Lyapunov's first method for strongly non-linear systems
Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 1, pp. 7–18
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Lyapunov's classical First Method is developed for strongly non-linear systems. Techniques for “truncating” strongly non-linear systems that possess a well-defined group of symmetries are described. Given such a group, it is possible, under fairly general assumptions, to determine, by purely algebraic methods, particular solutions of the truncated systems with prescribed asymptotic expansions. It is shown that these solutions can be extended to solutions of the full system by using certain series. Sufficient conditions for the existence of parametric families of solutions of the full system that possess certain asymptotic properties are also derived. The theory is illustrated by a wide range of examples. A new proof is given of one of the inversions of the Lagrange-Dirichlet theorem on the stability of equilibrium. It is shown that the method developed here may also be used to construct collision trajectories in problems of celestial mechanics in real time.
Kozlov V. V., Furta S. D., Lyapunov's first method for strongly non-linear systems, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 1, pp. 7–18
On solutions with generalized power asymptotics to systems of differential equations
Mathematical Notes, 1995, vol. 58, no. 6, pp. 1286–1293
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In the paper we study methods for constructing particular solutions with nonexponential asymptotic behavior to a system of ordinary differential equations with infinitely differentiable right-hand sides. We construct the corresponding formal solutions in the form of generalized power series whose first terms are particular solutions to the so-called truncated system. We prove that these series are asymptotic expansions of real solutions to the complete system. We discuss the complex nature of the functions that are represented by these series in the analytic case.
Kozlov V. V., Furta S. D., On solutions with generalized power asymptotics to systems of differential equations, Mathematical Notes, 1995, vol. 58, no. 6, pp. 1286–1293