Difference equation

A difference equation relates values of an unknown sequence at different discrete indices. It defines discrete evolution or constrains a sequence through shifts in its index.

Recurrence relation: An equation that defines terms of a sequence using other terms in that sequence. Difference equations commonly express sequence values through recurrence relations.
Sequence: An ordered list of values indexed by integers or another set. A difference equation takes a sequence as its unknown or as its solution.
Logistic map: A nonlinear recurrence that models population change and can produce period-doubling and chaos. It is a simple difference equation whose changing parameter reveals complex dynamics.
Differential equation: An equation involving an unknown function and one or more of its derivatives. It describes change through derivatives rather than shifts between discrete indices.
Initial value problem: A problem that specifies an equation and enough initial data to determine a solution. Initial sequence values select a particular solution among those allowed by the equation.
Linear difference equation: A difference equation in which the unknown sequence and its shifts appear linearly. Linearity enables superposition and systematic solution methods.
Cobweb plot: A graphical method for tracing successive iterates of a one-dimensional discrete map. It shows how solutions of a first-order difference equation move toward or away from fixed points.
Differential-difference equation: An equation combining derivatives of unknown functions with delayed or shifted values. It mixes continuous evolution with the discrete shifts central to difference equations.
Characteristic equation: An algebraic equation whose roots help determine solutions of a linear recurrence with constant coefficients. Its roots encode the exponential modes that form solutions of many linear difference equations.
Order of a difference equation: The largest index shift appearing in a difference equation, relative to its smallest shift. The order indicates how many independent starting values a typical solution requires.

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