An improved cell mapping method based on dimension-extension for fractional systems

Global analysis of fractional systems is a challenging topic due to the memory property. Without the Markov assumption, the cell mapping method cannot be directly applied to investigate the global dynamics of such systems. In this paper, an improved cell mapping method based on dimension-extension is developed to study the global dynamics of fractional systems. The evolution process is calculated by introducing additional auxiliary variables. Through this treatment, the nonlocal problem is localized in a higher dimension space. Thus, the one-step mappings are successfully described by Markov chains. Global dynamics of fractional systems can be obtained through the proposed method without memory losses. Simulations of the point mapping show great accuracy and efficiency of the method. Abundant global dynamics behaviors are found in the fractional smooth and discontinuous oscillator.