the following problems
are the particular cases:
- - about the pendulum of Kapitsa when E0<0 ,
- - dynamics of particles considered their characteristics
(charges and moments) in field's and when X is small.
In approximation k=2, using "OPUS" system to (1-3)
and expansion cos(X):
cos(X)=> {AS*[X-X**3/3]+AC*[1-X**2/2+X**4/24]},
where: a) AS=0, AC=1, expansion cos(X) near 0, b) AS=1, AC=0, expansion
cos(X) near PI/2 (90 degrees), we'll get the results for S-function
in form:
S = (3*E0*AC*Y(1,2,2)**4 + (12*E1*AC*Y(0,1,1)*Y(1,1,2) + 12*E1*AC*Y(0,1,2)*
Y(1,1,1) + 12*E1*AC*Y(0,2,2)*Y(1,0,1) + 24*E1*AS*Y(0,2,2))*Y(1,2,2)**3
+ ( 24*E0*AC*Y(1,2,1)**2 + ( - 24*E1*AC*Y(0,1,2)*Y(1,1,2) + 24*E1*AC*Y(0,1,1)*
Y(1,1,1) + 48*E1*AC*Y(0,2,1)*Y(1,0,1) + 96*E1*AS*Y(0,2,1))*Y(1,2,1)
+ ( - 12*E1*AC*Y(0,2,1) + 48*E0*AC)*Y(1,1,2)**2 + (36*E1*AC*Y(0,2,2)*Y(1,1,1)
+ 96*E1*AC*Y(0,1,2)*Y(1,0,1) + 192*E1*AS*Y(0,1,2))*Y(1,1,2) + (12*E1*AC*Y(0,
2,1) + 48*E0*AC)*Y(1,1,1)**2 + (96*E1*AC*Y(0,1,1)*Y(1,0,1) + 192*E1*AS*Y(0
,1,1))*Y(1,1,1) + 96*E0*AC*Y(1,0,1)**2 + 384*E0*AS*Y(1,0,1) - 192*E0*AC
+ 768)*Y(1,2,2)**2 + ((48*E1*AC*Y(0,1,1)*Y(1,1,2) + 48*E1*AC*Y(0,1,2)*Y(1,1,
1) + 48*E1*AC*Y(0,2,2)*Y(1,0,1) + 96*E1*AS*Y(0,2,2))*Y(1,2,1)**2
+ ( - 24* E1*AC*Y(0,2,2)*Y(1,1,2)**2 + 96*E1*AC*Y(0,2,1)*Y(1,1,1)*Y(1,1,2)
+ 24*E1* AC*Y(0,2,2)*Y(1,1,1)**2)*Y(1,2,1) + 32*E1*AC*Y(0,1,1)*Y(1,1,2)**3
+ (96*E1 *AC*Y(0,1,2)*Y(1,1,1) + 96*E1*AC*Y(0,2,2)*Y(1,0,1) + 192*E1*AS*Y(0,2,2))*Y
(1,1,2)**2 + (96*E1*AC*Y(0,1,1)*Y(1,1,1)**2 + (384*E0*AC*Y(1,0,1)
+ 768*E0 *AS)*Y(1,1,1) + 192*E1*AC*Y(0,1,1)*Y(1,0,1)**2 + 768*E1*AS*Y(0,1,1)*Y(1,0,
1) - 384*E1*AC*Y(0,1,1))*Y(1,1,2) + 32*E1*AC*Y(0,1,2)*Y(1,1,1)**3
+ (96*E1 *AC*Y(0,2,2)*Y(1,0,1) + 192*E1*AS*Y(0,2,2))*Y(1,1,1)**2
+ (192*E1*AC*Y(0,1 ,2)*Y(1,0,1)**2 + 768*E1*AS*Y(0,1,2)*Y(1,0,1)
- 384*E1*AC*Y(0,1,2))*Y(1,1, 1) + 64*E1*AC*Y(0,2,2)*Y(1,0,1)**3
+ 384*E1*AS*Y(0,2,2)*Y(1,0,1)**2 - 384* E1*AC*Y(0,2,2)*Y(1,0,1)
- 384*E1*AS*Y(0,2,2))*Y(1,2,2) + 48*E0*AC*Y(1,2,1) **4 + ( - 96*E1*AC*Y(0,1,2)*Y(1,1,2)
+ 96*E1*AC*Y(0,1,1)*Y(1,1,1) + 192*E1 *AC*Y(0,2,1)*Y(1,0,1) + 384*E1*AS*Y(0,2,1))*Y(1,2,1)**3
+ (( - 144*E1*AC*Y (0,2,1) + 192*E0*AC)*Y(1,1,2)**2 + (48*E1*AC*Y(0,2,2)*Y(1,1,1)
