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

By Tönu Puu

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 similar to bifurcations and deterministic chaos are given massive emphasis, either within the methodological half, and within the moment half, containing a variety of purposes in economics and in nearby technological know-how. Coexistence of attractors and the multiplicity of improvement paths in nonlinear structures are relevant themes. The functions concentrate on concerns similar to company cycles, oligopoly, interregional alternate dynamics, and fiscal improvement thought.

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

Sample text

2 Linear Systems 23 The difference between stability and instability is made by the sign of the trace, if negative we have stability, if positive instability. The sign of the discriminant determines whether we have a node or a spiral. The saddle, by its very nature, is unstable, but as we have seen it can become a watershed between different basins of attraction if there are several attractors. We just saw that in linear systems there are not, but in general there is such coexistence of attractors.

5. Stable node. / \ I \ I \ \" " . I Fig. 6. 26) As a matter of fact these coefficients are determined by the initial conditions, but they are only two and the coefficients are four. To determine them we need the characteristic directions which provide exactly the additional constraints we need. As a rule the eigenvalues are different. 21) that they have the same sign if Det>O, and opposite signs if Det

Such an annular region can be the area enclosed by for instance two ellipses. 42). Let us delimit the area by two circles with radii rand R respectively. The lower case radius is smaller than the upper case. A simple way of checking the direction of a vector field in relation to a boundary curve is to check the sign of the projection onto the normal of the boundary curve. If it is positive the direction is outward, if it is negative it is inward. For the boundary circles we deal with, the unit normals are just the vectors (x / r,y / r) and (x / R,y / R).

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