23 a sketch of several tangent vectors for example 23

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Fig. 2.3. A sketch of several tangent vectors for Example 2.3. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x(t) y(t) Direction Field Fig. 2.4. Direction field for Example 2.3. Of course, one can superimpose the orbits on the direction field. This is shown in Figure 2.5. Are these the patterns you saw in Figure 2.4? In this example we see all orbits “flow” towards the origin, or equilibrium point. Again, this is an example of what is called a stable node or a sink . (Imagine what happens to the water in a sink when the drain is unplugged.) Example 2.5. Saddle Consider the system x = x
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2.2 Equilibrium Solutions and Nearby Behaviors 31 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x(t) y(t) Fig. 2.5. Phase portrait for Example 2.3. y = y. (2.19) This is another uncoupled system. The solutions are again simply gotten by integration. We have that x ( t ) = c 1 e t and y ( t ) = c 2 e t . Here we have that x decays as t gets large and y increases as t gets large. In particular, if one picks initial conditions with c 2 = 0 , then orbits follow the x -axis towards the origin. For initial points with c 1 = 0 , orbits originating on the y -axis will flow away from the origin. Of course, in these cases the origin is an equilibrium point and once at equilibrium, one remains there. In fact, there is only one line on which to pick initial conditions such that the orbit leads towards the equilibrium point. No matter how small c 2 is, sooner, or later, the exponential growth term will dominate the solution. One can see this behavior in Figure 2.6. Similar to the first example, we can look at a variety of plots. These are given by Figures 2.6-2.7. The orbits can be obtained from the system as dy dx = dy/dt dx/dt = y x . The solution is y = A x . For different values of A negationslash = 0 we obtain a family of hyperbolae. These are the same curves one might obtain for the level curves of a surface known as a saddle surface, z = xy . Thus, this type of equilibrium point is classified as a saddle point. From the phase portrait we can verify that there are many orbits that lead away from the origin (equilibrium point), but there is one line of initial conditions that leads to the origin and that is the x -axis. In this case, the line of initial conditions is given by the x -axis.
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32 2 Systems of Differential Equations 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -5 0 5 10 15 20 25 30 t x(t),y(t) x(t) y(t) Fig. 2.6. Plots of solutions of Example 2.5 for several initial conditions. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x(t) y(t) Fig. 2.7. Phase portrait for Example 2.5, a saddle. Example 2.6. Unstable Node (source) x = 2 x y = y. (2.20)
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2.2 Equilibrium Solutions and Nearby Behaviors 33 This example is similar to Example 2.3. The solutions are obtained by replacing t with t. The solutions, orbits and direction fields can be seen in Figures 2.8-2.9. This is once again a node, but all orbits lead away from the equilibrium point. It is called an unstable node or a source . 0
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