1. Geometric Approach to Differential Equations 5 A point x∗ is called periodic for a system of differential equations ˙ = F(x), provided that there is some T 0 such that the solution x(t) with initial condition x(0) = x∗ has x(T ) = x∗ but x(t) = x∗ for 0 t T . This value T is called the period or least period. It follows that x(t + T ) = x(t) (i.e., it repeats itself after T units of time). The set of points {x(t) : 0 t T } is called the periodic orbit. v x 4 -4 2 -2 Figure 2. Solutions for the linear harmonic oscillator in (x, v)-space. The set of curves in the (x, v)-plane determined by the solutions of the system of differential equations is an example of a phase portrait, which we use throughout this book. These portraits are a graphical (or geometric) way of understanding the solutions of the differential equations. Especially for nonlinear equations, for which we are often unable to obtain analytic solutions, the use of graphical representations of solutions is important. Besides determining the phase portrait for nonlinear equations by an energy function such as the preceding, we sometimes use geometric ideas such as the nullclines introduced in Section 4.4. Other times, we use numerical methods to draw the phase portraits. Next, we consider the case for b 0. The solutions of the characteristic equa- tion (1.3) is λ = −b ± b2 4km 2m = −c ± i μ, where c = b 2 m and μ = k m c2. (These roots are also the eigenvalues of the matrix of the system of differential equations.) The general solution of the system of equations is x(t) = e−ct [A cos(μt) + B sin(μt)] , v(t) = e−ct [− (Aμ + Bc) sin(μt) + (Bμ Ac) cos(μt)] , where A and B are arbitrary constants. Another way to understand the properties of the solutions in this case is to use the “energy function”, E(x, v) = 1 2 v2 + ω2 2 x2,
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