8 1. Geometric Approach to Differential Equations x2 x1 Figure 5. Phase portrait of the Van der Pol oscillator Trajectories originating from nearby initial conditions move apart. Such systems are said to have sensitive dependence on initial conditions. See Figure 6 where two solutions are plotted, each starting near the origin. As these solutions are followed for a longer time, they fill up a set that is a fuzzy surface within three dimensions. (The set is like a whole set of sheets stacked very close together.) Even though the equations are deterministic, the outcome is apparently random or chaotic. This type of equation is discussed in Chapter 7. The possibility of chaos for nonlinear equations is an even more fundamental difference from linear equations. Figure 6. Two views of the Lorenz attractor Finally, we highlight the differences that can occur for time plots of solutions of differential equations. Figure 7(a) shows an orbit that tends to a constant value (i.e., to a fixed point). Figure 7(b) shows a periodic orbit that repeats itself after the time increases by a fixed amount. Figure 7(c) has what is called a quasiperiodic orbit it is generated by adding together functions of two different periods so it never exactly repeats. (See Section 2.2.4 for more details.) Figure 7(d) contains a chaotic orbit Section 7.2 contains a precise definition, but notice that there is no obvious regularity to the length of time of the different types of oscillation.
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