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Extract files with WinRAR Latest!Lyapunov equations of Hopfield type

In the study of neural networks, Lyapunov equations of Hopfield type play a special role. These equations are a generalization of Lyapunov equations and may be used to analyze stability of a Hopfield network (i.e. a network whose connectivity is sparse and exhibits global inhibition) with respect to a fixed point or a trajectory. Lyapunov equations of Hopfield type may be used to show stability of neural networks to inputs or the network’s dynamics.

Lyapunov equations

Consider an ordinary differential equation, where is a vector, is a matrix and is a scalar.

Consider the Lyapunov equation:

where is a symmetric matrix.

The matrix is a solution to the Lyapunov equation if:

where.

The Lyapunov equation is said to have a solution for some fixed. If there is a solution satisfying for all, then is called a Lyapunov matrix for.

Hopfield equation

Consider the Hopfield equation

where is a vector, is a matrix and is a scalar.

Consider the Lyapunov equation

where is a symmetric matrix.

The matrix is a solution to the Lyapunov equation if:

where.

Lyapunov equations of Hopfield type

Consider a Hopfield network consisting of units. The connectivity matrix is sparse and exhibits global inhibition; in other words, there exists a constant such that

where are all non-negative integers.

The vector is a solution to a Lyapunov equation of Hopfield type if:

where, and are all non-negative 4f8c9c8613

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