"OPUS" - analitical calculation systems
use for nonlinear particle dynamic analysis.

---"OPUS" has been written by means of ACS (analitical calculation systems) and based on the generalization of the corresponding methods to find and research the periodic solution stability of the usual differential equations using the critical points of the S-action function [1, 2, 3]. "OPUS" [2, 3] is based on two statements of Poincare:

  • - the first is "... periodic solutions are the only way we could try to enter the field concidered to be inaccessible",
  • -the second "...a periodical solutions can disappear in case of mergence with the other one ", e.i. " the periodic solutions disapear in pairs similarly the real roots of the algebraic equations" ([1], p.75).

---For n - approximation (n=1,2,...) in analitical form "OPUS" allows:

  • calculating the stable weighing zones of different bodies and particles (from elementary to macro-) without external feedback in and out of the resonance range,
  • finding the periodic solutions of the differential equations, not containing an explicit small parameter, for lagrange's and unlagrange's systems, where dissipation and disturbance are taken into consideration,
  • analyzing the nonlinear systems stability near the periodical solutions,
  • optimizing the system dynamics using parameters,
  • determining the spectrum and conditions of chaos beginning for nonlinear systems.

-----In the lagrange system case:

L = T{dX(1)/dt,...,dX(n)/dt} - U{X(1),...,X(n),t},

H = T + U,


-----Let U{X,f(t)}=U{X,f(t+T)} and F = F {dX(I1)/dt} - a dissipative function,

"OPUS" is based on going to the new Y(I1,I2,I3) variables from the previous X(I1), from H{X(I1),dX(I1)/dt,t} Hamiltonian with I1:=1,2,...n, where n - freedom degree number, X(I1), dX(I1)/dt - phase variables coordinates, velosity to the new "hamiltonian" S(Y(I1,I2,I3)) with the new dissipative function R{Y(I1,I2,2)} and "freedom degrees" number - n*(k + 1), where Y(I1,I2,I3) - the new phase variables I1=1,2,...n, I2:=0,1,...,k, where k - the corresponding approximation (quantity of harmonics taken into account), I3:=1,2, Y(I1,I2,I3=1) - coordinates, Y(I1,I2,I3=2) - velosity,

-----let f(t) = SUM(I2=0...w){Y(0,I2,1)*cos(I2*t) + [Y(0,I2,2)/I2]*sin(I2*t)}, X(I1)=SUM(I2=0...k){Y(I1,I2,1)*cos(I2*t) + [Y(I1,I2,2)/I2]*sin(I2*t)},

Y(I1,I2,I3) - slowly changing amplitudes, dY(I1,0,1)/dt = Y(I1,0,2). So in the systems without dissipation the problem lies in finding of S-function extremums relatively to Y(I1,I2,I3) and the task parameters: D1(J1) = dS/dY{J1}, D2(J1,J2) = dD(J1)/dY(J2) = d2S/dY{J1}*dY{J2}, J=f(I1,I2,I3). The S - function method is published in the journals and tested in the problems, some of them are represented as the examples in the paper a),b).

-----The examples of "OPUS" use for IBM PC/AT is represented below:

a) - for instable stationary state dynamic stability researching of the autonomoies system,

b) - for the pendulum of P.L. Kapitsa with the vibrating suspention point in and out of resonanse zone and for dynamics of particles considered their characteristics (charges and moments) in field's.

-----"OPUS" is also a method to calculation the stable confinement and levitation control of the bodies and particles movement. So the new "OPUS" technologies and available application fields are suggested to you [3]:

1. Development of the resonant traps for particles of different types and sizes (from elementary particles to macro-particles) using no feedback; studies of properties of individual particles and their dynamics in such traps (e.g. atoms and molecules) and packing them on the surfaces of solids - molecular technology); development of the resonant traps for plasma; obtaining the self-confined plasma.

2. Development of new contactless methods of treatment, preventive treatment and diagnostics, based on the resonant effects of electromagnetic fields on the biological systems; to regulate of biochemical reactions inside the biological organisms.

3. Separation of different kinds of powders (ferromagnetic, abrasive etc.), which are used, for example, in the production of the magnetic recording medium materials (for magnetic disks or tapes).

4. Development of new resonant supersensitive transducers of the fields (electromagnetic, acoustic, hydrodynamic, seismic, gravitational).

5. Robotization - electromagnetic suspention of the details, their contactless orientation in space and control during the assembling of different mechanisms and devices.

6. Suspension in the air and transference of the different bodies in the necessary direction (rotors of engines, toys, magnetically levitated trains).

SRC "IKAR" would be glad to cooperate with the researchers, who are interested in this technologies, in particular in technologies:

    • development of resonant traps where plasma is to be used;
    • self-confinement plasma - ball lightning;
    • development of the new non-contact methods for prophylactics, treatment and diagnostics on the basis of effect of resonant fields on biological systems; biochemical and chemical reaction's control.

---We already have received interesting outcomes in these directions.


    1. Poincare A. New methods of celestial mechanics.-M.: 1971, v.1.
    2. Shironosov V.G. About stability of unstable states, bifurcation and chaos of non-linear dynamic systems (Ob ustoichivosti neustoichivykh sostoyanii, bifurkatsii, khaose nelineinykh dinamicheskikh system) (in Russian). - DAN SSSR, 1990, v. 314, No. 2, p. 316-320.
    3. Shironosov V.G. Resonance in physics, chemistry and biology. Izhevsk. Publ. House "Udmurtia University", 2001. 92 p.