Irregularity indices

for linear systems of ordinary differential equations

Non-negative functions $\sigma$ on the space of mappings $A:{\mathbf{R}}^{+}\to \mathrm{Hom}({\mathbf{R}}^n,{\mathbf{R}}^n)$ (or ${\mathbf{R}}^{+}\to \mathrm{Hom}({\mathbf{C}}^n,{\mathbf{C}}^n)$), integrable on every finite interval, such that $\sigma (A)$ equals zero if and only if the system

$$\begin{array}{}\text{(*)}& \dot{x}=A(t)x\end{array}$$

is a regular linear system.

The best known (and easiest to define) such regularity indices are as follows.

1) The Lyapunov irregularity index [1]:

$${\sigma }_L(A)=\sum _{i=1}^n{\lambda }_i(A)-\underset{\overline{t\to +\mathrm{\infty }}}{lim}\frac{1}{t}\underset{0}{\overset{t}{\int }}\mathrm{tr}A(\tau )d\tau ,$$

where ${\lambda }_i(A)$ are the Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent) of the system (*), arranged in descending order, while $\mathrm{tr}A(t)$ is the trace of the mapping $A(t)$.

2) The Perron irregularity index [2]:

$${\sigma }_p(A)=\underset{1\le i\le n}{max}({\lambda }_i(A)+{\lambda }_{n+1-i}(-A^{\ast })),$$

where $A^{\ast }(t)$ is the adjoint of the mapping $A(t)$. If the system (*) is a system of variational equations of a Hamiltonian system

$$\dot{q}=\frac{\partial H}{\partial p},p\in {\mathbf{R}}^k,$$

$$\dot{p}=-\frac{\partial H}{\partial q},q\in {\mathbf{R}}^k,$$

then $n=2k$ and

$${\lambda }_i(-A^{\ast })={\lambda }_i(A),i=1\dots n.$$

Consequently, for a system of variational equations of a Hamiltonian system,

$${\lambda }_i(A)=-{\lambda }_{n+1}-i(A),i=1,\dots ,k,$$

is a necessary and sufficient condition for regularity (a theorem of Persidskii).

For other irregularity indices, see [4]–.

References[edit]

Comments[edit]

In the case of $A:{\mathbf{R}}^{+}\to \mathrm{Hom}({\mathbf{C}}^n,{\mathbf{C}}^n)$, read $\sum _{i=1}^n\mathrm{Re}{A}_{ii}(t)$ instead of $\mathrm{tr}A(t)$ in the definition of ${\sigma }_L$.