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2011 | OriginalPaper | Chapter

3. Analysis of Continuous-Time Nonlinear Systems

Authors : Prof. Laura Menini, Prof. Antonio Tornambè

Published in: Symmetries and Semi-invariants in the Analysis of Nonlinear Systems

Publisher: Springer London

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Abstract

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.

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Title: Analysis of Continuous-Time Nonlinear Systems
Authors: Prof. Laura Menini
Prof. Antonio Tornambè
Publisher: Springer London
Book: Symmetries and Semi-invariants in the Analysis of Nonlinear Systems
Print ISBN: 978-0-85729-611-5

Electronic ISBN: 978-0-85729-612-2

Copyright Year: 2011
DOI: https://doi.org/10.1007/978-0-85729-612-2_3