Nonlinear system

In mathematics together with science, a nonlinear system is the system in which the conform of the output is non proportional to the conform of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing turn in variables over time, maychaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Typically, the behavior of a nonlinear system is noted in mathematics by a nonlinear system of equations, which is a vintage of simultaneous equations in which the unknowns or the unknown functions in the case of differential equationsas variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of measure one. In other words, in a nonlinear system of equations, the equations to be solved cannot be or done as a reaction to a impeach as a linear combination of the unknown variables or functions thatin them. Systems can be defined as nonlinear, regardless of whether asked linear functionsin the equations. In particular, a differential equation is linear if it is for linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.

As nonlinear dynamical equations are unmanageable to solve, nonlinear systems are commonly approximated by linear equations linearization. This works living up to some accuracy and some range for the input values, but some interesting phenomena such(a) as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such(a) chaotic behavior may resemble random behavior, it is for in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple recast in one factor of the system score complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

Some authors use the term nonlinear science for the explore of nonlinear systems. This term is disputed by others:

Using a term like nonlinear science is like referring to the bulk of zoology as the explore of non-elephant animals.

Nonlinear differential equations

A system of differential equations is said to be nonlinear whether it is not a system of linear equations. Problems involving nonlinear differential equations are extremely diverse, and methods of written or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.

One of the greatest difficulties of nonlinear problems is that it is not broadly possible to companies so-called solutions into new solutions. In linear problems, for example, a vintage of linearly independent solutions can be used to develope general solutions through the superposition principle. A framework of this is one-dimensional heat transport with Dirichlet boundary conditions, the sum of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.

First cut ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation

has as a general solution and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity. The equation is nonlinear because it may be written as

and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u^{2 term were replaced with u, the problem would be linear the exponential decay problem.}

Second and higher formation ordinary differential equations more generally, systems of nonlinear equations rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.

Common methods for the qualitative analysis of nonlinear ordinary differential equations include:

The nearly 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 even linear. Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equations is solvable.

Another common though less mathematical tactic, often seen in fluid and heat mechanics, is to usage scale analysis to simplify a general, natural equation in aspecific boundary value problem. For example, the very nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis enable conditions under which the flow is laminar and one dimensional and also yields the simplified equation.

Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.

A classic, extensively studied nonlinear problem is the dynamics of a frictionless pendulum under the influence of gravity. Using Lagrangian mechanics, it may be present that the motion of a pendulum can be specified by the dimensionless nonlinear equation

where gravity points "downwards" and is the angle the pendulum forms with its rest position, as presented in the figure at right. One approach to "solving" this equation is to use as an integrating factor, which would eventually yield

which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral nonelementary unless .

Another way to approach the problem is to linearize all nonlinearities the sine function term in this case at the various points of interest through Taylor expansions. For example, the linearization at , called the small angle approximation, is

since for . This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at , corresponding to the pendulum being straight up:

since for . The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that will normally grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.

One more interesting linearization is possible around , around which :

This corresponds to a free fall problem. A very useful qualitative conception of the pendulum's dynamics may be obtained by piecing together such(a) linearizations, as seen in the figure at right. Other techniques may be used to find exact phase portraits and approximate periods.