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In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. [1] [2] Nonlinear problems are of interest to engineers, biologists, [3] [4] [5] physicists, [6] [7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature. [8]
An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory [5] [6] [7] Chrystal's equation: 1 + + + = Generalization of Clairaut's equation with a singular solution [8] Clairaut's equation: 1
An example would be petroleum product transport given a selection or combination of pipeline, rail tanker, road tanker, river barge, or coastal tankship. Owing to economic batch size the cost functions may have discontinuities in addition to smooth changes.
Other examples of nonlinear functions include exponential functions, logarithmic functions, trigonometric functions, power functions, Gaussian function, and Lorentz distributions. Some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear.
An example of a nonlinear control system is a thermostat-controlled heating system. A building heating system such as a furnace has a nonlinear response to changes in temperature; it is either "on" or "off", it does not have the fine control in response to temperature differences that a proportional (linear) device would have.
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations.
Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation
The following formula shows that a 4-ary function is bent when its nonlinearity is 6: = = In the mathematical field of combinatorics, a bent function is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables.