Download Phase Portraits of Linear Differential Equations and more Study notes Differential Equations in PDF only on Docsity! MATH 210 Differential Equations Fall 2005 Instructor: Georgi Medvedev Phase portraits. Consider a system of linear differential equations (1) where ! is a function of time. Let "$#%&')( ( ! , and * denote a ( +-,.+ ) matrix of coefficients * 0/ 2143 (2) Then system (1) can be written as 5 * 6 (3) By 7 7 denote the eigenvalues of (2), i.e., the roots of the characteristic equation,8888 &9 7 : 9 7 8888 7 9;$ < 7 9= ># 3 We suppose that 7 and 7 are distinct. First, we consider the case when 7 @? 7 are real. ByA 'A A ! A 'A A ! we denote the eigenvectors corresponding to 7 and 7 : * 9 7 BDC EA B F#G&A B ? #GIHKJ + 3 By a change of variables L>MON , where MK'A APQI / .1 we obtain the equation for N :NR>STN (4) where SU>MWV * MK / 7 ## 7 .1 X H%Y 7 Q 7 Q 3 We rewrite (4) in the coordinate form:N 7 N N 7 N Therefore, N N ( Z\[P]_^ (5)N N ( Z\[ `a^ (6) 1 where N ( / N ( N ( 1 >M-V ( 3 A N 9=N plane is called the phase plane of (4). For each N ( , Equations (5) and (6) define a curve in the phase plane: NGQ N NG $#% N $#% FN ( This curve is called an orbit. An orbit supplied with the sense of direction consistent with (5) and (6) is called a trajectory. It is easy to see that all orbits fill out the phase plane. A geometric representation of the trajectories in the phase plane is called a phase portrait (see Figure 1). Below, we describe phase portraits generated by (4) for different 7 and 7 . Since the trajectories of (3) and (4) are related by a linear transfor- mation MON , so are the phase portraits for these systems. Suppose 7 # 7 . Note that the origin $#G #%! is always a fixed point for (4), i.e., the right-hand side of (4) is equal to . Next, we consider two pairs of special trajectories: a positive and negative N \9 semiaxis, b positive and negative N 9 semiaxis. These trajectories are called separatrices. Note that since 7 # the trajectories lying in the N 9 semiaxis are directed toward the origin. Similarly, since 7 # the trajectories lying in the N 9 semiaxis diverge from the origin. One-dimensional subspaces of the phase space, which contain separatrices called are stable linear subspace $#G #% G #% and an unstable linear subspace $#G #% $#G% 3 A defining feature of is that for any point N ( , the trajectory passing through N ( lies in (i.e., is an invariant subspace) and ^ ! N B &>#G ^ V ! N B I>#% IH KJ + To describe the remaining trajectories of (4) we divide the right and the left sides of (6) by the right and the left sides of (5) taken to the power [ ][Q` K9#" # , respectively:N N V%$ NG(N ( V%$ & 3 Thus, N &ON V%$ '" #G (7) Therefore, all trajectories (other than separatrices) lie on the hyperbolas in the phase plane. The phase portrait for (4) is presented in Figure 1a. The phase portrait for (3) is obtained from that for (4) by applying linear transformation .>MON (see Figure 1b). A fixed point of a linear system eigenvalues having different signs is called a saddle. The same term is often used to describe the corresponding phase portraits (Figure ??1). Next, suppose 7 ( 7 ) # . Then +* . As before, we first plot two pairs of 2 −4 0 4 −4 0 4 y 1 y 2 −4 0 4 −4 0 4 x 1 x 2 Figure 4: A center. If # the trajectories of (4) lie on circles in the phase plane (Figure 4a). Therefore, those of (3) lie on ellipses (Figure 4b). The fixed point at the origin is called a center. If # the trajectories the original and auxiliary systems form spirals converging to the origin (stable focus)(see Figure 5). Similarly if # , trajectories diverge from the origin (an unstable focus)(Figure 6). 5 −0.8 0 0.8 −0.8 0 0.8 y 1 y 2 −0.8 0 0.8 −0.8 0 0.8 x 1 x 2 Figure 5: A stable focus. −0.8 0 0.8 −0.8 0 0.8 y 1 y 2 −0.8 0 0.8 −0.8 0 0.8 x 1 x 2 Figure 6: An unstable focus. 6
