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Introduction to Nonlinear Dynamics

In mathematics, a nonlinear system is one which does not satisfy the superposition principle, or one whose outcome is not directly proportional to its input; a linear system fulfils these conditions. In another word, a nonlinear system is any trouble where the variable(s) to be solved cannot be written as a linear combination of independent components. The non homogeneous system, that is linear apart from the presence of a function of the independent variables, is non-linear according to a strict definition; however such systems are generally studied alongside linear systems, as they can be transformed to a linear system of multiple variables.

In mathematics, a linear function (or map) *f(x)* is one that satisfies both of the following properties:

additivity, *f(x + y)\ = f(x)\ + f(y)*;

homogeneity, *f(αx) = αf(x)*.

(Additivity implies homogeneity for any rational α, and, for continuous functions, for any of real α. For complex α, the homogeneity does not follow from additivity; for illustration, an anti linear map is additive but not homogeneous.) The situations of additivity and homogeneity are frequently combined in the superposition principle.

*f(αx + βy) = αf(x) + βf(y)*,

The equation can be written as:

*f(x) = C*,

is known as linear if *f(x)* is a linear map and otherwise non-linear. The equation is known as homogeneous if *C = 0*.

The definition *f(x) = C* is very common in which x can be any sensible mathematical object (vector, number, function, etc.), and the function *f(x)* can literally be any mapping, involving integration or differentiation with related constraints (like boundary values). If *f(x)* consists of differentiation of x, then the outcome will be a differential equation.

Nonlinear algebraic equations that are also known as polynomial equation, and are defined by equating polynomials to zero. For illustration,

x^2 + x - 1 = 0 .

For a single polynomial equation, the root-finding algorithms can be used to find solutions to the equation (that is, sets of values for the variables which satisfy the equation). Though, systems of algebraic equations are more complex; their study is one motivation for the field of algebraic geometry, a hard branch of modern mathematics. It is still difficult to decide if a given algebraic system has complex solutions. Nevertheless, in the case of the systems with a finite number of complex answers, these systems of polynomial equations are now well understood and efficient techniques exist for solving them.

Nonlinear differential equations

A system of differential equation is said to be nonlinear if it is not a linear system. The problems including nonlinear differential equations are extremely diverse, and techniques of solution or analysis are problem dependent. Illustrations of nonlinear differential equations are the Navier-Stokes equations in fluid dynamics, Lotka-Volterra equations in biology, and Black-Scholes PDE in the finance.

One of the utmost difficulties of nonlinear problems is that it is not normally possible to combine known solutions into new solutions. In linear problems, for illustration, a family of linearly independent solutions can be used to construct common solutions through the superposition principle. A good illustration of this is one-dimensional heat transport with Dirichlet boundary conditions, the answer of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes the solution very flexible. It is frequently possible to find some very particular solutions to nonlinear equations; though the lack of a superposition principle avoids the construction of new solutions.

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