Multi-system interaction is an important and difficult problem in physics. Motivated by the experimental result of an electronic circuit element "Fractor", we introduce the concept of dynamicorder fractional dynamic system, in which the differential-order of a fractional dynamic system is determined by the output signal of another dynamic system. The new concept offers a comprehensive explanation of physical mechanism of multi-system interaction. The properties and potential applications of dynamic-order fractional dynamic systems are further explored with the analysis of anomalous relaxation and diffusion processes.Fractional dynamic system has been focused by physicists and mathematicians over the last decades, and has received great success in the analysis of anomalous diffusion [1-4], viscoelastic rheology [5,6], control systems [7,8], complex networks [9], wave dissipation in human tissue and electrochemical corrosion process [10], etc. [11-13].In the past several years, various physical applications have given birth to the variableorder fractional dynamic system [14][15][16]. However, the multi-system interaction must be considered in the physical mechanism analysis of the variable-order fractional dynamic system and its applications. Especially, the behavior of a dynamic system may change with the evolution of other dynamic systems in multi-system physical processes. How to characterize the interaction effect between these dynamic systems? Establishing a system of equations which includes intricate interaction terms, will cause great difficulties in modeling and computation. Even worse, since they may miss capturing the critical physical mechanism of the considered problems, the established model will produce incorrect results which greatly deviate from experimental results or field measurement data. Meanwhile, researchers have confirmed that the differential orders in some fractional dynamic systems are non-constant and are often functions of other variables or system outputs [16,17]. For instance, Glöckle and Nonnenmacher have found that the differential order of proteins relaxation is a function of temperature [18]. Therefore, in order to exploit the physical mechanism of a variableorder fractional dynamic system, another dynamic system usually should be considered. For simplicity, we name this type of variable-order fractional dynamic system as dynamic-order fractional dynamic system.The purpose of this letter is to make an innovative study of dynamic-order fractional dynamic systems. Since previous studies have indicated that the differential-order is a critical factor in the variable-order fractional dynamic system, it is necessary to analyze the physical mechanism about how environmental factors or system variables influence the system's differential order. From the analysis of fractional dynamic system, especially the variableorder dynamic system, it has been found the fractional differential order of one dynamic system usually stems from another dynamic system [16][17][18]. Hence, in many fraction...
