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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.
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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.
We use the term phase portrait to mean the graphs of enough trajecto ries to give a good sense of all the solutions to the system (1) 1. Critical Points. Definition. A critical point is a point where the derivatives are 0. There fore a point (x0, y0) is a critical point of the system (1) if. x x0 .
(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).
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 ...
Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, x˙ = f(x,y), y˙ = g(x,y). This system has an equilibrium point (or fixed point or critical point or singular point) (x 0,y 0) when f(x 0,y 0) = g(x 0,y 0) = 0. We can illustrate the behaviour of the ...
