Economics Dynamics Problems 163

Economics Dynamics Problems 163 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 147 It should be quite clear from these examples that independent homogeneous linear equation systems have a fixed point at the origin. Also, there is only the one fixed point. Having established that such a system has a fixed point, an equilibrium point, the next step is to establish whether such a point is stable or unstable. A trajectory that seems to approach a fixed point would indicate that the system was stable while one which moved away from a fixed point would indicate that the system was unstable. However, we need to be more precise about what we mean when we say ‘a fixed point ( x ∗ , y ∗ ) is stable or unstable’. A fixed point ( x ∗ , y ∗ ) which satisfies the condition f ( x , y ) = 0 and g ( x , y ) = is stable or attracting if, given some starting value ( x , y ) ‘close to’ ( x ∗ , y ∗ ), i.e., within some distance δ , the trajectory stays close to the fixed point, i.e., within some distance ε > δ . It is clear that this definition requires some measure of ‘distance’.....
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