The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). See also: List of nonlinear partial differential equations

Existence of solutions of Finite-Duration, which can happen under specific conditions for some non-linear ordinary differential equations.

Perturbation methods (can be applied to algebraic equations too).

Change of variables into something easier to study.

Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities.

Examination of any conserved quantities, especially in Hamiltonian systems.

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.Ĭommon methods for the qualitative analysis of nonlinear ordinary differential equations include: In mathematics, a linear map (or linear function) f ( x ), the problem would be linear (the exponential decay problem). Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. Some authors use the term nonlinear science for the study of nonlinear systems. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. Although such chaotic behavior may resemble random behavior, it is in fact not random.

It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.Īs nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ( linearization).

Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. 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.

Non physical phenomena definition