Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios
José Roberto Castilho Piqueira, Elbert E. N. Macau, Celso Grebogi · Mathematical Problems in Engineering · 2009
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Approximation methods and Lyapunov stability theorems elegantly solved the dynamical behavior analysis problems and gave hints about how to synthesize systems with some desired properties.At that time, bifurcation and chaos were not present in engineers' daily life.Chaotic dynamics was discovered at the end of the nineteenth century, by Henry Poincaré as a result from nonlinear phenomena subjected to qualitative changes in their behavior, provoked by bifurcations due to parameter variations.By studying the mathematical model of the circular restricted three-body problem, he was able to glimpse the chaotic motion that appears in complicated and apparently unpredictable trajectories that were close to periodic orbits, but spread fully in bounded regions of the phase space.Analyzing this motion, he concluded that ". . . it may happen that small differences in the initial conditions produce very great ones in the final phenomena.A small error in the former will produce an enormous error in the latter.Prediction becomes impossible, and we have the fortuitous phenomenon."In the last couple of decades of the twentieth century, the profound impact of chaos in science, engineering, and medicine was acknowledged.Fundamental experiments performed