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We provide geometric algorithms for checking the stability of matrix difference equations

The problem of the stability of the equation

The stability of matrix (

To the best of the authors' knowledge, the stability of (

In this paper we provide geometric algorithms for checking the stability of (

Matrices

The paper is organized as follows. In Section

Consider the scalar variant of (

Parameter

In this and the next sections we will consider only real positive values of

Let us point out a key property of symmetry of hodograph (

If

For coprime

From now and further we will assume that

The following lemma asserts that some part of the complex plane is free from points of the hodograph

Let

The function

For hodograph (

From Lemma

Considering Definition

The basic oval

Let

Let us fix

For the stability of (

Equation

If

Let us call the equation (asymptotically)

Theorems

If

The following theorems are based on the localization of roots of polynomial (

Let

If

If

If

If

(1) Let

(2) Let

(3) Let

If

(4) If

If in (

Let

If

If

If

(1) Let

(2) Let

CASE 1. Let

CASE 2. Let

Let us return to (

(3) The proof of Statement 3 of Theorem

Let us change the variables in (

Let

If

If

If

If

Let

If

If

If

Let

Let us consider a matrix equation

Obviously, matrix equation (

In this paper we consider (

Let us define a stability cone as the

Returning to Figure

Stability domains

To Example

The stability cones for

The stability cone as an intersection of three surfaces formed by the basic ovals,

Let us consider the simple case of a diagonal system

It follows from the definition of the

The natural extension of the the class of diagonal systems is that of systems with simultaneously triangularizable matrices. The following theorem is our main result.

Let

If some point

Let us make the change

Let us apply the results of the previous sections to the problem of the stability of discrete neural networks similar to continuous networks studied in [

A neural net.

Let

If for every

Let us proceed to the problem of stability of a neural network with a large number of neurons. The points

If

Let one consider system (

Let us consider the ring of neurons shown in Figure

For applications of Corollaries

To Example

Now let us consider Example

0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 | 1.3 | 1.5 | |

0.870 | 0.938 | 1.000 | 1.056 | 1.108 | 1.156 | 1.201 | 1.243 |

Let us consider the neuron chain shown in Figure

To Example

The condition

Stability cones for differential matrix equations

There are images of the stability domains in the space of parameters of scalar differential equations

The authors are indebted to K. Chudinov, A. Makarov, and D. Scheglov for the very useful comments.