Αποτελέσματα Αναζήτησης
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.
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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)
(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).
there is one idea, not mentioned in the book, that is very useful to sketching and analyzing phase planes, namely nullclines. Recall the basic setup for an autonomous system of two DEs: dx dt = f(x,y) dy dt = g(x,y) To sketch the phase plane of such a system, at each point (x0,y0)in the xy-plane, we draw a vector
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.
Phase plane: The phase plane is the diagram showing solutions on (x;y) plane. So-lutions (t;x(t);y(t)) are projected onto the (x;y) plane, so solutions move in the plane as t changes. Solution curves follow the vector eld (x0;y0) (or F). As an example, for the system x0= 0 1 1 0 x;
