Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d

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1 Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d July 6, 2009

2 Today s Session

3 Today s Session A Summary of This Session:

4 Today s Session A Summary of This Session: (1) More about eigenvalues and eigenvectors of a 2 2 matrix.

5 Today s Session A Summary of This Session: (1) More about eigenvalues and eigenvectors of a 2 2 matrix. (2) 2 2 systems of differential equations (not necessarily linear)

6 Today s Session A Summary of This Session: (1) More about eigenvalues and eigenvectors of a 2 2 matrix. (2) 2 2 systems of differential equations (not necessarily linear) (3) Phase-plane method (types of nodes)

7 Today s Session A Summary of This Session: (1) More about eigenvalues and eigenvectors of a 2 2 matrix. (2) 2 2 systems of differential equations (not necessarily linear) (3) Phase-plane method (types of nodes)

8 Today s Session A Summary of This Session: (1) More about eigenvalues and eigenvectors of a 2 2 matrix. (2) 2 2 systems of differential equations (not necessarily linear) (3) Phase-plane method (types of nodes)

9 Critical points, critical point set, equilibrium point(s) Example 1: Consider the system: x = x y y = x + y 2

10 Critical points, critical point set, equilibrium point(s) Example 1: Consider the system: x = x y y = x + y 2 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set.

11 Critical points, critical point set, equilibrium point(s) Example 1: Consider the system: x = x y y = x + y 2 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, x = 0 y = 0

12 Critical points, critical point set, equilibrium point(s) Example 1: Consider the system: x = x y y = x + y 2 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, This means: x = 0 y = 0 x y = 0 x + y 2 = 0

13 Critical points, critical point set, equilibrium point(s) Example 1: Consider the system: x = x y y = x + y 2 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, This means: So the critical point is (1,1). x = 0 y = 0 x y = 0 x + y 2 = 0

14 Example 1: Case of a Spiral To find the eigenvalues, we first put the system in the form v = A v + v 0 Where ( ) ( ) x 1 1 v =, A =, and ( ) 0 v y =. Note 2 the book (in section 9.5) uses x for the vector v. The eigenvalues of A are: λ = 1 ± i. This means the node (1,1) is an unstable spiral. Graph this on pplane.

15 Example 2: Case of an Asymptotically Stable Node (Stable Spiral) Consider the system: x = x 2y + 2 y = 8x y + 1

16 Example 2: Case of an Asymptotically Stable Node (Stable Spiral) Consider the system: x = x 2y + 2 y = 8x y + 1 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set.

17 Example 2: Case of an Asymptotically Stable Node (Stable Spiral) Consider the system: x = x 2y + 2 y = 8x y + 1 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, x = 0 y = 0

18 Example 2: Case of an Asymptotically Stable Node (Stable Spiral) Consider the system: x = x 2y + 2 y = 8x y + 1 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, This means: x = 0 y = 0 x 2y = 2 8x y = 1

19 Example 2: Case of an Asymptotically Stable Node (Stable Spiral) Consider the system: x = x 2y + 2 y = 8x y + 1 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, This means: So the critical point is (0,1). x = 0 y = 0 x 2y = 2 8x y = 1

20 Example 2, cont d To find the eigenvalues, we first put the system in the form v = A v + v 0 Where ( ) ( ) x 1 2 v =, A =, and ( ) 2 v y =. The 1 eigenvalues of A are: λ = 1 ± 4i. We have an asymptotically stable spiral at (0,1). Graph this in pplane.

21 Example 3: Determine the nature of nodes for x = 2x 5y y = x + 2y

22 Example 3: Determine the nature of nodes for x = 2x 5y y = x + 2y To find the critical points, one needs to solve, simultaneously, x = 0 y = 0

23 Example 3: Determine the nature of nodes for x = 2x 5y y = x + 2y To find the critical points, one needs to solve, simultaneously, x = 0 y = 0 The critical point is the origin (0,0).

24 Example 3, cont d To find the eigenvalues, we first put the system in the form v = A v ) ( 2 5, A = 1 2 Where ( x v = y λ = ±i. Since both eigenvalues are pure imaginary numbers. We have an center at (0,0). We have periodic orbits for trajectories. ). The eigenvalues of A are:

25 Example 3, cont d Question: Use the eigenvalues and eigenvectors to solve this system.

26 Example 3, cont d Question: Use the eigenvalues and eigenvectors to solve this system. The eigenvector associated with λ 1 = i is ( 2 + i v 1 = 1 eigenvector associated with λ 2 = i is ( 2 i v 2 = 1 solution is given by: v = c1 e it v 1 + c 2 e it v 2. ). The ). The or v = c1 e it ( 2 + i 1 ) + c 2 e it ( 2 i 1 ).