Other examples of oscillatory systems are certain chemical reactions with multiple steps, some of which involve dynamic equilibria rather than reactions that go to completion. This is useful in determining if the dynamics are stable or not. In these models the phase paths can "spiral in" towards zero, "spiral out" towards infinity, or reach neutrally stable situations called centres where the path traced out can be either circular, elliptical, or ovoid, or some variant thereof. In this way, phase planes are useful in visualizing the behaviour of physical systems in particular, of oscillatory systems such as predator-prey models (see Lotka–Volterra equations). The flows in the vector field indicate the time-evolution of the system the differential equation describes. a path always tangent to the vectors) is a phase path. The entire field is the phase portrait, a particular path taken along a flow line (i.e.
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. Vectors representing the derivatives of the points with respect to a parameter (say time t), that is ( dx/ dt, dy/ dt), at representative points are drawn. Graphically, this can be plotted in the phase plane like a two-dimensional vector field. The solutions to the differential equation are a family of functions. The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation. It is a two-dimensional case of the general n-dimensional phase space. 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 ( x, y), or ( q, p) etc.
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