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In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say , or etc.. It is a two-dimensional case of the general n-dimensional phase space.
The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation.
The solutions to the differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional vector field. Vectors representing the derivatives of the points with respect to a parameter , that is , at representative points are drawn. With enough of these arrows in place the system behaviour over the regions of plane in analysis can be visualized and limit cycles can be easily identified.
The entire field is the phase portrait, a particular path taken along a flow line is a phase path. The flows in the vector field indicate the time-evolution of the system the differential equation describes.