. Actualmente varios grupos de investigación se han enfocado en el control, la implementación de filtros, controladores PID, la sincronización, la implementación de circuitos de sistemas caoticos de orden fraccional, etc. En la actualidad, el número de aplicaciones de cálculo fraccional crece rápidamente, estos fenómenos matemáticos han permitido describir y modelar un objeto real de manera más precisa que los métodos clásicos "enteros". Junto con el desarrollo del cálculo fraccional, se demostró que muchos sistemas dinámicos no lineales de orden fraccionario se comportan de manera caótica. Este es el tipo de sistemas no lineales que se abordan en este tema de investigación con el enfoque en las derivadas de orden arbitrario, donde se presentan las simulaciones numéricas del comportamiento caótico en modelos autónomos no lineales de orden fraccional. Los casos de estudio son seis osciladores caóticos de orden fraccional; Los sistemas de Lorenz, Rӧssler, Financiero, Lui, Chen y Lü, cuyos atractores se obtienen aplicando las aproximaciones de las definiciones de Grünwald-Letnikov y el método predictor corrector de Adams-BashforthMoulton. Chaos, Fractional order systems, Oscillators Citation: SILVA-JUÁREZ, Alejandro, SALAZAR-PEDRAZA, Miguel De Jesús, PONCE-MELLADO, Juan Jorge and HERRERA-SÁNCHEZ, Gustavo. Simulation of Chaotic Oscillators of Fractional Order. ECORFAN Journal-Taiwan. 2019 3-6: 11-17 * Correspondence to Author (Email: alejandro.silva@utpuebla.edu.mx) † Researcher contributing as first author. © ECORFAN Journal-Taiwan www.ecorfan. org/taiwan 12 Article ECORFAN JournalTaiwan December, 2019 Vol.3 No.6 11-17 ISSN 2524-2121 ECORFAN® All rights reserved SILVA-JUÁREZ, Alejandro, SALAZAR-PEDRAZA, Miguel De Jesús, PONCE-MELLADO, Juan Jorge and HERRERA-SÁNCHEZ, Gustavo. Simulation of Chaotic Oscillators of Fractional Order. ECORFAN Journal-Taiwan. 2019 Introduction With the development of fractional calculus and chaos theory, chaotic systems of fractional order have become a subject of study to assess their characteristics, design them with analog and digital electronics, and propose applications [1]. Models of fractional order systems allow describing and modeling a real object more accurately than using classic "integer" methods. The biggest difference between the models of fractional orders and of whole order is that those of fractional order depend on the history of the system, since they have memory [2]. That is why traditional numerical methods must adapt or propose new methods to simulate oscillators of fractional order [3-7]. Approximation of fractional order systems Fractional calculation is a generalization of integration and differentiation to the fundamental operator of non-integer order D a t q described in (1), where a and t are the limits of the operation and q ∈ R. In this way, the numerical calculation of the derivatives of fractional order are made with approximations and in this work the definition of GrünwaldLetnikov (G-L) [1,3], described in (2) is applied, D a t q = { dq dtq , q > 0, 1, q = 0, ∫ (dτ) , q t a < 0. (1) D (k−Lm/h) tk q f(t) ≈ 1 h q ∑ (−1) j ( q j ) f(tk−j) k j=0 (2) Where Lm is the memory length, tk = kh, h is the step size of the calculation and (−1) ( q j ) are the binomial coefficients denoted by cj (q) (j = 0,1, . . . ). The following expression can be used for its calculation [4]: c0 (q) = 1, cj (q) = (1 − 1+q j ) cj−1 (q) (3) Then, the general numerical solution of the fractional differential equation D a t q y(t) = f(y(t), t), can be expressed as y(tk) = f(y(tk), tk)h q − ∑ cj (q) y(tk−j) k j=v (4) For the term of memory expressed by the summation, the principle of "short memory" can be used. Then the lower index of the sums in the relationships (4) will be v = 1 for k < (Lm/h) y v = k − (Lm/h) for k > (Lm/h), or without using the principle of short memory, v = 1 is made for all k. Obviously, truncating the memory length implies an inaccuracy. If f(t) ≤ M, The following estimate can be established to determine the memory length Lm, providing the required accuracy ε: Lm ≥ ( M ε|Γ(1−q| ) 1/q (5) An evaluation of the effect of short memory and the convergence ratio of error between short and long memory are described in [7]. For the numerical simulation of the fractional order system, a method based on the predictor-corrector scheme of type ABM [4] has also been proposed, the method is based on the fact that the fractional differential equation Dt q y(t) = f(y(t), t), y0 (k) , k = 0,1, . , m − 1 is equivalent to the integral volterra equation y(t) = ∑ y0 (k) t k k! [q]−1 k=0 + 1 Γ(q) ∫ (t − τ)f(τ, y(τ))dτ. t 0 (6) Discretizing (6) to tn = nh (n = 0,1, . . . , N), h = Tsim/N and using the principle of short memory as in [8] the numerical approximation of the solution is obtained y(tn) and the order of precision is preserved. The corrective scheme is shown in (7).