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by: Moshe Swift III

DifferentialEquations MATH210

Marketplace > Drexel University > Mathematics (M) > MATH210 > DifferentialEquations
Moshe Swift III
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This 6 page Class Notes was uploaded by Moshe Swift III on Wednesday September 23, 2015. The Class Notes belongs to MATH210 at Drexel University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/212305/math210-drexel-university in Mathematics (M) at Drexel University.

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Date Created: 09/23/15
MATH 210 Differential Equations Fall 2005 Instructor Georgi Medvedev Phase portraits Consider a system of linear differential equations 351 21111351 a12 521 1 352 I a211 11223521 where ct x1tcgtT is a function oftime Let 320 32 323 and A denote a 2 X 2 matrix of coef cients 0611 112 A 2 0621 0622 gt Then system 1 can be written as 5 Am 3 By Al Ag denote the eigenvalues of 2 ie the roots of the characteristic equation all A all AZ a a99a aov aoar 0 ml a22gt 11 11 1 1 e We suppose that Al and Ag are distinct First we consider the case when A1 7E A2 are real By 211 2 0111012 212 212112122112 we denote the eigenvectors corresponding to A1 and Ag A AiIhlq 0 2 0 221 By a change ofvariables e 2 Pg where P 2 211212 2 j 2 we obtain the equation for y 21 22 1 I Dye 4 where D P IAP lt 31 f gt diaqu Ag We rewrite 4 in the coordinate form 2 311 I Myra 112 I A2112 Therefore 111 2 we 5 w 2 2136 6 where 07 yP l 0 11 00 37 7 A yl 3 plane is called the phase plane of 4 For each 3 E lRQ Equations 5 and 6 de ne a curve in the phase plane 91612126 E R 2110uyz0 2 goat E 1R 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 ll 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 A1 and Ag Since the trajectories of 3 and 4 are related by a linear transfor mation a Py so are the phase portraits for these systems Suppose A1 lt 0 lt Ag Note that the origin 0 001 is always a xed point for 4 ie the righthand side of 4 is equal to 0 Next we consider two pairs of special trajectories a positive and negative yl semiaxis b positive and negative yg semiaxis These trajectories are called separatrices Note that since A1 lt 0 the trajectories lying in the yl semiaxis are directed toward the origin Similarly since Ag gt U the trajectories lying in the yg semiaxis diverge from the origin Onedimensional subspaces of the phase space which contain separatrices called are stable linear subspace 33010 I 61016 E 111 and an unstable linear subspace E UaU I 01016 E 1R A de ning feature of EquotEquot is that for any point 3 E EquotEquotquot the trajectory passing through 3 lies in EquotE 39 ie EquotEquot 39 is an invariant subspace and 33102110 I 01 gglxmi I UL 1 11 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 I u gt 0 respectively L 2 y 2 c 1117 31271393 Thus 212 I 031 u gt U 7 Therefore all trajectories other than separatrices lie on the hyperbolas in the phase plane The phase portrait for 4 is presented in Figure la The phase portrait for 3 is obtained from that for 4 by applying linear transformation a Pg see Figure lb A xed 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 771 Next suppose M lt Ag lt 0 Then Equot lRZ Equot I As before we rst plot two pairs of 08 08 v E V u yz 4 x2 G s MES o lt 0 6 8 0 08 0458 08 a yl b x1 Figure l A saddle the phase portraits for 4 a and for 3 b when A1 lt 0 lt Ag separatrices corresponding to trajectories passing through 3 iy and g 0tyg 316313 34 0 The equation for the remaining trajectories is obtained by dividing 6 by 5 taken to the power u gt U y220yi 0ltult1 The corresponding phase portraits for 4 and 5 are shown in Figure 2 The xed point at the origin is called a stable node If A1Q gt U the phase portrait is constructed in complete analogy to the previous case In this case the equilibrium at the origin is called an unstable node Figure 3 Finally we consider the case when A1 and Ag are complex conjugate A12aibgtga ib By 101 u i2 21 u in where um E lRQ we denote the corresponding eigenvectors The eigenvectors can be chosen complex conjugate Next we form a matrix P 2 The change of variables as Pg yields 311 I 0631153121 8 322 I 5211 602 9 To study 8 and 9 it is convenient to use the polar coordinates T lt15 7 2 and lt15 arctan 10 311 By differentiating 10 and using 8 and 9 the expression for 7quot in 10 we obtain 7 ylgl 3123 mquot gt iquot am 11 Similarly 92919291 2 jgt 12 1 7quot y2 x2 0 0 Figure 2 A stable node 0 8 0 8 V1 A y2 x2 v V V2 0 0 08 0 8 0 8 0 8 y1 x1 Figure 3 An unstable node 9 Figure 4 A center If a U the trajectories of 4 lie on circles in the phase plane Figure 4a Therefore those of 3 lie on ellipses Figure 4b The xed point at the origin is called a center If a lt 0 the trajectories the original and auxiliary systems form spirals converging to the origin stable focussee Figure 5 Similarly if a gt U trajectories diverge from the origin an unstable focusFigure 6 f Figure 5 A stable focus 5 3 L Figure 6 An unstable focus


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