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

By Tönu Puu

The current publication depends on numerous variations of my prior publication "Nonlinear financial Dynamics", first released in 1989 within the Springer sequence "Lecture Notes in Economics and Mathematical Systems", and republished in 3 extra, successively revised and accelerated variations, as a Springer monograph, in 1991, 1993, and 1997, and in a Russian translation as "Nelineynaia Economicheskaia Dinamica". the 1st 3 variations have been excited by purposes. The final used to be fluctuate­ ent, because it additionally integrated a few chapters with mathematical history mate­ rial -ordinary differential equations and iterated maps -so as to make the publication self-contained and compatible as a textbook for economics scholars of dynamical structures. To an analogous pedagogical function, the variety of illus­ trations have been elevated. The publication released in 2000, with the identify "A ttractors, Bifurcations, and Chaos -Nonlinear Phenomena in Economics", used to be a lot replaced, that the writer felt it moderate to provide it a brand new identify. there have been new math­ ematics chapters -on partial differential equations, and on bifurcations and disaster thought -thus making the mathematical history fabric rather whole. the writer is excited that this new publication did very well, yet he most well liked to rewrite it, instead of having only a new print run. fabric, stemming from the 1st types, was once greater than ten years previous, whereas nonlinear dynamics has been a quick constructing box, so a few analyses regarded relatively out of date and pedestrian. the mandatory revision became out to be particularly substantial.

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Any initial deviation from it will be enlarged in the course of time. So, the unstable limit cycle would act as a watershed between the attractors at the origin and at infinity. 5. 9 Stability: Lyapunov's Method versus Linearisation We should check that the origin indeed is an attractor while the limit cycle repels, and a repellor while the limit cycle attracts. 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.

16. Closed orbits obtained by the Poincare-Lindstedt method. unlike the central part of Fig. 1 above. More about the Poincare-Lindstedt and other perturbation methods, of which we will present one more below, can be found in Stoker, Hayashi, and Jordan and Smith. 101) which is the much studied van der Pol's equation. We see that it is a harmonic oscillator which includes a friction term. , 48 2 Differential Equations: Ordinary antidamping for small. Accordingly, close to the unstable singular point at the origin of phase space there is a region where the system is supplied with energy, whereas it loses energy further off in phase space.

Even though our purpose is not computational, it is instructive to follow the details. 47) which has the variables separated. 49) Of course K is a constant of integration, and it is entered by its logarithm and with the minus signs for convenience. 51) 34 2 Differential Equations: Ordinary Fig. 13. Approach to limit cycle. This is the well known solution to the logistic equation. 52) where ¢J is another integration constant, an arbitrary phase lead. 52) indeed corroborate what we inferred from general considerations about the phenomena in the phase diagram.

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