Nonlinear science refers to the study mathematical systems and natural phenomena which are not linear. The paradigms of nonlinearity manifest that both the interests and power of nonlinear science arise in a precise way since common concepts are discovered about systems in diverse areas of mathematics and natural sciences. Such paradigms, or common concepts, provide insights into nonlinear problems within a broad array of disciplines where the understanding of the paradigms can pave the way toward the essence of nonlinearity and consequences thereof [Campbell, D. K. (1987)]. Advances in computer science, allowing for voluminous quantitative numerical simulations of nonlinear systems as well as related computer-based investigations into problems which could be inaccessible to analytic methods underscore the critical position nonlinearity holds since most systems in nature are inherently nonlinear.
A nonlinear system, to put differently, is defined as a system where the change of output is disproportional to the change in the input. Nonlinear dynamical systems which also describe the changes in variables over time may appear unpredictable, chaotic or counterintuitive, in contrast with further simpler linear systems. Behaviors of a nonlinear system are described by mathematics as a nonlinear system of equations, namely a set of simultaneous equations where the unknown functions with differential equations are manifested in the form of variables of a polynomial of degree higher than one or in the argument of a function that is not a polynomial of degree one. Nevertheless, systems can be defined as nonlinear irrespective of whether the known linear functions appear in the equations. A differential equation is linear if it is linear in terms of the unknown function and its derivatives even though nonlinear in terms of the other variables apparent in it.
A nonlinear system of equations is made up of a set of equations in several variables in such a way that one of them at least is not a linear equation. Regarding nonlinear recurrence relation, the definition is put forth as terms of a sequence as a nonlinear function of preceding terms, with some examples being the logistic map and the relations which define the various Hofstadter sequences. Nonlinear discrete models representing a wide class of nonlinear recurrence relationships include the Nonlinear Autoregressive Moving Average with eXogenous inputs model as well as the related nonlinear system identification and analysis procedures [Billings, S.A. (2013)]. Such approaches can be employed for studying an extensive class of complex nonlinear behaviors in the domains concerning time, space and frequency.
Nonlinear differential equations and problems concerning them are diverse and methods of solution or analysis are dependent on the problem. Some examples of nonlinear differential equations can be listed as the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology. One important challenge of nonlinear problems is that it would not always be possible to combine known solutions into new solutions.
Nonlinearity generally emerges through the collective behaviors of even the simplest systems. It is not sufficient to linearly or simply add the effects of the components. This means the interactions between the components bring about the emergent phenomena like solitons, chaos, fractals as well as meta- or multi-stability, to name some. The dynamics that result from the interactions may be unpredictable at high levels despite the underlying physics being deterministic while resulting in nonequilibrium pattern formation. While deterministic chaos and coherent structures are referred to as two paradigms of nonlinear science as they characterize typical aspects of nonlinear phenomena. The other two paradigms are pattern formation, competition, and selection as well as adaptation, evolution, and learning, characterized by interdisciplinary relevance [“Chapter 2-Background” National Research Council. (1997)]. The inherent interdisciplinary relevance of nonlinear science is observed in its impact on traditional sciences, mathematics, engineering as well as the social sciences, particularly economics and demographics. A proper pursuit of nonlinear science entails the blending of four separate methodological approaches. These are modeling (to improve the analytical foundation of the problem for the emerging areas of nonlinear science), experimental mathematics (the use of ingeniously conceived computer-based numerical simulations for qualitative insights into analytically intractable problems), novel analytical mathematical methods (for handling functional recursion relations, solving nonlinear PDEs or describing complex structures that arise in chaotic systems) and experimental observations with similar nonlinear phenomena in natural and man-made systems.
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