There are some methods for determination of nuclear reactor stability. Among methods which are applied for this aim are as following:

Liapunov method — Lagrange method — Popov method — Pontryagin method.

There is Lyapunov’s method to determine the stability of nonlinear reactor dynamics by constructing certain positive definite functions of the reactor variables and parameters (Pankaj and Vivek, 2011).

There has been calculated the Lyapunov exponents from a time series of the excess neutron population of a boiling water reactor (BWR) and used it to conclude about the stability of the steady state operation of that particular BWR (Munoz et al., 1992).

There has also discussed the application of topological methods in reactor kinetics study (Smets and Giftopoulos, 1959).

The topological and Lyapunov methods were compared with Aizermann-Rosen methods for analyzing a point reactor model (Devooght and Smets, 1967).

The Pade approximations has been used to obtain solutions for point kinetic equations (Aboanberand, 2002).

Perturbation theory has also been widely used in studying reactor dynamics. There has been obtained specific types of steady solutions to study power oscillations in a reactor results from a Hopf bifurcation (Pandey, 1996; Munoz and Verdu, 1991; Tsuji et al., 1993; Konno et al., 1994). The KBM theory has been used for the nonlinear analysis of a reactor model with the effect of time-delay in the automated control system (Konno et al, 1992).

A singular perturbation has been used to study relaxation oscillations in typical nuclear reactors (Ward and Lee, 1987).

The point kinetic equations in the presence of delayed neutrons with one temperature reactivity coefficient for a step input of reactivity have analytically been solved by applying the perturbation theory (Gupta and Trasi, 1986).

The regular perturbations to obtain an analytical solution for general reactivity have been used (Nahla, 2009).

The multiple time-scales expansion to obtain analytical solutions of the neutron kinetic equations has been applied (Merk and Cacuci, 2005).

The variation methods in conjunction with the Hopf bifurcation theory for a BWR with one group delayed neutron have also been applied (Munoz and Verdu, 1991).

A combination of the center-manifold reduction (CMR) and the method of normal forms have already been applied widely for nonlinear analysis of nuclear reactor dynamics (Pandey, 1996; Tsuji et al., 1993; Konno et al., 1994).

In the Liapunov model, the dynamically equations may be either differential or non differential equations. But the method of problem evaluation is not based on solving the equations. This method is based on energy in classic mechanical. In one of mechanical system the stability condition is when the total energy of a system decreases.

Liapunov applied this property and based the stability function. This function is entitled: V(x) Function.

This function has features as following:

1. V (x)has definite positive value.

2. The partial differential of V(x) is continued.

3. Where the p is momentum, V(p) is negative quasi relatively.

4. The V (x) function can be written as following:

V(x) =Э/х) dx1 + dV(x) dx1 + + ЭУ(х) dxn_ (69)

дхг dt dx2 dt dxn dt

It can also write:

In fact the Liapunov function is a function that considers either state variables or variables which cause to imbalance state. Due to the potential function is able to do a process, so the Liapunov function is as V function. One can write:

V = f (xu x2,…, xn) (71)

If: V = 0 then the system will be steady state.

This method revolved about the determination of a V function, which satisfies certain requirements of the stability theorem. In an initial reactor model a point reactor with constant power removal and without delayed neutrons is considered.

Due to p is a variable which causes unbalancing the system, therefore it can be considered as an x variable. Then it can be written:

yp) = d V(p) dpi +dV(p)dp1 + _ + dV(p) dpn (72)

dpi dt dp2 dt dpn dt

and:

V (p) = wT (p).p