Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in by Tönu Puu

By Tönu Puu

Attractors, Bifurcations, & Chaos - now in its moment version - begins with an creation to mathematical tools in glossy nonlinear dynamics and offers with differential equations. Phenomena akin to bifurcations and deterministic chaos are given substantial emphasis, either within the methodological half, and within the moment half, containing numerous purposes in economics and in nearby technological know-how. Coexistence of attractors and the multiplicity of improvement paths in nonlinear platforms are primary issues. The functions concentrate on matters resembling company cycles, oligopoly, interregional exchange dynamics, and fiscal improvement concept.

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Extra info for Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics

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As a rule the eigenvalues are different. 21) that they have the same sign if Det>O, and opposite signs if Det

6 The Saddle-Node Bifurcation We can now use the same model to illustrate another important bifurcation, the saddle-node bifurcation. 42), so that we deal with an attractive limit cycle. 44). 51). 46). 67). 68) we p = 1 . 44) we have x = y = 0, which is the known fixed point at the origin. 73) Because the signs are always reversed for x and y this in all yields four different points on the unit circle. That is: provided that k > 1, otherwise there are no more fixed points than the origin. The emergence of the new fixed points is a sign of a bifurcation.

129). 131) which can be readily solved. 133) Provided lei < 2 the discriminant is negative, and we have a spiral. This is actually the case we have been dealing with. What happens in the linearised system when the friction term becomes so large that the inequality lei < 2 is not fulfilled, is that the spiral is converted into a node. The interesting thing is the term e / 2 . If it is positive then the origin is unstable, if it is negative then the origin is stable. This corroborates what was concluded by Lyapunov's direct method, but yields no new results as anticipated.

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