hw3(2) - Homework assignment 3 Due date: Monday April 23,...

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Homework assignment 3 Due date: Monday April 23, 2007. 1. Find the matrix exponentials of the following matrices (Don’t use software for your calcula- tions! You can use only use it to verify your results.): p 0 1 1 0 P , 0 1 1 1 0 1 1 1 0 , 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 , 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 , 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 . 2. Give an example of two 2 × 2 matrices A and B such that e A + B n = e A e B . 3. Determine stability of the zero solution of the following equations: ¨ x + ˙ x + sin x = 0 . ¨ x + ˙ x cos x + sin x = 0 . Note that these are problems 12 . 5 # 13 and # 14 from our text. Ignore the hint given there, and instead try the following function: V ( x,y ) = 2 sin 2 ± x 2 ² + y 2 2 . 4. Show that the competition model ˙ x = x ( ax by + r 1 ) ˙ y = y ( cx dy + r 2 ) , where all parameters a,b,c,d,r 1 and r 2 are positive, does not have a non-constant periodic solution in the region D : D = { ( x,y ) | x > 0 ,y > 0 } . Hint: Apply Dulac’s criterion with appropriately chosen positive function
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