Symmetries and Semi-invariants in the Analysis of Nonlinear Systems

The book begins by a quick review of elementary properties about analytic and meromorphic functions, differential and difference equations, and differential forms. This part is not intended to explain concepts in depth; it indicates how much it is assumed about the background of the reader, and also serves to fix notation. After this review, the Cauchy–Kovalevskaya and the Frobenius theorems are discussed with some details. The last section of the chapter defines semi-simple, normal and nilpotent square matrices, and gives some first properties of them; many other properties that belong to the linear algebra field are actually reported in the second chapter, since there they can be explained or proven with reference to the central topics in this book.

The goal of this chapter is to introduce two of the central topics in the book, i.e., symmetries and semi-invariants, for both continuous-time and discrete-time linear systems. The idea is that, being simpler to carry out computations in closed form for linear systems, this way the reader can become confident with such concepts before dealing with the unavoidable obstacles brought in by nonlinearities. The chapter is composed of two sections, the first one dealing with the concepts of centralizer and normalizer of a square matrix, and the second one dealing with first integrals and Darboux polynomials; Darboux polynomials are polynomial semi-invariants and, for linear systems, they constitute a significant subset of all the possible semi-invariants. A large portion of the first section is actually used to state and proof some important results of linear algebra that are central for the subsequent chapters.

In this chapter, the concepts of semi-invariants (an extension of first integrals) and symmetries (and orbital symmetries) are introduced for continuous-time systems, and their relations with known results in nonlinear systems theory are explored. Such two concepts are strongly related for planar systems, and their relation can be extended to higher order systems through the concept of the inverse Jacobi last multiplier. Before exploring such relations, the concepts of homogeneity and homogeneous systems (with their characteristic solutions) have been introduced. With all such a machinery available, it is possible to detail many properties of two very well known normal forms for autonomous systems: the Poincaré–Dulac and the Belitskii normal forms. Dual semi-invariants are strictly related with semi-invariants and, in turn, they are related with invariant distributions. Both symmetries and invariant distributions can be used to obtain a reduction of the order of a given system, and, in particular, the reduction based on invariant distributions is useful to study structural properties of the system, possibly endowed with inputs and outputs. The last sections of this chapter propose a brief review of other uses of the concept of symmetry; such ideas, although almost out of the scope of the book, are inherently related with the topics studied here and are therefore useful to deepen the comprehension.

The goal of this chapter is to extend, as much as possible, the results about symmetries, semi-invariants, homogeneity and normal forms, valid for continuous-time systems, to discrete-time ones. Although there are important obstructions (e.g., it seems not possible to define a concept analogous to the one of orbital symmetry), most of the material is derived without important difficulties, sometimes arriving at a result that is slightly weaker than the corresponding one in continuous-time.

After a first section dealing with the Euler–Lagrange equations, modeling mechanical systems, the chapter deals with the natural extension of the Euler–Lagrange systems constituted by the Hamiltonian ones. The Hamiltonian systems are classically defined by using the Poisson brackets, which are strictly connected with first integrals; several properties of the Poisson brackets, as their relation with changes of coordinates, are explored in detail. In view of the historical importance of the Hamiltonian systems, their symmetries and their normal forms were studied independently and have therefore names referring to researchers that studied them; e.g., a Hamiltonian symmetry of a Hamiltonian system is named a Noether symmetry and the Poincaré–Dulac normal form of a Hamiltonian system is named the Birkhoff–Gustavson form. The chapter is completed by a study about how wide is the class of the Hamiltonian systems; it is shown that every system having an inverse Jacobi last multiplier equal to one can be written as a Hamiltonian system, if the Poisson bracket is properly defined (this can be done in general with the help of the Nambu bracket).

Lie algebras are receiving increasing attention in the field of systems theory, because they can be used to represent many classes of physically motivated nonlinear systems and also switched systems. It turns out that some of the concepts studied in this book, such as the Darboux polynomials and the Poincaré–Dulac normal form, are particularly interesting when referred to Lie algebras. An important application of the concepts of this book is the computation of nonlinear superposition formulas for finite dimensional Lie algebras; such computations are based on first integrals. The last sections of this chapter, based on the exponential notation, deal with the Wei–Norman equations and their use, both for the computation of the solution of systems described by Lie algebras and for the derivation of commutation rules.

In this chapter, symmetries, homogeneity and semi-invariants are used as tools to solve the problem of determining a state immersion that renders linear (and, in general, of higher order) a given nonlinear system. The problem is dealt with both for continuous-time and discrete-time systems. The proposed technique allows, under some assumptions, also the computation of a linearizing diffeomorphism (that does not alter the dimension of the state vector), when it exists. The last two sections particularize the previous results to Hamiltonian systems, for which, due to their structure, a very strong characterization of the problem can be given.

This chapter deals with stability analysis of the origin, both for continuous-time and discrete-time systems. After a brief review of some well known results, a detailed study of the scalar case is carried out. Then, for planar systems, first, the connection between the center manifold theory and semi-invariants is pointed out; secondly, the critical cases for which it is possible to derive stability conditions that are easy to check are studied. It turns out that only some critical cases remain difficult to solve in general, in particular those with a zero linear part and a subclass of those with nilpotent but non-zero linear part (confirming the difficulty of distinguishing a center from a focus). Finally, the last sections deal with the use of semi-invariants as elementary bricks for the construction of Lyapunov functions; such methods are developed both for continuous-time and discrete-time systems of arbitrary order.