The examples of "OPUS" use for IBM is represented below
for instable stationary state dynamic stability researching of the autonomoies system.

Let's take an example -

a)-"OPUS" using example on IBM PC/AT to research dynamic confinement of the saddle point in the independent system:

• T = (d2X(1)/dt2)**2 + (d2X(2)/dt2)**2.
• U = C20*X(1)**2 + C02*X(2)**2 + C40*X(1)**4 + C22*X(1)**2*X(2)**2 + C04*X(1)**4.

Let's set a limit on the first approximation (k=1). Using "OPUS" we'll have the following expressions for S(C,Y) -functions:

S=( - 3*C04*Y(2,1,2)**4 + ( - 6*C04*Y(2,1,1)**2 - 24*C04*Y(2,0,1)**2 - 3*C22* Y(1,1,2)**2 - C22*Y(1,1,1)**2 - 4*C22*Y(1,0,1)**2 - 4*C02 + 2)*Y(2,1,2)**2+ ( - 4*C22*Y(1,1,1)*Y(1,1,2)*Y(2,1,1) - 16*C22*Y(1,0,1)*Y(1,1,2)*Y(2,0,1))* Y(2,1,2) - 3*C04*Y(2,1,1)**4 + ( - 24*C04*Y(2,0,1)**2 - C22*Y(1,1,2)**2 - 3*C22*Y(1,1,1)**2 - 4*C22*Y(1,0,1)**2 - 4*C02 + 2)*Y(2,1,1)**2 - 16*C22* Y(1,0,1)*Y(1,1,1)*Y(2,0,1)*Y(2,1,1) - 4*Y(2,0,2)**2 - 8*C04*Y(2,0,1)**4 + ( - 4*C22*Y(1,1,2)**2 - 4*C22*Y(1,1,1)**2 - 8*C22*Y(1,0,1)**2 - 8*C02)* Y(2,0,1)**2 - 3*C40*Y(1,1,2)**4 + ( - 6*C40*Y(1,1,1)**2 -24*C40*Y(1,0,1)**2- 4*C20 + 2)*Y(1,1,2)**2 - 3*C40*Y(1,1,1)**4 + ( - 24*C40*Y(1,0,1)**2 - 4*C20+ 2)*Y(1,1,1)**2 - 4*Y(1,0,2)**2 - 8*C40*Y(1,0,1)**4 - 8*C20*Y(1,0,1)**2) /8,

Let's take an example with Y(1,1,1)=Y111, all the other Y(1,I2,I3)=0. Then from extremum statement S we'll have all D1(j1)=0 and Y111**2=( - 2*C20 + 1)/(3*C40) or Y111=0. For roots LX from D2(I1,I2) the equation will be DT=DET(D2(LX))=0. Solving DT relatively to LX through FACTORIZE(DT) for LX(I1) will be:

• LX(1)= - 2*(-C20 + 1).
• LX(2)=-1.
• LX(3)= 2*C20 - 1.
• LX(4)= 0.
• LX(5)= ( 2*C20*C22 - 6*C40*C02 - C22)/(3*C40).
• LX(6)= (-1).
• LX(7)= ( 2*C20*C22 - 4*C40*C02 + 2*C40 - C22)/(4*C40).
• LX(8)= ( 2*C20*C22 - 12*C40*C02 + 6*C40 - C22)/(12*C40).

From LX(1) => C20<1 and from LX(3) => C20<1/2 and from X111>0 => (e.g. X111=Y111**2) => C20<1/2 =>, e.g. LX(I1) =< 0 , then system with L=T-U has stable solutions for the following cases:

a) C02<1/2 , C22 > 6*abs(C40)*(-2*C02+1)/(-2*C20+1);

b) C02 >1/2, C22 >- 2*abs(C40)*(-2*C02+1)/(-2*C20+1),

consequently:

system with L has the stable solutions not only for U with C20,C02>0 (minimum point of potentiol energy (b), but and with C20>0, C02<0 saddle point (a) in dynamics!

Using numerical and analog modding the results were checked up on hybrid complex.

In trivial case, when all Y(I1,I2,I3)=0 it follows from DT form for receved solutions stability C20, C02>0, condition is necessary, i.e. U - minimum point when X(1)=X(2)=0.