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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 “

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 in all areas of science allowed us to finally understand a realistic scenario in which the chaotic behavior not only occurs in Nature, but it is also a fundamental mechanism that Nature accounts for to mediate its phenomena. Thus, the chaotic behavior can be observed, for example, in the electrical activity from biological systems, in the transition of a fluid to turbulent motion, and in the motion of the moons of the giant planets.

Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,” allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation Theory, Dynamical Systems, and Chaos started to be part of the mandatory set of tools for design engineers.

Consequently, an important interaction between Mathematics and Engineering started to gain importance and, in this scenario, the blending of three distinct methodological approaches, analytical, numerical, and experimental needs to be improved as follows.

Numerical experiments that cleverly and precisely use tailored computer-based numerical simulations to give insights into problems that are analytically intractable so far.

Analytical methods to deal with nonlinear differential equations and to geometrically describe structures that arise as a consequences of the chaotic dynamics.

High precision experiments, which involve up-to-date measurement technology and procedures to capture the chaotic behavior in specific fields.

This special edition of the

Following these lines, this special issue contains 25 papers organized as follows. Concerning an overview of modeling nonlinear dynamics

As guest editors for this special issue, we wish to thank all those who submitted manuscripts for consideration. We also would like to thank the many individuals who served as referees. We hope that these articles can motivate and foster further scientific works that will allow for a continuous and better understand of the role of the chaotic dynamics in our world.