By Moshe Gitterman
Pendulum is the best nonlinear process, which, besides the fact that, offers the ability for the outline of alternative phenomena in Nature that take place in physics, chemistry, biology, medication, communications, economics and sociology. The chaotic habit of pendulum is generally linked to the random strength performing on a pendulum (Brownian motion). one other kind of chaotic movement (deterministic chaos) happens in nonlinear platforms with in simple terms few levels of freedom. This booklet provides a entire description of those phenomena happening in underdamped and overdamped pendula topic to additive and multiplicative periodic and random forces. No initial wisdom, corresponding to complicated mathematical or numerical equipment, is needed from a reader except undergraduate classes in mathematical physics. a large team of researchers, in addition to scholars and academics will, hence, make the most of this definitive booklet on nonlinear dynamics.
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Extra info for The Chaotic Pendulum
1. Reprinted with permission from , Copyright (1999), American Institute of Physics. A different intermittent (or tangent bifurcation) transition to chaos was found  from the numerical and analog solutions of Eq. 47. 875 mA), such that for A < AC , one obtains stable oscillations having the driving frequency. For A slightly larger than AC , the system enters the chaotic regime via an intermittent transition as A is decreased. At this transition, dφ/dt becomes unstable and periodic oscillations occur, randomly interrupted by bursts of dφ/dt.
In summary, the regular orbits appear as points of a smooth area whereas the chaotic trajectories appear as a platter of points filling a certain area. 62) In a chaotic regime, λ is positive, whereas in the regular regime, λ is negative or zero, implying that the initial separation either diminishes or remain constant. For a system of first-order differential equations, there are several Lyapunov exponents, and we are interested in the largest of these. 4 Correlation function The analysis of the autocorrelation function of a trajectory shows the difference between regular and chaotic regimes.
8, we show  the efficiency of the four methods of detecting chaos described above, using Eq. 59) as an example. A comparison between Figs. 8 shows that although some methods (power spectrum) are ambiguous, one can see a clear distinction between regular (Fig. 7) and chaotic (Fig. 8) behavior. August 4, 2010 14:36 WSPC/Book Trim Size for 9in x 6in Pendulum Equations master 23 Fig. 04875. Reprinted with permission from , Copyright (1994), Institute of Physics. 6 Period doubling and intermittency Starting from the periodic regime, the change of some parameters in the dynamic equations may move the system into the chaotic regime .
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