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Nonlinear Dynamics

Nonlinear dynamics deals with problems that can be described by nonlinear differential equations. The main topics of interest are

1. How can we solve nonlinear differential equations?

2. What can be deduced about the stability of systems described by these nonlinear equations?

A nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfils these conditions. In other words, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A non-homogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables.

Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos. Some aspects of the weather (although not the climate) are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. A nonlinear system is not random.

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behaviour, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behaviour has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

It deals with the long-term qualitative behaviour of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behaviour of the system depend on its initial condition?"

Note that the chaotic behaviour of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behaviour. Chaos theory has been so surprising because chaos can be found within almost trivial systems.

Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials to zero. For example, x² + x – 1 = 0

Nonlinear recurrence relations

A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences.

Nonlinear differential equations

A system of differential equations is said to be nonlinear if it is not a linear system. 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, the Lotka–Volterra equations in biology, and the Black–Scholes PDE in finance.

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.