# T r t r e c e c 2 1 2 2 1 1 ξ ξ x case 1 nodal sink

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t r t r e c e c 2 1 ) 2 ( 2 ) 1 ( 1 ξ ξ x + =

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Case 1: Nodal Sink (2 of 3) If the solution starts at an initial point on the line through ξ (1) , then c 2 = 0 and the solution remains on this line for all t . Similarly if the initial point is on the line through ξ (2) . The solution can be rewritten as Since r 1 - r 2 < 0, for c 2 0 the term c 1 ξ (1) e ( r 1 - r 2) t is negligible compared to c 2 ξ (2) , for large t . Thus all solutions are tangent to ξ (2) at the critical point x = 0 except for solutions that start exactly on the line through ξ (1) . This type of critical point is called a node or nodal sink . ( 29 [ ] ) 2 ( 2 ) 1 ( 1 ) 2 ( 2 ) 1 ( 1 2 1 2 2 1 ξ ξ ξ ξ x c e c e e c e c t r r t r t r t r + = + = -
Case 1: Nodal Source (3 of 3) The phase portrait along with several graphs of x 1 versus t are given below. The behavior of x 2 versus t is similar. If 0 < r 2 < r 1 , then the trajectories will have the same pattern as in figure (a) below, but the direction will be away from the critical point at the origin. In this case the critical point is again called a node or a nodal source .

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Case 2: Real Eigenvalues of Opposite Sign (1 of 3) Suppose now that r 1 > 0 and r 2 < 0, with general solution and corresponding eigenvectors ξ (1) and ξ (2) as shown below. If the solution starts at an initial point on the line through ξ (1) , then c 2 = 0 and the solution remains on this line for all t . Also, since r 1 > 0, it follows that || x || as t . Similarly if the initial point is on the line through ξ (2) , then || x || 0 as t since r 2 < 0. Solutions starting at other initial points have trajectories as shown. , 2 1 ) 2 ( 2 ) 1 ( 1 t r t r e c e c ξ ξ x + =
Case 2: Saddle Point (2 of 3) For our general solution the positive exponential term is dominant for large t , so all solutions approach infinity asymptotic to the line determined by the eigenvector ξ (1) corresponding to r 1 > 0. The only solutions that approach the critical point at the origin are those that start on the line determined by ξ (2) . This type of critical point is called a saddle point . , 0 , 0 , 2 1 ) 2 ( 2 ) 1 ( 1 2 1 < + = r r e c e c t r t r ξ ξ x

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Case 2: Graphs of x 1 versus t (3 of 3) The phase portrait along with several graphs of x 1 versus t are given below. For certain initial conditions, the positive exponential term is absent from the solution, so x 1 0 as t . For all other initial conditions the positive exponential term eventually dominates and causes x 1 to become unbounded. The behavior of x 2 versus t is similar.
Case 3: Equal Eigenvalues (1 of 5) Suppose now that r 1 = r 2 = r . We consider the case in which the repeated eigenvalue r is negative. If r is positive, then the trajectories are similar but direction of motion is reversed. There are two subcases, depending on whether r has two linearly independent eigenvectors or only one.

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