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Phase plane plotter. dx/dt=. dy/dt=. The direction field solver knows about trigonometric, logarithmic and exponential functions, but multiplication and evaluation must be entered explicitly (2*x and sin (x), not 2x and sin x). The Display:
- Direction Field Plotter
Licensing: This web page is provided in hopes that it will...
- Direction Field Plotter
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.
Phase Plane analysis. Linearization. Aleksei Tepljakov, Ph.D. November 25, 2015. Recall: State Space Representation of Dynamic Systems. Suppose that a system has p inputs ui(t), i = 1, 2, . . . , p, and q outputs yk(t), k = 1, 2, . . . , q, and there are n system states that make up a state variable vector x = x1 x2 · · · xn T.
A quick guide to sketching phase planes. Section 5.2 of the text discusses equilibrium points and analysis of the phase plane. However, there is one idea, not mentioned in the book, that is very useful to sketching and analyzing phase planes, namely nullclines.
Recall that the phase plane is a plane with coordinates x and y, and a trajectory is obtained by plotting all the points (x(t);y(t)) of some solution to the system. There are two critical facts about trajectories in the phase plane.
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. See Figure 1 for an illustration of this procedure.
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)
