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

By Tönu Puu

ISBN-10: 3662040948

ISBN-13: 9783662040942

ISBN-10: 3662040964

ISBN-13: 9783662040966

Attractors, Bifurcations, & Chaos - now in its moment version - begins with an creation to mathematical tools in smooth nonlinear dynamics and offers with differential equations. Phenomena akin to bifurcations and deterministic chaos are given enormous emphasis, either within the methodological half, and within the moment half, containing quite a few functions in economics and in nearby technological know-how. Coexistence of attractors and the multiplicity of improvement paths in nonlinear structures are valuable themes. The functions specialise in matters equivalent to company cycles, oligopoly, interregional exchange dynamics, and monetary improvement concept.

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Additional resources for Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics

Sample text

As a pair of complex conjugate numbers. 31) where the imaginary numbers in the solutions cancel out against suitably conjugate complex coefficients. 2 Linear Systems Fig. 7. Stable spiral. 21 Fig. 8. Unstable spiral. 33) Thus we see that in the case of a negative discriminant we have a spiralling movement, defined by the trigonometric functions, with a contraction or expansion of the spiral defined by the exponential depending on whether the coefficient p is negative or positive. 28), so the sign of the trace again differentiates between stability and instability.

4 we see the saddle. 20 2 Differential Equations: Ordinary In the case both eigenvalues have the same sign we deal with a node, stable if both eigenvalues are negative, unstable if both eigenvalues are positive. The former is hence the only stable case dealt with until now, whereas the latter is even more unstable than the saddle, having two unstable directions. 21) the sign of the trace makes the difference between a stable and an unstable node. Given Det>O and TrO and Tr>O we have an unstable node.

Considering small amplitudes, Ix! 125) entirely depends on the sign of £ . If it is positive then the system gains energy, if it is negative, then the system loses energy. Constant energy would be represented by closed orbits, to be exact, circles in phase space. When the system loses energy, then the spiralling trajectories always cross the circles of constant energy in an inward motion, and thus approach the origin asymptotically. This is to say that the origin is an attractor, a spiral sink when £ < O.

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

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