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| {{differential equations}}
| | Myrtle Benny is how I'm known as and I feel comfortable when people use the complete title. Minnesota is exactly where he's been living for years. His spouse doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for quite a while. Bookkeeping is my profession.<br><br>Look into my website: [http://www.asianas.com/index.php?m=member_blog&p=view&id=132737&sid=54346 asianas.com] |
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| In [[applied mathematics]], in particular the context of [[nonlinear systems|nonlinear system analysis]], a '''phase plane''' is a visual display of certain characteristics of certain kinds of [[differential equation]]s; 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 space|two-dimensional]] case of the general ''n''-dimensional [[phase space]].
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| The '''phase plane method''' refers to graphically determining the existence of [[limit cycle]]s in the solutions of the differential equation.
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| The solutions to the differential equation are a family of [[function (mathematics)|function]]s. Graphically, this can be plotted in the phase plane like a two-dimensional [[vector field]]. Vectors representing the [[derivative]]s 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.
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| 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.
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| In this way, phase planes are useful in visualizing the behaviour of [[physical system]]s; in particular, of oscillatory systems such as predator-prey models (see [[Lotka–Volterra equation]]s). 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.<ref name="Jordan, Smith">{{citebook|title=Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers|edition=4th|author=D.W. Jordan, P. Smith|publisher=Oxford University Press|year=2007|isbn=978-0-19-902825-8}}</ref>
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| 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.<ref>{{citebook|title=Chaos: An Introduction to Dynamical Systems|author=K.T. Alligood, T.D. Sauer, J.A. Yorke|publisher=Springer|year=1996|isbn=978-0-38794-677-1}}</ref>
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| ==Example of a linear system==
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| A two-dimensional system of [[linear differential equation]]s can be written in the form:<ref name="Jordan, Smith"/>
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| :<math> \begin{align}
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| \frac{dx}{dt} & = Ax + By \\
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| \frac{dy}{dt} & = Cx + Dy
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| \end{align}</math>
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| which can be organized into a [[matrix (mathematics)|matrix]] equation:
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| :<math> \begin{align}
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| & \frac{d}{dt} \begin{pmatrix}
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| x \\
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| y \\
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| \end{pmatrix} = \begin{pmatrix}
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| A & B \\
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| C & D \\
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| \end{pmatrix}\begin{pmatrix}
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| x \\
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| y \\
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| \end{pmatrix} \\
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| & \frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x}.
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| \end{align}</math>
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| where '''A''' is the 2 × 2 [[coefficient matrix]] above, and '''x''' = (''x'', ''y'') is a [[coordinate vector]] of two [[independent variable]]s. | |
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| Such systems may be solved analytically, for this case by integrating:<ref>{{citebook|title=Elementary Differential Equations and Boundary Value Problems|edition=4th|author=W.E. Boyce, R.C. Diprima|publisher=John Wiley & Sons|year=1986|isbn=0-471-83824-1}}</ref>
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| <math>\frac{dy}{dx} = \frac{Cx+Dy}{Ax+By}</math>
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| although the solutions are [[implicit function]]s in ''x'' and ''y'', and are difficult to interpret.<ref name="Jordan, Smith"/>
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| ===Solving using eigenvalues===
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| More commonly they are solved with the coefficients of the right hand side written in matrix form using [[eigenvalues]] λ, given by the [[determinant]]:
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| :<math>\det(\mathbf{A}- \lambda \mathbf{I})=0</math>
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| and [[eigenvectors]]:
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| :<math> \mathbf{A}\mathbf{x}=\lambda\mathbf{x}</math>
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| The eigenvalues represent the powers of the exponential components and the eigenvectors are coefficients. If the solutions are written in algebraic form, they express the fundamental multiplicative factor of the exponential term. Due to the nonuniqueness of eigenvectors, every solution arrived at in this way has undetermined constants ''c''<sub>1</sub>, ''c''<sub>2</sub>, ... ''c<sub>n</sub>''.
