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Economics Dynamics Problems 192

# Economics Dynamics Problems 192 - r = β i and s = − β...

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176 Economic Dynamics Figure 4.26. in polar coordinates of the original system. Since β > 0 then θ decreases over time, and so the motion is clockwise . Furthermore, as t → ∞ then either R 0 if α < 0 or R → ∞ if α > 0. Consequently, the trajectories spiral either towards the origin or away from the origin depending on the value of α . The two possibilities are illustrated in figure 4.26. The critical point in such situations is called a spiral point . Case 5 (Complex roots, α = 0 and β > 0 ) In this case we assume the roots λ = r and λ = s are complex conjugate with
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Unformatted text preview: r = β i and s = − β i (i.e. α = 0). In line with the analysis in case 4, this means ± ˙ x ˙ y ² = ± β − β ²± x y ² resulting in ˙ R = 0 and ˙ θ = − β , giving R = c and θ = − β t + θ , where c and θ are constants. This means that the trajectories are closed curves (circles or ellipses) with centre at the origin. If β > 0 the movement is clockwise while if β < 0 the movement is anticlockwise. A complete circuit around the origin denotes the phase...
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