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????????? ??????????? ???????

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??????????? ??????? ?????? ?????????? ??????

??????? 14

 

 

 

 

 

 

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?????????:

???????
????? ??-41

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?.?.

?????? ?. ?.

 

 

 

 

 

 

 

 

???? —
2017

 
?????
 
 
??????.. 3
??????? ???????.. 5
1 ??????????? ????’???? ???????
??????????????? ???????. 5
2 ??????????? ??????? ?????????? ??????? ??
????????? MATLAB.. 8
??????? ?.. 14
 

 

 

??????

 

????? ? ????????? ????????? ???????
??????????????? ???????

                                                                         (1)

 ?? , x = x(t), y = y(t)— ??????????, t— ?????? ????.
??? ???????? ??????? ?????? (??????????) ????? ???????????? ????????????
??????????? ?????????:

                                                         (2)

??  — ????? ????????, N — ?????????
????????,  — ???????? ????? (??? ????????? ????????????
?? ??????? ????????? (2)),  — ?????????? ?????? ????????? ?????? ? ?????  (?????, , , ?? , ), ,  – ????????? ?????? ?? ????????? ????? x ??
y . ????????? ???????? ???????? ???? ????? ? ???? ?? ?????????? ??? ????
???? ?????????? (????? ? ????????? ” (k – 1) -?? ?????????”
????????????? ? ???????? ” k -?? ?????????” ?????? ???????,
???? ????????? ???????? ????????? ? ???????? ?????). ?????, ??? ?????????
?????????? ?????????? ???????, ?? ???????? ? ????? , ????? ??? ????????? ??????????
(?????,  ?? ) ???? ?’?????? ????????? ??????.
??? ?????, ???????? ????? ????????? (; (??? ???????? ? ?? ????????, ???
??????? ???? ????, ? ??? ?????? ???? ???????? ??????????? ??????????). ???????? ????????? ?????????? ??????????
??????? ??????????????? ??????? (???? ?????????, ???????????? ?????????,
???????????) ????? ??????????, ?? ????’?????? ???????, ?? ?????? ???????
??????? ??????? ??????? ????? (?????????????? ??????? ???????). ?????  – ?????? ??????? ?????, ????? ??????
??????????? ???????? (1).

,

???? ??????? ?????????????? ??????? ?? ?????????, ????? , ?? ????????????? ???????? ??????? (??????? (*)):

??? ???????

????? ???????????
???????

??????? ???????

??? ????????? ?????

????? ?????

?????

?????????? ? ???’?????
??????? ????????

??????? ?????

?????????? ? ?????????
??????? ????????

????????? ?????

?????? ???’????

??????? ?????

?????? ???????

????????? ?????

?????? ?????? ??????

?????

 

??????? ???????

1
??????????? ????’???? ??????? ???????????????
???????

 

????????? ???????
??????? ?????????? ??????? (1.1) ?????? ???????? 14:

                                          
(1.1)

???????????
????? ??????? ???? ??????? ??????? (1.1) ?? ???? ???????? ???????? (???????????) ????? ???????: ?(4; 1), ?(2; 2),
C(-2; -2).
??????? ?????(???????? ????????) ?????????????? ??????? ??? ??????:

 

J =                                           (1.2)

 

J(?) = J(4, 1) =                                       (1.3)

?????? ?????
??????? (1.3): ?1,2 = -1±.

??????????????
??????? (*), ????????, ?? ?. A – ??????? ?????.

????????? ???????? ???????????? ???????? ?? ????????? ??????? ?????????:

 

 

???? ??? ??????????? ?? ???????????? ????????.

 

J(B) = J(2, 2) =                                         (1.4)

 

?????? ?????
??????? (1.4): ?1,2 =±.

??????????????
??????? (*), ????????, ?? ?. B – ?????.

?????? ???????
??????? (1.4): S1 = (-, 1); S2 = ();

 

J(?) = J(-2, 2) =                                     (1.5)

?????? ?????
??????? (1.4): ?1,2 = -4±.

??????????????
??????? (*), ????????, ?? ?. C – ??????? ?????.

????????? ???????? ???????????? ???????? ?? ????????? ??????? ?????????:

 

 

???? ??? ??????????? ????? ???????????? ???????.

 

??????? 1.1 – ??????? ??????? ??????? Wolfram (1.1)

 

??????? 1.2 – ????? ????????? ??????? ???????
??????? Graph Phase Plane (1.1)

 

2
??????????? ??????? ?????????? ??????? ??
????????? MATLAB

 

 

????????? ?????? ?????, ??????
???????? ?? ?’?????? ????????? ??????? ?????? ? ?????? ????? ??????
?????????????? ?????????? ????????? ???????(2).

?? ??????? 2.1. ????????? ?????? ????? ? ??????????
?????? (4;2). ?????? ????? ?????????? ?? ????????? ?????  ???  =  =0,01.

 

 

???????
2.1 – ????? ????? ? ?????????? ?????? (4;2)

 

???????? ????????? ????? (0;0)
??? ??? ?? ??????. ????????? ?? ??????? 2.2.

