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To draw the phase plane, we must calculate and plot the nullclines, fill in enough direction-field arrows to see how solutions move through the phase plane, identify steady states and determine their stability graphically (if possible) and plot a few example solution curves.
Dfield & pplane are programs designed for phase plane analysis of differential equations. Dfield is used on first order differential equations of the form x´ = f(t,x), while pplane is used for a system of differential equations of the form x´ = f(x,y), y´ = g(x,y).
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1. Non-linear planar systems. With LCC systems detailed, we now move on to the general non-linear planar system1. x0 = f(x; y); y0 = g(x; y) and, with x = (x; y) and F = (f; g), the vector form. (S) x0 = F(x): The path of a solution in the phase plane is called a solution curve or an orbit.
(x1-x2)-plane is called state plane or phase plane. Using vector notation x˙ =f(x), (7) where f(x)=(f1(x),f2(x))we consider f(x)as a vector field on the state plane. The family of all trajectories is called the phase portrait of the system (6).
The Phase Plane. The sort of system for which we will be trying to sketch solutions can be written in the form. x = ax + by. ⇔ y x + dy = c. where a, b, c, d are constants. x = Ax, where. a b . = , (1) d. solution of this system has the form (we write it two ways) x(t) . x(t) = , y(t) = x(t) = y(t). (2)
If the space of unknown functions of t is not a line or a plane then we talk about the phase space, a term that also encompasses the notions of phase line and phase plane. The phase portrait of the system x′ = f(x, y), y′ = together with several solutions of the given system. g(x, y) is the phase plane. Example of phase portrait for the van ...
