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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 wikiPageLength "8579".
- Phase_plane wikiPageOutDegree "38".
- Phase_plane wikiPageRevisionID "660545740".
- Phase_plane wikiPageWikiLink Applied_mathematics.
- Phase_plane wikiPageWikiLink Category:Nonlinear_control.
- Phase_plane wikiPageWikiLink Category:Ordinary_differential_equations.
- Phase_plane wikiPageWikiLink Characteristic_polynomial.
- Phase_plane wikiPageWikiLink Chemical_kinetics.
- Phase_plane wikiPageWikiLink Coefficient_matrix.
- Phase_plane wikiPageWikiLink Coordinate_vector.
- Phase_plane wikiPageWikiLink Dependent_and_independent_variables.
- Phase_plane wikiPageWikiLink Derivative.
- Phase_plane wikiPageWikiLink Determinant.
- Phase_plane wikiPageWikiLink Differential_equation.
- Phase_plane wikiPageWikiLink Eigenvalues.
- Phase_plane wikiPageWikiLink Eigenvalues_and_eigenvectors.
- Phase_plane wikiPageWikiLink Eigenvectors.
- Phase_plane wikiPageWikiLink Function_(mathematics).
- Phase_plane wikiPageWikiLink Implicit_function.
- Phase_plane wikiPageWikiLink Independent_variable.
- Phase_plane wikiPageWikiLink Limit_cycle.
- Phase_plane wikiPageWikiLink Limit_cycles.
- Phase_plane wikiPageWikiLink Linear_differential_equation.
- Phase_plane wikiPageWikiLink Lotka–Volterra_equation.
- Phase_plane wikiPageWikiLink Lotka–Volterra_equations.
- Phase_plane wikiPageWikiLink Matrix_(mathematics).
- Phase_plane wikiPageWikiLink Node_(autonomous_system).
- Phase_plane wikiPageWikiLink Nonlinear_system.
- Phase_plane wikiPageWikiLink Nonlinear_systems.
- Phase_plane wikiPageWikiLink Phase_line_(mathematics).
- Phase_plane wikiPageWikiLink Phase_portrait.
- Phase_plane wikiPageWikiLink Phase_space.
- Phase_plane wikiPageWikiLink Physical_system.
- Phase_plane wikiPageWikiLink Predator-prey_model.
- Phase_plane wikiPageWikiLink Quadratic_equation.
- Phase_plane wikiPageWikiLink Quadratic_formula.
- Phase_plane wikiPageWikiLink Saddle_point.
- Phase_plane wikiPageWikiLink Sine.
- Phase_plane wikiPageWikiLink Trace_(linear_algebra).
- Phase_plane wikiPageWikiLink Trigonometric_function.
- Phase_plane wikiPageWikiLink Trigonometric_functions.
- Phase_plane wikiPageWikiLink Two-dimensional_space.
- Phase_plane wikiPageWikiLink Vector_field.
- Phase_plane wikiPageWikiLink File:Phase_plane_nodes.svg.
- Phase_plane wikiPageWikiLinkText "Phase plane".
- Phase_plane wikiPageWikiLinkText "phase plane".
- Phase_plane hasPhotoCollection Phase_plane.
- Phase_plane wikiPageUsesTemplate Template:Differential_equations.
- Phase_plane wikiPageUsesTemplate Template:Reflist.
- Phase_plane subject Category:Nonlinear_control.
- Phase_plane subject Category:Ordinary_differential_equations.
- Phase_plane hypernym Display.
- Phase_plane type Article.
- Phase_plane type Work.
- Phase_plane type Article.
- 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 sameAs Фазалық_жазықтық.
- Phase_plane sameAs Fazinė_plokštuma.
- Phase_plane sameAs Płaszczyzna_fazowa.
- Phase_plane sameAs m.06mvlb.
- Phase_plane sameAs Фазовая_плоскость.
- Phase_plane sameAs Q2088081.
- Phase_plane sameAs Q2088081.
- Phase_plane wasDerivedFrom Phase_plane?oldid=660545740.
- Phase_plane isPrimaryTopicOf Phase_plane.