 

???????
2.2 – ????? ????? ? ?????????? ?????? (0;0)

 

???????? ????????? ??????  ?  
???????? ??? ???????? ???????? ???????? ? ??????????? ?? ?? 10-2,
????????? ????? ?????? ???????? . ????????????????? ?????? ???? ?????? ????? (3;3). ??????? ??????? ?????????
?? ???. 2.3.

 

??????? 
2.3 – ??????? ??????? ??? ??????  ?

??? ??? ?????
????? ????????  ?  = 10-1 . ????????? ????? ???????? ?????????? ???
???????? ? ????? ???????????? ????? (2;2), ???. ??????? 2.4.

 

??????? 
2.4 – ??????? ??????? ??? ????????  ?

????? ??????? ????????? ????????: N=100 ?? ?????????  ?  = 10-3.

?????????? ????????? ?? ??????? 2.5 ? ???????? ??????????? ?? ??????
??????? ????????? ???????.

????? 
2.5 – ??????? ??????? ??? ????????  ?  ?? N

 

?? ??????? 2.6 ?? 2.7 ?? ????????? ?????????
??????? ?????? ? ????? ????????? ????? ????? ?(4;1).

??????? 2.6 – ????????? ??????? ?????? ? ?????
????? ?(4;1).

??????? 2.7 – ????????? ??????? ?????? ? ?????
?????(????????? ???????) ?(4;1).

 

?? ??????? 2.8 ????????? ????????? ??????? ?????? ? ?????
????? ?(2;2).

 

??????? 2.8 – ????????? ??????? ?????? ? ?????
????? ?(2;2).

?? ??????? 2.9 ????????? ????????? ???????
?????? ? ????? ????? ?(-2;-2).

??????? 2.9 – ????????? ??????? ?????? ? ????? ????? ?(-2;-2).

????????

 

??????????? ?????????
???????????? ?????? ? ??????????? ????’???? ???????,

??????????? ?????
??????: ?(4; 1), ?(2; 2), C(-2; -2), ?????????? ?? ???? (??? ?????? ?????? ?? ?????). ???????  ??????? ???? ?????????? ?? ????????? Wolfram ?? Graph Phase Plane.

????? ?????????? ???????? ??? ???????? ???????
?????? ? ??????? ?????, ???????? ???????? ??????? ??? ?????? ?????????? ????? (????? ???? ??????????? ????????? ???
????? ?????) ?? ??? ????????? ????????? ??????? ? ????? ????? ??????.

 

 

??????? ?

 

Listing

mathanalys.m

a=1;

while a>0

    disp(‘1. Build the path
from the given point’)

    disp(‘2. Build phase
portrait ‘)

    disp(‘3. Show the system
behaviour around peculiar points’)

    a=input(‘Chose your number: ‘);

    if a==1

         x0=input(‘Input starting point for x: ‘);

         y0=input(‘Input starting point for y: ‘);

         N=input(‘Input
a number of points ‘);

         parametr1 =input(‘Input parametr for x ‘);

         parametr2 =input(‘Input parametr for y ‘);

         x,y =
phase(x0,y0,N,parametr1,parametr2);

         figure

         plot(x,y)

    elseif a==2

         x0=input(‘Input starting point for x: ‘);

         y0=input(‘Input starting point for y: ‘);

         N=input(‘Input
a number of points ‘);

         Nx=input(‘Input a number of points for x: ‘);

         Ny=input(‘Input a number of points for y: ‘);

         dx=input(‘Input a step between starting points for x: ‘);

         dy=input(‘Input a step between starting points for y: ‘);

         parametr1 =input(‘Input parametr for x ‘);

         parametr2 =input(‘Input parametr for y ‘);

        
phase(x0,y0,N,Nx,Ny,dx,dy,parametr1,parametr2) ;

      elseif a==3

          x0=input(‘Input starting point for x: ‘);

         y0=input(‘Input starting point for y: ‘);

         N=input(‘Input
a number of points ‘);

         parametr1 =input(‘Input parametr for x ‘);

         parametr2 =input(‘Input parametr for y ‘);

        
phase(x0,y0,N,20,20,0.1,0.1,parametr1,parametr2) ;

    end

end

 

 

 

 

phase.m:

function phase(x00,y00,N,nx,ny,dx,dy,parametr1,parametr2)

 

x0=
x00 – (nx/2)*dx;

y0=
y00 – (ny/2)*dy;

 

x,y=path(x0,y0,N,parametr1,parametr2);

 

figure

plot(x,y) 

hold
on

 

p1
= x0;

for k=2:nx

    p1 = p1 + dx;

    p2 = y0;

        for j=2:ny

            p2 = p2 + dy;

           
x,y=path(p1,p2,N,parametr1,parametr2);

            plot(x,y)

        end

end

hold
off

 

 

path.m:

function x,y = path(x0,y0,N,parametr1,parametr2)

x =
zeros(N,1);

y =
zeros(N,1);

x(1)
= x0;

y(1)
= y0;

 

for i=2:N

    Vx = ((x(i-1))*(y(i-1)))-4;

    Vy = (x(i-1)-4)*(y(i-1)-x(i-1));

    Vvx = Vx/power(Vx*Vx+Vy*Vy,1/2);

    Vvy = Vy/power(Vx*Vx+Vy*Vy,1/2);

    x(i)= x(i-1) + parametr1*Vvx;

    y(i)= y(i-1) + parametr2*Vvy;

end