Economics Dynamics Problems 169

# Economics Dynamics Problems 169 - Systems of rst-order...

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Systems of first-order differential equations 153 Figure 4.8. No matter what the initial point, it will be found that each trajectory approaches the fixed point ( x , y ) = (0 , 0). In other words, the fixed point (the equilibrium point) is globally stable. Considering the vector of forces for this system captures this feature. The equilibrium solution lines are y = 3 x for ˙ x = 0 y = x 3 for ˙ y = 0 with a fixed point at the origin. To the right of the x -line we have y < 3 x or 3 x + y < 0 implying ˙ x < 0 so x is falling While to the left of the x -line we have y > 3 x or 3 x + y > 0 implying ˙ x > 0 so x is rising Similarly, to the right of the y -line we have y < x 3 or 0 < 3 y + x implying ˙ y > 0 so y is rising While to the left of the y -line we have y > x 3 or 0 > 3 y + x implying ˙ y < 0 so y is falling All this information, including the vectors of force implied by the above results,
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