+ 384*E1*AC *Y(0,1,2)*Y(1,0,1) + 768*E1*AS*Y(0,1,2))*Y(1,1,2) +
(144*E1*AC*Y(0,2,1) + 192*E0*AC)*Y(1,1,1)**2 + (384*E1*AC*Y(0,1,1)*Y(1,0,1)
+ 768*E1*AS*Y(0,1,1) )*Y(1,1,1) + 384*E0*AC*Y(1,0,1)**2 + 1536*E0*AS*Y(1,0,1)
- 768*E0*AC + 3072)*Y(1,2,1)**2 + ( - 128*E1*AC*Y(0,1,2)*Y(1,1,2)**3
+ ((384*E1*AC*Y(0,2 ,1) - 384*E0*AC)*Y(1,0,1) + 768*E1*AS*Y(0,2,1)
- 768*E0*AS)*Y(1,1,2)**2 + ( - 384*E1*AC*Y(0,1,2)*Y(1,0,1)**2 -
1536*E1*AS*Y(0,1,2)*Y(1,0,1) + 768*E1 *AC*Y(0,1,2))*Y(1,1,2) + 128*E1*AC*Y(0,1,1)*Y(1,1,1)**3
+ ((384*E1*AC*Y(0, 2,1) + 384*E0*AC)*Y(1,0,1) + 768*E1*AS*Y(0,2,1)
+ 768*E0*AS)*Y(1,1,1)**2 + (384*E1*AC*Y(0,1,1)*Y(1,0,1)**2 + 1536*E1*AS*Y(0,1,1)*Y(1,0,1)
- 768*E1 *AC*Y(0,1,1))*Y(1,1,1) + 256*E1*AC*Y(0,2,1)*Y(1,0,1)**3
+ 1536*E1*AS*Y(0,2 ,1)*Y(1,0,1)**2 - 1536*E1*AC*Y(0,2,1)*Y(1,0,1)
- 1536*E1*AS*Y(0,2,1))*Y(1, 2,1) + ( - 32*E1*AC*Y(0,2,1) + 48*E0*AC)*Y(1,1,2)**4
+ (32*E1*AC*Y(0,2,2)* Y(1,1,1) + 192*E1*AC*Y(0,1,2)*Y(1,0,1) + 384*E1*AS*Y(0,1,2))*Y(1,1,2)**3
+ (96*E0*AC*Y(1,1,1)**2 + (192*E1*AC*Y(0,1,1)*Y(1,0,1) + 384*E1*AS*Y(0,1,
1))*Y(1,1,1) + ( - 192*E1*AC*Y(0,2,1) + 384*E0*AC)*Y(1,0,1)**2 +
( - 768* E1*AS*Y(0,2,1) + 1536*E0*AS)*Y(1,0,1) + 384*E1*AC*Y(0,2,1)
- 768*E0*AC + 768)*Y(1,1,2)**2 + (32*E1*AC*Y(0,2,2)*Y(1,1,1)**3
+ (192*E1*AC*Y(0,1,2)*Y( 1,0,1) + 384*E1*AS*Y(0,1,2))*Y(1,1,1)**2
+ (192*E1*AC*Y(0,2,2)*Y(1,0,1)**2 + 768*E1*AS*Y(0,2,2)*Y(1,0,1)
- 384*E1*AC*Y(0,2,2))*Y(1,1,1) + 256*E1*AC* Y(0,1,2)*Y(1,0,1)**3
+ 1536*E1*AS*Y(0,1,2)*Y(1,0,1)**2 - 1536*E1*AC*Y(0,1, 2)*Y(1,0,1)
- 1536*E1*AS*Y(0,1,2))*Y(1,1,2) + (32*E1*AC*Y(0,2,1) + 48*E0* AC)*Y(1,1,1)**4
+ (192*E1*AC*Y(0,1,1)*Y(1,0,1) + 384*E1*AS*Y(0,1,1))*Y(1,1 ,1)**3
+ ((192*E1*AC*Y(0,2,1) + 384*E0*AC)*Y(1,0,1)**2 + (768*E1*AS*Y(0,2,
1) + 1536*E0*AS)*Y(1,0,1) - 384*E1*AC*Y(0,2,1) - 768*E0*AC + 768)*Y(1,1,1)
**2 + (256*E1*AC*Y(0,1,1)*Y(1,0,1)**3 + 1536*E1*AS*Y(0,1,1)*Y(1,0,1)**2
- 1536*E1*AC*Y(0,1,1)*Y(1,0,1) - 1536*E1*AS*Y(0,1,1))*Y(1,1,1) -
1536*Y(1,0, 2)**2 + 128*E0*AC*Y(1,0,1)**4 + 1024*E0*AS*Y(1,0,1)**3
- 1536*E0*AC*Y(1,0, 1)**2 - 3072*E0*AS*Y(1,0,1)) / (3072),
Let's take an example with
X=>Y(1,1,1)*cos(t), cos(X) ~(1-X**2/2+X**4/24) and AS=0, AC=1,
Y(1,1,1)=Y111, Y(0,2,1)=1, Y(1,0,1)=Y(1,0,2)=Y(1,1,2)=Y(1,2,1)=Y(1,2,2)=Y(0,1,1)=Y(0,1,2)=Y(0,2,2)=0.
