As well-known, asymptotic behavior of solutions of a system of ordinary differential equations in a small neighborhood of an equilibrium is determined in general by its linear part. The first Lyapunov method provides us with a very powerful tool for constructing a complete family of solutions of that system of equations in a form of exponential series if the linear cut of the system has only eigen-values with negative real parts. What can happen in a very degenerate case when all the eigen-values of the linearization are equal to zero? We will call such a system a strongly non-linear one. It seems that by dealing with those systems we confine ourselves to a very particular case. But it turns out that application of a combination of the center manifold theorem with Poincare normal form technique (an elementary explanation what that method means will be given during the lecture) allows ones to reduce any so-called critical case of stability to the above situation. Critical case of stability means that the linearization of the system under consideration has at least one eigen-value with zero real part. That means that we can try to cut this strongly non-linear system up to certain non-linear terms so that the truncation will have an ‘elementary’ particular solution like an exponential one in the linear case. Similarly to classic Lyapunov theory we can build up that particular solution to a particular solution of the whole system in a form of generalized power series with respect to the independent variable.
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