Editing Nonlinear system (section)

==Nonlinear differential equations==
A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[system of linear equations]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution 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 generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary condition]]s, the solution 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.

===Ordinary differential equations===
First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation
:<math>\frac{d u}{d x} = -u^2</math>

has <math>u=\frac{1}{x+C}</math> as a general solution (and also the special solution <math>u = 0,</math> corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as
:<math>\frac{du}{d x} + u^2=0</math>

and the left-hand side of the equation is not a linear function of <math>u</math> and its derivatives. Note that if the <math>u^2</math> term were replaced with <math>u</math>, the problem would be linear (the [[exponential decay]] problem).

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.

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

*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities
*Linearization via [[Taylor expansion]]
*Change of variables into something easier to study
*[[Bifurcation theory]]
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)
*Existence of solutions of Finite-Duration,<ref>{{cite book |author = Vardia T. Haimo |title = 1985 24th IEEE Conference on Decision and Control |chapter = Finite Time Differential Equations |year = 1985 |pages = 1729–1733 |doi = 10.1109/CDC.1985.268832 |s2cid = 45426376 |chapter-url=https://ieeexplore.ieee.org/document/4048613}}</ref> which can happen under specific conditions for some non-linear ordinary differential equations.

===Partial differential equations===
{{main|Nonlinear partial differential equation}}
{{See also|List of nonlinear partial differential equations}}
The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.

Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[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 provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.

Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.

===Pendula===
{{Main|Pendulum (mathematics)}}

[[File:PendulumLayout.svg|thumb|Illustration of a pendulum|right|200px]]
[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum|right|200px]]

A classic, extensively studied nonlinear problem is the dynamics of a frictionless [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation
:<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math>

where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield
:<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math>

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 <math>C_0 = 2</math>).

Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is
:<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math>

since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:
:<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math>

since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually 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 <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:
:<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>

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