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This editorial provides a brief review of some concepts related to the subject of the papers published in this special issue devoted to the onset of nonlinear dynamics in systems of the applied sciences. Nonlinear dynamics is currently an active and fashionable discipline that is having a profound effect on a wide variety of fields, including populations dynamics, physics, biology, economics, and sociology.

The origin of

However, the development of a mathematical apparatus for irregular (hyperbolic) dynamics comes from mathematicians and theoretical physicists of the Russian school; see the review paper [

Nowadays, nonlinear dynamics can be found in almost every branch of the applied science. It includes systems in which feedback, iterations, nonlinear interactions, and the general dependency of each part of the system upon the behavior of all other parts demand the use of nonlinear differential equations rather than the well-known linear differential equations, for example, Bellman equation [

The motions involved in nonlinear equations are not simply combinations of a bunch of simpler motions. Moreover the dynamics involving nonlinear (ordinary, partial, or integro) differential equations are extremely different, and the related mathematical methods and analysis are problem dependent. Numerical simulations are also carried out for supporting the results.

The qualitative analysis of nonlinear ordinary differential equations is usually performed by searching conserved quantities (an approach that is typically used in Hamiltonian systems) and/or dissipative quantities. Linearization of the equations by Taylor expansion, change of variables, bifurcation theory, and perturbation methods is the most used approaches.

Nonlinear partial differential equations are qualitatively analyzed by using change the variables, separation of variables, and integral transforms. Other methods include the examination of the characteristics curves, and scale analysis (typically in fluid and heat mechanics) that allows, in some cases, for simplifing the nonlinear Navier-Stokes equations.

The nonlinear dynamics that appear in the above mentioned equations, and in the papers of this special issues, include highly sensitive to initial conditions (chaos dynamics, see; among others, papers [

In this special issue, the tools of nonlinear dynamics have been used in attempts to better understand irregularity in diverse mathematical models of population dynamics, physics, biology, and economy. The interested reader is addressed to explore these interesting and fascinating results further. Moreover applications can refer to more research fields.

The guest editors of this special issue hope that problems discussed and investigated in the papers by the authors of this issue can inspire and motivate researchers in these fields to discover new, innovative, and novel applications in all areas of pure and applied mathematics.

The guest editors would like to express their gratitude to the reviewers for their serious evaluation of the papers submitted in this special issue. Thanks also go to the authors for their interesting papers. Carlo Bianca acknowledges the support by the FIRB project RBID08PP3J (Metodi matematici e relativi strumenti per la modellizzazione e la simulazione della formazione di tumori, competizione con il sistema immunitario, e conseguenti suggerimenti terapeutici).