Symmetries and Semi-invariants in the Analysis of Nonlinear Systems by Laura Menini & Antonio Tornambè

Author:Laura Menini & Antonio Tornambè
Language: eng
Format: epub
Publisher: Springer London, London

which implies . □

Example 4.10

Consider again the vector function F introduced in Example 4.9. A symmetry g of F is

4.7 Lax Pairs for Discrete-Time Nonlinear Systems

The concept of the Lax pair, which is classical in the continuous-time case, can be extended to the discrete-time case (see [94]). The notation in this section is somewhat different from the one in the rest of the book, e.g., matrices A and B are not constant here.

Let a vector function F(x)∈ℝ n be given. Given a matrix function A(x)∈ℝ ν×ν , with entries A i,j (x), the symbol A○F clearly denotes the matrix function having A i,j ○F as entries.

Definition 4.4

Given a vector function F(x)∈ℝ n , a DT-Lax pair (briefly, a Lax pair if no confusion can arise) associated with F(x) is an ordered pair of matrix functions (A,B), with A(x),B(x)∈ℝ ν×ν , ν 2≥n, and B being invertible over the field of meromorphic functions, such that

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