Algebraic Methods in Nonlinear Perturbation Theory

Abstract

Algebraic methods in noolinear penurbation theory I Bogaevski. A. Povzner.p.em. ~ (Applied mathematical sciences)Translation of: Algebraicheski metody v nelineinoi tcori vozmusbchenii.Includes bibliographical references and inde~.

Flatness and defect of non-linear systems: introductory theory and examples

“…We exploit here an idea due to the Russian physicist Kapitsa (Bogaevski andPovzner 1991, Landau andLifshitz 1982) for stabilizing these three systems in the neighborhood of quite arbitrary positions and trajectories, and in particular positions which are not equilibrium points. This idea is closely related to a curiosity of classical mechanics that a double inverted pendulum (Stephenson 1908), and even the N linked pendulums which are inverted and balanced on top of one another (Acheson 1993), can be stabilized in the same way.…”

“…More precisely, we develop on three examples an idea due to the Russian physicist Kapitsa (Bogaevski and Povzner 1991, Landau and Lifshitz 1982, Sagdeev et al 1988. He considers the motion of a particle in a highly oscillating field and proposes a method for deriving the equations of the averaged motion and potential.…”

Feedback Classification of Nonlinear Single-Input Control Systems with Controllable Linearization: Normal Forms, Canonical Forms, and Invariants

“…Consider the variable length pendulum of Bressan and Rampazzo [5] (see also [2] and [8]). We denote by ξ 1 the length of the pendulum, by ξ 2 its velocity, by ξ 3 the angle with respect to the horizontal, and by ξ 4 the angular velocity.…”

“…Indeed, the function x 2 4 x1 g 2 −x 2 3 starts with third order terms, which corresponds to the fact that the invariants a [2]j,i+2 vanish for any 1 ≤ j ≤ 2 and any 0 ≤ i ≤ 2 − j. The only nonzero component of f [3] is f …”

Dynamical reduction of discrete systems based on the renormalization-group method

“…There are several methods for the dynamical reduction such as the multiple-scale methods (including the reductive perturbation method [2,3]), the average methods (including Whitham method [4]), the method of normal forms [5] and so on. We are usually interested in asymptotic behavior of the system after a long time.…”