Phase Plane Method

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 (say time t), that is (dx/dt, dy/dt), 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 (i.e. a path always tangent to the vectors) is a phase path. The flows in the vector field indicate the time-evolution of the system the differential equation describes.

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). 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. This is useful in determining if the dynamics are stable or not.

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. In such cases one can model the rise and fall of reactant and product concentration (or mass, or amount of substance) with the correct differential equations and a good understanding of chemical kinetics.

Read more about Phase Plane Method:  Example of A Linear System, See Also

Other articles related to "phase, plane, phase plane method":

Interferometry - Categories - Wavefront Splitting Versus Amplitude Splitting
... In 1834, Humphrey Lloyd interpreted this effect as proof that the phase of a front-surface reflected beam is inverted ... By adjusting the tilt, which adds a controlled phase gradient to the fringe pattern, one can control the spacing and direction of the fringes, so that one may obtain an easily ... be adjusted so that they are in focus in any desired plane ...
Phase Plane Method - See Also
... Phase line, 1-dimensional case Phase space, n-dimensional case Phase portrait ...

Famous quotes containing the words method, phase and/or plane:

    No method nor discipline can supersede the necessity of being forever on the alert. What is a course of history or philosophy, or poetry, no matter how well selected, or the best society, or the most admirable routine of life, compared with the discipline of looking always at what is to be seen? Will you be a reader, a student merely, or a seer? Read your fate, see what is before you, and walk on into futurity.
    Henry David Thoreau (1817–1862)

    I had let preadolescence creep up on me without paying much attention—and I seriously underestimated this insidious phase of child development. You hear about it, but you’re not a true believer until it jumps out at you in the shape of your own, until recently quite companionable child.
    Susan Ferraro (20th century)

    It was the most ungrateful and unjust act ever perpetrated by a republic upon a class of citizens who had worked and sacrificed and suffered as did the women of this nation in the struggle of the Civil War only to be rewarded at its close by such unspeakable degradation as to be reduced to the plane of subjects to enfranchised slaves.
    Anna Howard Shaw (1847–1919)