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- Phase_plane abstract "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. (any pair of variables). 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 (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.".
- Phase_plane wikiPageExternalLink phaseplane.aspx.
- Phase_plane wikiPageExternalLink systemsde.aspx.
- Phase_plane wikiPageExternalLink node8.html.
- Phase_plane wikiPageID "2110756".
- Phase_plane wikiPageRevisionID "594804786".
- Phase_plane hasPhotoCollection Phase_plane.
- Phase_plane subject Category:Nonlinear_control.
- Phase_plane subject Category:Ordinary_differential_equations.
- Phase_plane type Abstraction100002137.
- Phase_plane type Communication100033020.
- Phase_plane type DifferentialEquation106670521.
- Phase_plane type Equation106669864.
- Phase_plane type MathematicalStatement106732169.
- Phase_plane type Message106598915.
- Phase_plane type OrdinaryDifferentialEquations.
- Phase_plane type Statement106722453.
- Phase_plane comment "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. (any pair of variables).".
- Phase_plane label "Phase plane".
- Phase_plane label "Płaszczyzna fazowa".
- Phase_plane label "Фазовая плоскость".
- Phase_plane sameAs Płaszczyzna_fazowa.
- Phase_plane sameAs m.06mvlb.
- Phase_plane sameAs Q2088081.
- Phase_plane sameAs Q2088081.
- Phase_plane sameAs Phase_plane.
- Phase_plane wasDerivedFrom Phase_plane?oldid=594804786.
- Phase_plane isPrimaryTopicOf Phase_plane.