Analysing periodicity, nonlinearity and transitional characteristics of nonlinear dynamic systems with Periodicity Ratio (PR)

The multiple-periodicity, nonlinearity and transitional characteristics of nonlinear dynamic systems subjected to external excitations are studied in this research. Diagnoses of the number and changing multiple-periodicities of Duffing’s systems are performed with implementation of the Periodicity Ratio (PR). The multiple-periodicity diagram is generated such that the periodicities and nonlinearity of the systems with respect to the system parameters can be graphically studied. The stability and convergence of the systems are investigated. The results of the research show that the number of period of periodicity of the systems increases continuously when certain system parameters increase. Transitional characteristics of the systems are also investigated. Both Lyapunov Exponents and Periodicity Ratio are implemented to diagnose the transitional routes of the systems. New symmetrical transition characters from periodicity to quasi-periodicity and chaos are displayed in terms of PR values. Comparing to Lyapunov Exponents, the Periodicity Ratio discloses more detailed and accurate transition information.