Chapter7diffeq - CHAPTER 7 Nonlinear Systems of...

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682 CHAPTER 7 Nonlinear Systems of Differential Equations 7.1 Nonlinear Systems ± Review of Classifications 1. 2s i n xx t y yx y t γ =+ + Dependent variables: , x y Parameter: Nonautonomous linear system Nonhomogeneous (s i n) t 2. 34 i n uu ut υ ′=− + Dependent variables: , u Parameters: none Nonautonomous linear system Nonhomogeneous (sin ) t 3. 12 21 = =—sin x x κ Dependent variables: , x x Parameter: Autonomous nonlinear 1 (sin ) x system 4. sin pq qp q t ′ = =− Dependent variables: p , q Parameters: none Nonautonomous nonlinear ( ) pq system 5. Sr S I Ir S II RI ′ = − ′ = Dependent variables: R , S , I Parameters: , r Autonomous nonlinear ( ) SI system ± Verification Review 6. = x x yy ′ = Substituting t x ye = = into the two differential equations, we get = = tt ee 7. x y ′ = ′ = − Substituting sin x t = and sin yt = into the two differential equations, we get cos cos sin sin = −= 8, 9. Use direct substitution, as in Problems 6 & 7.
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SECTION 7.1 Nonlinear Systems 683 ± A Habit to Acquire For the phase portraits of Problems 10-13 we focus on the slope information contained in the DEs, using the following general principles: Setting 0 x ′ = gives the v-nullcline of vertical slopes Setting 0 y ′ = gives the v-nullcline of vertical slopes The equilibria are located where an h-nullcline intersects a v-nullcline, i.e., where 0 x ′ = and 0 y ′ = simultaneously In the regions between nullclines the DEs tell whether trajectories move left or right (sign of x ), up or down (sign of y ) The direction picture that results shows the stability of the equilibria Note: if the trajectories circle around an equilibrium, further argument is necessary to distinguish between a center and a spiral (which could be either stable or unstable). Note: For computer-drawn trajectories, the Runge-Kutta method will be far more accurate than Euler's method at answering these questions. Note: Recall from Section 6.5 that an equilibrium with trajectories that head toward it in one direction and others that head away in another direction is a saddle . Unique trajectories ( separatrices ) head to or from a saddle and separate the behaviors. 10. (1 ) xy yx x ′ = =− Equilibria: (0,0), (1,0) v-nullcline: 0 y = h-nullclines: 0 x = and 1 x = Because some direction arrows point away from (1,0) the second equilibrium is unstable; in fact because other direction arrows point toward it, this equilibrium is more precisely a saddle. Because direction arrows circle around (0,0) the first equilibrium could be either a center or a spiral. We argue that the symmetry of the direction field for positive and negative y implies trajectories must circle rather than spiral, so the equilibrium is a center point. The trajectories and vector field confirm all of the above information; see figures. Most trajectories come from the lower right, bend around the equilibria, and leave at the upper right, except those that circle through points between (0,0) and (1,0), the equilibria, the separatrices of the saddle.
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This note was uploaded on 04/07/2008 for the course MA 232 taught by Professor Toland during the Fall '08 term at Clarkson University .

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Chapter7diffeq - CHAPTER 7 Nonlinear Systems of...

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