Determination of risky and safe boundaries of the stability region of discrete systems (Q1324049)

This paper deals with the problem of determining the risky and safe boundaries of the equilibrium stability region for discrete systems given by \[ \overline {x}_ j= \sum_{l=1}^ n a_{jl} x_ l+ F_ j (x_ 1, x_ 2, \dots, x_ n), \qquad j=1,2, \dots, n, \tag{1} \] where \(n\geq 2\) and \(a_{jl}\) are real coefficients, \(F_ j (x_ 1, x_ 2, \dots, x_ n)\) are analytical or sufficiently smooth functions, assuming that \(A= [a_{jl} ]_{n\times n}\) and the characteristic equation \(\Delta (\lambda)= | A-\lambda I|=0\) has roots \(\lambda_{1,2}= \exp(\pm i\varphi)\), \(0< \varphi< \pi\), \(\varphi\neq \pi/2\), \(2\pi/3\), and the remaining roots \(\lambda_ k\) satisfy \(| \lambda_ k |<1\). For the discrete systems (1), the paper proposes a constant similar to the Lyapunov values for differential equations as follows: \[ g_ 1= \begin{cases}\text{Re } \lambda_ 2 \biggl[ A_{112}+ {{2(2 \lambda_ 1-1)} \over {\lambda_ 1 (1- \lambda_ 1)}} A_{11} A_{12} \biggr] -2 | A_{12} |^ 2- | A_{22}|^ 2, \;n=2\\ \text{Re } \lambda_ 2 \biggl[ A_{112}+ {{4A_{113} A_{312}} \over {1- \lambda_ 3}} + {{4\lambda_{123} A_{311}}\over {\lambda_ 1^ 2- \lambda_ 3}} + {{2(2 \lambda_ 1-1)} \over {\lambda_ 1 (1- \lambda_ 1)}} A_{111} A_{112} \biggr] - 2| A_{12} |^ 2- | A_{22} |^ 2, \;n=3\\ \text{Re } \lambda_ 2 \sum_{j=1}^ n \Delta_{j1} (\lambda_ 1) \{ \sum_{1\leq k\leq l\leq n} a_{jkl} (\sigma_{l1} \gamma_{11}^{(k)}+ \sigma_{k1} \gamma_{11}^{(l)}+ \sigma_{k2} \gamma_{20}^{(l)}+ \sigma_{l2} \gamma_{20}^{(k)})+\\ \sum_{1\leq k\leq l\leq n}a_{jkls} [\sigma_{k1} \sigma_{l1}\sigma_{s2}+ \sigma_{21}(\sigma_{k1} \sigma_{2l}+\sigma_{k2} \sigma_{l1})]\}/ \sum_{k=1}^ n\Delta_{1k} (\lambda_ 1)\Delta_{k1} (\lambda_ 1),\;n>3 \end{cases} \tag{2} \] where all the coefficients \(A_{kl}\), \(A_{kls}\) and \(A_{jkls}\) are expressed in terms of \(a_{jl}\), and \(\sigma_{kl}\) in terms of both \(a_{jl}\) and \(\lambda_ k\). Then the conclusion on the boundaries of the stability regions of (1) is that \(\bullet\) The boundaries are safe, if \(g_ 1<0\); \(\bullet\) The boundaries are risky, if \(g_ 1>0\).