Research Of Chaos Theory And Its Application Based On Sine Map

In recent years, scholars discover lots of fast-slow oscillation phenomena in the continuous differential dynamic system, such as the neuron system, chemical and power system. Bursting as the main mode of neuron activity alternating between the rest and spiking state was first founded in the neuronal systems. Many decades of computational and theoretical study have proposed a large number of bursting patterns, which belong to the 1-branch bursting. Based on these work, Q S Bi and his partners observed some new bursting patterns of focus-focus type in a self-excited Lorenz system. Up to now, many types of 1-branch bursting patterns can be modeled by a low dimensional map, but few papers are reported on the unified model which can model the multi-branch bursting patterns such as the 2-branch fold/fold bursting pattern. In this study, we will introduce a three dimensional(3D) discrete model to model the new multi-branch bursting patterns.Based on a one-dimensional sinusoidal map, many nonlinear phenomena, such as period-doubling and symmetry breaking bifurcation have been found. For building the multi-branch bursting phenomena, a new two-dimensional sinusoidal discrete map is achieved by nonlinearly coupling a sinusoidal map and a cubic map. The research results indicate that there are complex nonlinear physical phenomena in this discrete map, such as symmetry breaking bifurcation, hopf bifurcation, period doubling bifurcation, periodic oscillation fast-slow effect, etc. Furthermore, we construct a three-dimensional discrete system. The fixed points and the corresponding eigenvalues are obtained based on its fast subsystem, and the stability of the system is analyzed to study the complex nonlinear dynamic behavior of the fast subsystem and the evolution of their attractors. The results indicate three are lots of bursting patterns in this system. The bursting patterns of different branches are not the same. In particular, a variety of theoretical tools and numerical simulation methods have been developed to explore the mechanism of bursting patterns. What’s more, we provide the electronic circuit experiment method to verify the bursting patterns in an analog oscilloscope.