The “perturbation method” is, probably, the most powerful and most widely used method for the determination of the existence and stability of periodic solutions. It goes back to Poincaré  and is often called “Poincaré’s method”. The essence of the method can be explained easily. A one parameter family of systems of differential equations is considered and it is assumed that for a certain value of the parameter μ, say, for μ = 0 the system has a periodic solution. Under some non-criticality assumptions it can be proved that for sufficiently small |μ| the system has a periodic solution too, close to the periodic solution of the µ = 0 case. One may show also that under some non-criticality conditions the stability of the “unperturbed (μ = 0) periodic solution” is inherited by the perturbed one. If the family depends analytically on the parameter μ, then for sufficiently small |μ| the “perturbed periodic solution” can be developed into a power series with respect to μ. The coefficients of this series can be determined successively by solving systems of linear differential equations. It is to be noted that the perturbation method is applied in many situations concerning solutions of algebraic equations, difference equations, etc.; however, in this book its application to the existence and stability problem of periodic solutions of ODEs is treated only.
KeywordsPeriodic Solution Implicit Function Theorem Fundamental Matrix Relaxation Oscillation Unperturbed System
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