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Homotopy analysis method constructs a homotopy with an embedding parameter, which is considered as a 'small parameter' to adjust and control the convergence of approximation series.

This method is used to remove secular terms and obtain periodic solutions of differential equations by adjusting the frequency.

Regular perturbation is used when the solution can be expanded in a power series of the small parameter.

Perturbation theory deals with finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.

Singular perturbation is applied when the perturbation parameter causes a change in the number or nature of solutions, often leading to boundary layer effects.

An adiabatic invariant is a quantity that remains constant when changes occur slowly. This concept is used in physics to study systems with slowly changing parameters.

The method of multiple scales involves using different scales for different terms in the differential equation to approximate a solution.

The RG method investigates the changes of physical systems as viewed at different scales, eliminating divergences arising from perturbation series by rescaling the system.

A semi-classical approximation method for solving differential equations with a slowly varying potential.

The averaging method simplifies equations by averaging the effects of rapidly oscillating terms over one or more periods.

This technique matches inner and outer expansions by overlapping asymptotic sequences to approximate solutions.