A
system of
differential equations 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 conditions, 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 equations are often exactly solvable by
separation of variables, especially for autonomous equations. For example, the nonlinear equation :\frac{d u}{d x} = -u^2 has u=\frac{1}{x+C} as a general solution (and also the special solution u = 0, corresponding to the limit of the general solution when
C tends to infinity). The equation is nonlinear because it may be written as :\frac{du}{d x} + u^2=0 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 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. Common methods for the qualitative analysis of nonlinear ordinary differential equations include: • Examination of any
conserved quantities, especially in
Hamiltonian systems • 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 methods (can be applied to algebraic equations too) • Existence of solutions of Finite-Duration, which can happen under specific conditions for some non-linear ordinary differential equations.
Partial differential equations 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). 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 equation(s) is solvable. Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use
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
characteristics and using the methods outlined above for ordinary differential equations.
Pendula A classic, extensively studied nonlinear problem is the dynamics of a frictionless
pendulum under the influence of
gravity. Using
Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the
dimensionless nonlinear equation :\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0 where gravity points "downwards" and \theta 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 d\theta/dt as an
integrating factor, which would eventually yield :\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1 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 C_0 = 2). 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 expansions. For example, the linearization at \theta = 0, called the small angle approximation, is :\frac{d^2 \theta}{d t^2} + \theta = 0 since \sin(\theta) \approx \theta for \theta \approx 0. This is a
simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at \theta = \pi, corresponding to the pendulum being straight up: :\frac{d^2 \theta}{d t^2} + \pi - \theta = 0 since \sin(\theta) \approx \pi - \theta for \theta \approx \pi. The solution to this problem involves
hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that |\theta| 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 \theta = \pi/2, around which \sin(\theta) \approx 1: :\frac{d^2 \theta}{d t^2} + 1 = 0. 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 portraits and approximate periods. ==Types of nonlinear dynamic behaviors==