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Nonlinear Systems Nonlinear Systems In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, Is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables. Nonlinear problems are of interest to engineers, physicists and mathematicians because most physical systems are inherently nonlinear in nature.

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Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos. Some aspects of the weather (although not the climate) are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. A nonlinear system is not random. In mathematics there are two types of systems; one is linear system and second is nonlinear system. In mathematics we call system as nonlinear if it is not linear and do not support the properties of linear system. Non linear system does not support the property of superposition principle. Output of nonlinear system is never directly proportional to input. We cannot write non linear system in form of independent components but we can convert nonlinear system into linear system. We can find solution of nonlinear systems. Let us understand it with help of an example: Assume we have an equation as x2 + x - 3 = 0. We can write it in form of function as well. Say f(x) = a where f(x) = x2 + x and a = 3, From above it is clear that it is a non linear system as it is not supporting two basic properties of linear system, which are:

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Additivity: Which states that function f(x + y) is equals to f(x) + f(y). Homogeneity: Which states that function f(ρx) then it is equals to function ρ f(x). Above equation does not follow two properties of linear system so it is a nonlinear system. We can solve above nonlinear equation by Quadratic Formula. If it is difficult to get exact output of nonlinear equation then we use graph to examine roots of nonlinear system. For this we use the property f(x) – a = 0.

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Nonlinear Systems