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The renormalization method based on the Newton-Maclaurin expansion is appliedto study the transient behavior of the solutions to the difference equations asthey tend to the steady-states. The key and also natural step is to make therenormalization equations to be continuous such that the elementary functionscan be used to describe the transient behavior of the solutions to differenceequations. As the concrete examples, we deal with the important second ordernonlinear difference equations with a small parameter. The result shows thatthe method is more natural than the multi-scale method.

Transient behavior of the solutions to the second order difference equations by the renormalization method based on Newton-Maclaurin expansion