Get Nonlinear Systems: Analysis, Stability, and Control PDF

By Shankar Sastry

There has been loads of pleasure within the final ten years over the emer­ gence of recent mathematical recommendations for the research and keep an eye on of nonlinear structures: Witness the emergence of a suite of simplified instruments for the research of bifurcations, chaos, and different advanced dynamical habit and the enhance­ ment of a complete conception of geometric nonlinear keep an eye on. Coupled with this set of analytic advances has been the titanic elevate in computational strength to be had for either the simulation and visualization of nonlinear structures in addition to for the implementation in actual time of subtle, real-time nonlinear regulate legislation. therefore, technological advances havebolstered the influence of analytic advances and produced an immense number of new difficulties and purposes which are nonlinear in a necessary approach. Nonlinear controllaws were carried out for classy flight keep watch over structures on board helicopters, and vertical take offand touchdown airplane; adaptive, nonlinearcontrollaws havebeen implementedfor robotic manipulators working both singly, or in cooperation on a multi-fingered robotic hand; adaptive regulate legislation were applied forjetengines andautomotive gas injection structures, in addition to for automatic road structures and air site visitors administration platforms, to say a number of examples. Bifurcation concept has been used to give an explanation for and comprehend the onset of fiutterin the dynamics of airplane wing buildings, the onset of oscillations in nonlinear circuits, surge and stall in plane engines, voltage cave in in an influence transmission network.

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Extra resources for Nonlinear Systems: Analysis, Stability, and Control

Sample text

The value h) is called a period-doubling bifurcation point. 4 involving the population alternating between two values x*, and xz. Further, the period 2 limit cycle is stable; that is, all nonzero initial populations tend to an asymptotic pattern of alternating between the two values, x*, and xz. Another way of understanding these results is by examining the dynamics of the two-step evolution, namely, xt+2 = f(f(xt)) =: f2(xt). 3. 5. In fact, f 2 acquires its double hump character precisely when the value of the parameter h = 3.

5. 564 and so on. , lim k-oo hk+1 - hk so that the sequence of period doubling bifurcations gets closer geometrically rapidly with periodic points of all periods 2" for n = 0, 1, 2.... 570. The appearance of stable limit cycles of period 2", n = 1, 2, ... , and the corresponding destabilization of the limit cycles of periods 2n-1 are referred to as the sequence of period-doubling bifurcation. 6. 6692. , not only for the specific logistic map that we have considered, but also for other one parameter maps that are "close" to it.

However, to model the fact that a steady state prey population is required to sustain growth, this term is modified using an offset in x as (x -d/c). Admittedly, the model is oversimplified but is nevertheless ecologically meaningful. A more meaningful model is given in the exercises as a refinement of this one. Other scenarios could occur as well: for some values of the parameters, the Volterra-Lotka equations predict convergence of the solutions to a single stable equilibrium population. 13).

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