nagle_differential_equations_ISM_Part63

nagle_differential_equations_ISM_Part63 - Chapter 12 14....

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Chapter 12 14. (3 , 3) is an unstable saddle point; ( - 2 , - 2) is an asymptotically stable spiral point. See Fig. 12–E on page 310. 16. (0 , 0) is an unstable saddle point; ( - 4 , - 2) is an asymptotically stable spiral point. See Fig. 12–F on page 310. EXERCISES 12.4: Energy Methods 2. G ( x ) = sin x + C ; E ( x,v ) = ± 1 2 ² v 2 + sin x 4. G ( x ) = ± 1 2 ² x 2 - ± 1 24 ² x 4 + ± 1 720 ² x 6 + C ; E ( x,v ) = ± 1 2 ² v 2 + ± 1 2 ² x 2 - ± 1 24 ² x 4 + ± 1 720 ² x 6 6. G ( x ) = e x - x + C ; E ( x,v ) = ± 1 2 ² v 2 + e x - x - 1 8. See Fig. 12–G on page 311. 10. See Fig. 12–H on page 311. 12. See Fig. 12–I on page 312. 14. vh ( x,v ) = v 2 , so energy decreasing along a trajectory. See Fig. 12–A on page 312. 16. vh ( x,v ) = v 2 , so energy decreasing along a trajectory. See Fig. 12–B on page 313. 18. See Fig. 12–C on page 313. EXERCISES 12.5: Lyapunov’s Direct Method 2. Asymptotically stable 4. Stable 6. Unstable 8. Asymptotically stable 10. Stable 306
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Exercises 12.7 12. Stable 14. Stable EXERCISES 12.6: Limit Cycles and Periodic Solutions
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nagle_differential_equations_ISM_Part63 - Chapter 12 14....

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