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An innovative box-counting method is developed in this research for diagnozing the nonlinear characteristics of dynamical systems. With the method developed, an approach that depicts the evolutionary process on Poincaré maps is established such that the nonlinear dynamical characteristics of the transient and stable process of the system can be graphically and quantitatively identified. A Duffing–van der Pol system is adopted in the research to demonstrate an application of the method. A diagram graphically describing the periodic, quasiperiodic, chaotic, and transient chaotic regions of the system's responses is constructed based on the method. Furthermore, the nature of different box-point curves is explained based on the topology of chaos and quasiperiodicity. The method developed shows innovation and efficiency in diagnozing nonlinear dynamical systems based on the topological properties of general nonlinear systems. [ABSTRACT FROM AUTHOR]

The nonlinear response of neuron firing under external electromagnetic radiation as well as transmembrane current, as two kinds of external forces, are studied in an improved Fitzhugh–Nagumo (FHN) model. The control effects of external forces to neuron firing are measured by winding number Ω during mode transition of motion types. The phenomenon of match and mismatch between the angular frequency ω out and the angular frequency of external forcing can be explained by whether the winding number is 1 / 1 or not. Ω remains as constants like 1 / 1 , 1 / 2 and 1 / 3 , etc. The amplitude of transmembrane current I 0 is increased so that it distributes in a stair-like structure in parameter space before the system enters chaotic state when I 0 is large enough. The scenarios of motion type switching are different between and beyond in the stairs. Besides, the occurrence and the disappearance of chaos are accompanied with the destruction of the orbit with the variation of parameters which temporarily spoils the toroidal topology and induces chaotic oscillation. Indeed, nonlinear response to external forces is pivotal to neuron firing and its control. [ABSTRACT FROM AUTHOR]