1. Geometric Approach to Differential Equations 7 decreasing. We can use the level curves and trajectories in the phase space (the (x, v)-space) to get information about the solutions. Even this simple nonlinear equation illustrates some of the differences between linear and nonlinear equations. For the linear harmonic oscillator with b = 0, the local behavior near the fixed point determines the behavior for all scales for the pendulum, there are periodic orbits near the origin and nonperiodic orbits farther away. Second, for a linear system with periodic orbits, all the periods are the same the period varies for the pendulum. The plots of the three solutions which are time periodic are given in Figure 4. Notice that the period changes with the amplitude. Finally, in Section 2.2 we give an algorithm for solving systems of linear differential equations with constant coefficients. On the other hand, there is no simple way to obtain the solutions of the pendulum equation the energy method gives geometric information about the solutions, but not explicit solutions. x t 4 2 0 -2 -4 16 Figure 4. Time plots of solutions with different amplitudes for the pendulum showing the variation of the period Besides linear systems and the pendulum equation, we consider a few other types of systems of nonlinear differential equation including two species of popu- lations, which either compete or are in a predator–prey relationship. See Sections 4.6 and 5.1. Finally, the Van der Pol oscillator has a unique periodic orbit with other solutions converging to this periodic orbit. See Figure 5 and Section 6.3. Basically, nonlinear differential equations with two variables cannot be much more complicated than these examples. See the discussion of the Poincar´e–Bendixson Theorem 6.1. In three and more dimensions, there can be more complicated systems with apparently “chaotic” behavior. Such motion is neither periodic nor quasiperiodic, appears to be random, but is determined by explicit nonlinear equations. An example is the Lorenz system of differential equations ˙ = −10 x + 10 y, ˙ = 28 x y xz, (1.5) ˙ = 8 3 z + xy.
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