Asymptotic expansions of solutions of strongly nonlinear systems of differential equations. (Асимптотики решений сильно нелинейных систем дифференциальных уравнений.) (Russian) Zbl 0949.34003

Moskva: Izdatel’stvo Moskovskogo Universiteta. 244 p. (1996).
The subject of this book is related to the problem of constructing asymptotic expansions for solutions to systems of ordinary differential equations near nonhyperbolic fixed points. The most important results have been previously published by the authors [J. Appl. Math. Mech. 60, No. 1, 7–18 (1996); translation from Prikl. Mat. Mekh. 60, No. 1, 10–22 (1996; Zbl 0885.34044)]. According to the first Lyapunov method, to the eigenvalues with nonzero real parts there correspond exponential in time expansions with coefficients polynomial in time. In the book under review, attention is focused on nonexponential expansions. The basic approach is to consider the so-called semiquasihomogeneous system and the corresponding “truncated” quasihomogeneous system and to complete a “quasihomogeneous ray” solution to the truncated system up to a formal asymptotic expansion solution to the original system. Apparently, the authors are not aware of the works by A. D. Bryuno [Power geometry in algebraic and differential equations. Moskva: Nauka. Fizmatlit (1998; Zbl 0903.34001)], who proposed a universal iterative algorithm based on power geometry. So, instead of general power transformations applied at each step of the algorithm, they use a single, more particular, time-dependent linear transformation of the phase variables. This strongly restricts and complicates the presentation and the calculation methods, although it could be preferable in some cases for numerical realization. On the other hand, Bryuno does not discuss the presence of actual solutions corresponding to the formal expansions found. Truncated systems of Poincaré normal forms (see Chapter 2) denoted as “model systems” have been treated by many other authors.
Chapter 1 contains basic results. Autonomous ordinary differential equations with right-hand sides expanded in formal power series are considered and formal asymptotic expansion solutions as time tends to infinity or to a finite limit are sought. Basic definitions of semiquasihomogeneous systems based on a group-theoretic approach are given. A result on the existence of expansions in powers of the time variable \(t\) with coefficients polynomial in the logarithm of \(t\) is proved. By using the usual tools of functional analysis, it is established that to each of these expansions there corresponds an actual solution, and that for analytic systems the expansions converge in the absence of logarithms. The latter case is typical because the logarithms can appear only in initial steps of the expansion procedure under some degeneration conditions. Then, more general nonautonomous systems admitting expansion in formal series in phase variables with coefficients that are smooth in time are considered. A theorem on the existence of a multiparameter family of solutions (in the spirit of Lyapunov’s result) with nonexponential asymptotics is proved. Here, appropriate conditions are stated in terms of the impropriety measure of the truncated system (where time in the right-hand side is considered as a parameter) linearized near a ray solution. Some examples are presented. Finally, a group-theoretic approach is discussed which is based on the usage of an arbitrary one-parameter group of phase space transformations instead of particular quasihomogeneous groups.
Chapter 2 is devoted to neutral systems with only zero and purely imaginary eigenvalues. In 1, autonomous systems with right-hand sides expanded in formal Maclaurin series are considered and conditions for the existence of asymptotic solutions as time tends to infinity are given in terms of quasihomogeneous truncations of the Poincaré normal forms. Critical cases of two pairs of purely imaginary eigenvalues for general four-dimensional systems are discussed in detail. In 2, the results are generalized to systems that are periodic in time and the analogous theory is presented for systems possessing invariant tori. Here, for simplicity, it is assumed that the corresponding linearized quasiperiodic system is reducible and that some Diophantine nonresonance conditions are satisfied. In 3, the authors discuss the specific character of the problems treated in 1 which is caused by the Hamiltonian nature of the equations under investigation. In this way, well-known theorems on the instability of equilibria in Hamiltonian systems with two degrees of freedom in the presence of resonances are obtained [see also A. D. Bryuno, loc. cit. 1998].
Chapter 3 is devoted to a class of autonomous problems where the desired asymptotic expansions diverge even if the system is analytic. At first, systems having both zero eigenvalue and an eigenvalue with nonzero real part are considered. The basic approach is to establish the existence of a formal invariant manifold corresponding to all the zero eigenvalues, and then to apply the technique developed to the system reduced to this manifold. The nonanalyticity of this manifold causes the divergence of the asymptotic expansions constructed. The existence of the corresponding actual solutions is immediately justified by A. N. Kuznetsov’s theory [Funct. Anal. Appl. 23, No. 4, 308–317 (1989); translation from Funkts. Anal. Prilozh. 23, No. 4, 63–74 (1989; Zbl 0717.34004)] (the technique of Chapter 1 is also not applicable), whose main results are described. The existence conditions are given for hybrid solutions whose asymptotic expansions contain both exponents and negative powers of time. The appearance of iterated logarithms in the asymptotic expansions is also discussed. Systems that are not solved with respect to highest derivatives are considered. Here, some derivatives can be lost in the quasihomogeneous truncation which leads, generally speaking, to the divergence of the expansions. Some examples are treated, in particular, Euler-Poinsot equations describing dynamics of a heavy rigid body. Most of these topics are also discussed by A. D. Bryuno [op. cit. 1998].
Chapter 4 is more illustrative and is devoted to questions related to the inversion of the Lagrange theorem on the stability of an equilibrium. Some energy criteria for the stability and instability of an equilibrium are discussed for various classes of systems. The influence of imposing additional nonholonomic constraints is briefly considered. Then instability theorems based on constructing asymptotic solutions are presented.
Appendix 1 transfers the method developed in Chapter 1 to other dynamical systems, such as systems of ordinary differential equations with deviating argument and some class of integro-differential equations. The appropriate concepts of quasihomogeneity and semiquasihomogeneity are introduced. Then sufficient conditions are established for the existence of asymptotic solutions as time tends to infinity. However, in the case of integro-differential equations, the authors are able to detect only formal asymptotic expansions since Kuznetsov’s theory has been developed only for ordinary differential equations. Finally, in Appendix 2, the influence of the presence and the structure of ray solutions of the truncated quasihomogeneous systems on the integrability is discussed. Two theorems on the nonexistence of an additional integral polynomial and expansion in formal Maclaurin series are proved.


34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34Exx Asymptotic theory for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34K40 Neutral functional-differential equations