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Economics Dynamics Problems 165

Economics Dynamics Problems 165 - Systems of rst-order...

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Systems of first-order differential equations 149 Mathematicians have demonstrated a number of properties for the trajectories of autonomous systems. Here we shall simply list them. (1) There is no more than one trajectory through any point in the phase plane (2) A trajectory that starts at a point that is not a fixed point will only reach a fixed point in an infinite time period (3) No trajectory can cross itself unless it is a closed curve. If it is a closed curve then the solution is a periodic one. 4.3 Vectors of forces in the phase plane We established in chapter 2, when considering single autonomous differential equations, that we could establish the direction of x when t varies from the sign of ˙ x . In the case of a system of two differential equations we can establish the direction of x from the sign of ˙ x and the direction of y from the sign of ˙ y . Such movements in x and y give us insight into the dynamics of the system around the equilibrium.
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