Chaotic systems are dynamical systems that are highly sensitive to initial conditions. Such systems are used to model many real-world phenomena in science and engineering. The main purpose of this paper is to present several efficient numerical treatments for chaotic systems involving fractal-fractional operators. Several numerical examples test the performance of the proposed methods. Simulations with different values of the fractional and fractal parameters are also conducted. It is demonstrated that the fractal-fractional derivative enables one to capture all the useful information from the history of the phenomena under consideration. The numerical schemes can also be implemented for other chaotic systems with fractal-fractional operators.

混沌系统是对初始条件高度敏感的动力学系统。此类系统用于模拟科学和工程学中的许多现实世界现象。本文的主要目的是为涉及分形-分形算子的混沌系统提供几种有效的数值处理方法。几个数值示例测试了所提出方法的性能。还进行了分数和分形参数的不同值的仿真。证明了分形-分数阶导数可以使人们从正在考虑的现象历史中捕获所有有用的信息。数值方案也可以用于具有分形-分数分数算子的其他混沌系统。