Theory Of Integral Manifolds Research Articles

Applications of integral manifolds to power system studies

This paper gives an introduction to integral manifold theory as applied to reduced order modeling of power systems. A multimachine example of the elimination of a fast voltage control system is given.

Application of sign-constant functions to the theory of integral manifolds

Application of sign-constant functions to the theory of integral manifolds

Oscillations of nonlinear feedback systems which contain tightly coupled subsystems in cascade

We formulate and prove a theorem which gives a rigorous theoretical justification for the use of describing functions to predict the existence of limit cycles in a multiple nonlinear feedback system and to predict the stability properties of these limit cycles. Our approach uses the classical sinusoidal-input describing function and the theory of integral manifolds. We demonstrate the applicability of our result by means of two specific examples.

Qualitative analysis of oscillations in nonlinear control systems: A describing function approach

We present a rigorous analysis of the stability of oscillations in a wide class of nonlinear control systems with numerator dynamics. The analysis employs the classical sinusoidual-input describing function, elementary control theory, and the theory of integral manifolds. We demonstrate, by means of specific examples, how the present results can be used to obtain detailed information concerning the behavior of solutions.

Integral manifolds for non‐linear circuits

AbstractAlthough the theory of integral manifolds is well known among applied mathematicians as a powerful tool in non‐linear oscillations, it is relatively unknown, let alone applied, among circuit engineers. the purpose of this mostly tutorial paper is to illustrate the applications of integral manifolds to explain various non‐linear phenomena widely observed in non‐linear circuits. Numerous examples and graphical illustrations are included in order to present the theory with a minimum amount of mathematics.

Stability Analysis of Limit Cycles in Non Linear Feedback Systems Using Describing Functions: Improved Results

We formulate and establish a rigorous stability analysis for limit cycles for a large class of nonlinear feedback systems. Our approach uses the classical sinusoidal-input describing function and the theory of integral manifolds. The present results constitute an improvement of earlier results by the present authors. We demonstrate the usefulness of the present results by means of specific examples (in the complete version of this paper).

Stability analysis of limit cycles in nonlinear feedback systems using describing functions: Improved results

We formulate and establish a rigorous stability analysis for limit cycles for a large class of nonlinear feedback systems. Our approach uses the classical sinusoidal-input describing function and the theory of integral manifolds. The present results constitute an improvement of earlier results by the present authors. We demonstrate the usefulness of the present results by means of a specific example.

Theory of integral manifolds of singularly perturbed differential-difference equations

Theory of integral manifolds of singularly perturbed differential-difference equations

Heavens and their integral manifolds

By using the apparatus of exterior forms, a new spinorial notation and Cartan’s theory of integral manifolds, some new results concerning complex strong heavenly metrics are established. In particular, the study of a subfamily of two-variable heavens (type G-[−], D-[−], and N-[−]) is reduced to linear equations, a prolongation process related to first and second heavenly equaions is studied leading to a (presumably) infinite heirarchy of 1−forms, and finally, the symmetries of the studied structure are investigated from the point of view of its description by Pfafian forms, elucidating in this way previous results concerning Killing vectors.

A simple proof of Frobenius’s integration theorem

The connection between Stokes's Integral Theorem and the Frobenius-Cartan Integration Theorem concerning Pfaffian systems has been noted a long time. In this note, we generalize Stokes's theorem to implicit vector valued differential forms and derive from it a general Frobenius theorem concerning mappings in Banach spaces. The only difficulty in the proof arises in the need to show differentiability with respect to a parameter of solutions of a certain differential equation, but is is easily overcome. The generality of the theorem seems to be necessary for applications to the new subjects of infinite groups and of differential geometry in infinitely many dimensions. E.g., it allows us to associate a local group to any infinite-dimensional Lie algebra in a Banach space. For finite dimensional vector spaces we obtain the classical theorem with nearly minimal differentiability conditions [4]. Also for finite dimensional spaces, one might derive from it parts of the Cartan-Kahler theory of integral manifolds [3 ] for not completely integrable CI systems.