Equilibrium stability for the discrete diamond-alpha operator
Equilibrium stability for the discrete diamond-alpha operator
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离散菱形-alpha 算子的平衡稳定性
DOI：
10.1007/s40840-022-01417-7
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发表时间：
2023
影响因子：
1.2
通讯作者：
D. R. Anderson and M. Onitsuka
中科院分区：
文献类型：
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作者：
Izuki Mitsuo;Nakai Eiichi;Sawano Yoshihiro;大塚浩史;D. R. Anderson and M. Onitsuka
我们建立了离散菱形-α差分算子的平衡（平凡）解的李雅普诺夫稳定性，使用虚构的菱形-α椭圆。这统一和扩展的平衡分析一阶向前（Delta）和向后（nabla）差分方程与一个常数复系数。证明了当系数为负椭圆真实的部分时方程是渐近稳定的，当系数为零椭圆真实的部分时方程是稳定的，当系数为正椭圆真实的部分时方程是不稳定的，但α = 1/2的临界情形除外.我们还提供了一个渐近稳定的结果，在相应的非齐次方程的情况下的原点。
We establish the Lyapunov stability of the equilibrium (trivial) solution for the discrete diamond–alpha difference operator, using the imaginary diamond–alpha ellipse. This unifies and extends equilibrium analysis for first-order forward (Delta) and backward (nabla) difference equations with a constant complex coefficient. We prove that for coefficients with negative elliptical real part the equation is asymptotically stable, with zero elliptical real part the equation is stable, and with positive elliptical real part the equation is unstable, except in the critical case of alpha equals one half. We also provide an asymptotic stability result for the origin in the case of the corresponding inhomogeneous equation.