Then from exetremum condition S, D1(J1)=0, obtain Y111**2=f(E0,E1)={12*(2*E0+E1-2)/(3*E0+2*E1)}.
For roots LX from D2(J1,J2) we'll obtain equation: DT = DET{D2(LX,E0,E1,Y111)}=0.
Solving DT relatively to LX through FACTORIZE(DT) we can obtain
corresponding equation to find LX :
{LX**2+Y111**2*(E0/2+E1/2+1)/24}*f(LX) = 0, (4)
where Y111**2={12*(2*E0+E1-2)/(3*E0+2*E1)}, (5)
and f - a complex form function of LX, E0,E1. Existence of inverted
pendelum (E0<0) stable state follows from the analysis (4) in
parametrical resonanse zone …1>2*(1-…0). Calculating analogically
for X ~ PI/2+Y(1,2,1)*cos(2*t), cos(X) ~ (X-X**3/3) for Y121=Y(1,2,1)
and DT(LX)=0 we can obtain rather awkward expressions. In the simpler
case with E0=0 from DT(LX) analysis it is not different to show
that solution with
Y121={2*[SQRT(3*E1**2+16)-4]/(3*E1)}, (6)
is instaible on Y(1,0,1), Y(1,0,2). Dependence plots Y111, Y121
of E1 when E0=0 are given in pic.
It's easy to realise that solving jointly (5,6) the
corresponding bifurcation point
from the statement (see pic.) can be determined: Y111(E1)+Y121(E1)=PI/2,
e.i. Y111 ~ 59 grade, Y121 ~ 31 grade, E1 ~ 2.44. Simultaneously
in the case (with E0=0) bifurcation appearance can lead to chaos
in the system (1-3). Fluctuations, errors from macrosystem used
in physical analogue or numerical modeling of determined system
described in (1-3) can be the reasons of bifurcation. As a result
cascades of transitions between different types of periodic motions
when …1 ~ 2.44 (oscillatory 1:2, 1:1, rotatory and so on) which
are taken as chaos will be observed. Mashine
modeling of the given system and hybrid complex and natural modeling
on the magnetic compass needle which is set in magnetic field confermed
the received results correctness.
Using "OPUS" in the trivial case when all Y(1,I2,I3)=0,
the corresponding equation for LX has the following form: {LX**2+[(1-E0)**2-E1**2/4]/16}*{LX**4+LX**2*(1+E0/4)**2/4
+(1-E0/4)/8*[E1**2/16+E0*(1-E0/4)/2]/8} (7) from the 1st and 2nd
bracket the solution of (7) stable solution upper and lower bound
{E1**2/4<(1-E0)**2, E1**2>8*abs[E0*(1-E0/4)]} for the inverted
pendulum (…0<0) out of parametric resonance 1:2 zone and stable
solution absence in resonance zone. Non - linearity existance
of principle in (3) for dynamic stability of instable states in
resonanse zone is interesting to note. Doing the same calculations
for cos(X)=> with the help of "OPUS" cos(X) ~(1-X**2/2+m*X**4/4)
we can obtain for Y111 in parametric resonanse zone 1:2 : Y111**2={4*(1-E0-E1/2)/[m*(2*E1+3*E0)]}
(8) m =>0 Y111=>00 follows from (8) expression it's characteristic
for the linear system in parametric resonance zone.
Using "OPUS"
system we could demonstrate theoreticaly and experementally the
posibility of body and particles selective spatial confinement,
their characteristics taken into consideration (charges, moments
- mechanical, electric, magnetic) in resonance electromagnetic non
- uniform fields [2] unlike confinement out of resonance zone [3-4].
...
......
Confinement of a
grain Sm-Co in a resonance
trap.
Literature:
Russian