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| The general solution is:
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| :<math>x = \begin{bmatrix} k_{1} \\ k_{2} \end{bmatrix} c_{1}e^{\lambda_1 t} + \begin{bmatrix} k_{3} \\ k_{4} \end{bmatrix} c_{2}e^{\lambda_2 t}. </math>
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| where λ<sub>1</sub> and λ<sub>2</sub> are the eigenvalues, and (k<sub>1</sub>, k<sub>2</sub>), (k<sub>3</sub>, k<sub>4</sub>) are the basic eigenvectors. The constants ''c''<sub>1</sub> and ''c''<sub>2</sub> account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.
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| The above determinant leads to the [[characteristic polynomial]]:
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| :<math>\lambda^2 - (A+D)\lambda + (AD-BC)=0</math>
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| which is just a [[quadratic equation]] of the form:
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| :<math>\lambda^2 - p\lambda + q=0</math>
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| where;
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| ::<math>p = A+D = \mathrm{tr}(\mathbf{A}) \,,</math> | |
| ("tr" denotes [[trace (linear algebra)|trace]]) and
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| ::<math>q=AD-BC=\det(\mathbf{A})\,.</math> | |
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| The explicit solution of the eigenvalues are then given by the [[quadratic formula]]:
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| :<math>\lambda = \frac{1}{2}(p\pm \sqrt{\Delta})\,</math>
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| where
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| ::<math>\Delta=p^2-4q \,.</math>
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| ===Eigenvectors and nodes===
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| The eigenvectors and nodes determine the profile of the phase paths, providing a pictorial interpretation of the solution to the dynamical system, as shown next.
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| [[File:Phase plane nodes.svg|thumb|300px|Classification of equilibrium points of a [[linear differential equation|linear]] [[Autonomous system (mathematics)|autonomous system]].<ref name="Jordan, Smith"/> These profiles also arise for non-linear autonomous systems in linearized approximations.]]
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| The phase plane is then first set-up by drawing straight lines representing the two eigenvectors (which represent stable situations where the system either converges towards those lines or diverges away from them). Then the phase plane is plotted by using full lines instead of direction field dashes. The signs of the eigenvalues will tell how the system's phase plane behaves:
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| *If the signs are opposite, the intersection of the eigenvectors is a [[saddle point]].
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| *If the signs are both positive, the eigenvectors represent stable situations that the system diverges away from, and the intersection is an '''unstable [[Node (autonomous system)|node]]'''.
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| *If the signs are both negative, the eigenvectors represent stable situations that the system converges towards, and the intersection is a '''stable node'''.
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| The above can be visualized by recalling the behaviour of exponential terms in differential equation solutions.
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| ===Repeated eigenvalues=== | |
| This example covers only the case for real, separate eigenvalues. Real, repeated eigenvalues require solving the coefficient matrix with an unknown vector and the first eigenvector to generate the second solution of a two-by-two system. However, if the matrix is symmetric, it is possible to use the orthogonal eigenvector to generate the second solution.
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| ===Complex eigenvalues===
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| Complex eigenvalues and eigenvectors generate solutions in the form of [[sine]]s and [[Trigonometric function|cosines]] as well as exponentials. One of the simplicities in this situation is that only one of the eigenvalues and one of the eigenvectors is needed to generate the full solution set for the system.
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| ==See also==
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| *[[Phase line (mathematics)|Phase line]], 1-dimensional case
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| *[[Phase space]], ''n''-dimensional case
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| *[[Phase portrait]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://tutorial.math.lamar.edu/classes/de/phaseplane.aspx Lamar University, Online Math Notes - ''Phase Plane'', P. Dawkins]
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| *[http://tutorial.math.lamar.edu/classes/de/systemsde.aspx Lamar University, Online Math Notes - ''Systems of Differential Equations'', P. Dawkins]
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| *[http://virtual.cvut.cz/dynlabmodules/ihtml/dynlabmodules/nonlin/node8.html Overview of the phase plane method]
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| [[Category:Nonlinear control]]
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| [[Category:Ordinary differential equations]]
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| [[fr:Méthode du plan de phase]]
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| [[pl:Metoda płaszczyzny fazowej]]
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| [[ru:Метод фазовой плоскости]]
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Myrtle Benny is how I'm known as and I feel comfortable when people use the complete title. Minnesota is exactly where he's been living for years. His spouse doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for quite a while. Bookkeeping is my profession.
Look into my website: asianas.com