Validity of fractal derivative to capturing chaotic attractors

Suggested independently with different definitions, fractal derivative and conformable derivative are alpha proportional. They have been applied in quit a few problems in many field of sciences in the last few years with great success. However, some researchers have pointed out some criticisms and even concluded that they were flawed. In this paper, while confirm the validity of the conformable and fractal derivatives and we present their applications to chaotic attractors. We considered a general non-linear Cauchy problem where the differential operator is that of fractal and conformable and present the derivation of conditions for which the existence and the uniqueness of the exact solution are reached. Several examples are considered, solved and numerical simulations depicting real world observations. (C) 2019 Elsevier Ltd. All rights reserved.