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

By Tönu Puu

The current ebook is dependent upon a variety of versions of my prior booklet "Nonlinear fiscal Dynamics", first released in 1989 within the Springer sequence "Lecture Notes in Economics and Mathematical Systems", and republished in 3 extra, successively revised and elevated variants, as a Springer monograph, in 1991, 1993, and 1997, and in a Russian translation as "Nelineynaia Economicheskaia Dinamica". the 1st 3 variants have been inquisitive about purposes. The final used to be fluctuate­ ent, because it additionally integrated a few chapters with mathematical historical past mate­ rial -ordinary differential equations and iterated maps -so as to make the ebook self-contained and appropriate as a textbook for economics scholars of dynamical platforms. To an identical pedagogical goal, the variety of illus­ trations have been extended. The publication released in 2000, with the name "A ttractors, Bifurcations, and Chaos -Nonlinear Phenomena in Economics", was once lots replaced, that the writer felt it moderate to provide it a brand new name. there have been new math­ ematics chapters -on partial differential equations, and on bifurcations and disaster idea -thus making the mathematical heritage fabric particularly entire. the writer is worked up that this new publication did very well, yet he most popular to rewrite it, instead of having only a new print run. fabric, stemming from the 1st models, was once greater than ten years previous, whereas nonlinear dynamics has been a quick constructing box, so a few analyses appeared particularly out of date and pedestrian. the required revision grew to become out to be fairly substantial.

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

Example text

They, however, do not spiral towards a fixed point, but towards the limit cycle. The limit cycle thus is a new sort of attractor present in two dimensions along with the fixed points. Unlike the case of a centre with a family of concentric orbits, the limit cycle is a structurally stable object in the phase space. Accordingly, it too is admitted along with the various fixed points in structurally stable systems according to Peixoto's characterisation theorem. There is a powerful tool in connection with the limit cycles, which normally goes under the name of the Poincare-Bendixon theorem.

102) by x, .... e. 1 d ( X. 2 -- 2 dt +x 2 ) =£ (1 -x 2) x. 125) For an oscillator this nonnegative sum of squares represents total energy, kinetic plus potential. As, unlike the case of the undamped pendulum, the system is not conservative, energy will change during the course of time. 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.

106) for the nonlinear friction term. 109) The differential equation we have solved is actually a partial differential equation with respect to fast time, and though the arbitrary coefficients of the sine and cosine terms are constant with respect to fast time, they may depend on slow time, which has been taken in consideration in writing the solution. We will next use the second equation for determining these slowly varying coefficients, though we will not even attempt to solve that equation itself.

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