Chaos in the forced pendulum

# METHODS OF MATHEMATICS / MATHEMATICAL PHYSICS # http://www1.uprh.edu/rbaretti # http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysics.htm # http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart2.htm # http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart3.htm # http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart4.htm # http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart5.htm 

[[phi5 == 25/2, phi6 == 25/2, phi9 == 75/2, phi10 == 75/2]]

[[phi5 == 25/2, phi6 == 25/2, phi9 == 75/2, phi10 == 75/2]]

R=RealField(20); R(25/2);R(75/2); 

12.500
37.500

12.500
37.500

# exact answers #x,y,Phiexact= 0.667E+00 0.667E+00 0.119E+02 #x,y,Phiexact= 0.667E+00 0.133E+01 0.381E+02 #x,y,Phiexact= 0.133E+01 0.667E+00 0.119E+02 # x,y,Phiexact= 0.133E+01 0.133E+01 0.381E+02 

# www.geocities.com/serienumerica4 # The H Matrix Laplace/Poisson problem in a square mesh from Numerical solutions of Diff # #Equations # W. E. Milne p.203 var('phi1,phi2,phi4,phi5,phi6,phi7'); var('phi8,phi9,phi10,phi11,phi13,phi14'); var('b1,b2,b3,b4'); phi1=0;phi2=0;phi3=0;phi4=0;phi7=1/9;phi8=0;phi11=4/9;phi13=1/9;phi14=4/9; b1=-4.94e-2; b2=-1.23e-2 ;b3=-1.23e-2 ; b4=.691; solve([4*phi5-phi6-phi9+0*phi10==b1,-phi5+4*phi6+0*phi9-phi10==b2,-phi5+0*phi6+4*phi9- phi10==b3,0*phi5-phi6-phi9+4*phi10==b4],phi5,phi6,phi9,phi10) 

[[phi5 == 37/3000, phi6 == 1481/30000, phi9 == 1481/30000, phi10 ==
5923/30000]]

[[phi5 == 37/3000, phi6 == 1481/30000, phi9 == 1481/30000, phi10 == 5923/30000]]

R=RealField(20); R(37/3000 ) ;R(1481/30000 ) ; R(1481/30000);R(5923/30000 ) 

0.012333
0.049367
0.049367
0.19743

0.012333
0.049367
0.049367
0.19743

# 10 Φ5 - 4.5 Φ6 +0 Φ9 + 0 Φ10 = sin(Ï€/3) (9) #-4.5Φ5 +10 Φ6 +0 Φ9 + 0 Φ10 = sin(2Ï€/3) # -Φ5 +0 Φ6 + 10 Φ9 – 4.5Φ10 = 0 # 0 Φ5 - Φ6 - 4.5 Φ9 + 10Φ10 = #0 . var('phi1,phi2,phi4,phi5,phi6,phi7'); var('phi8,phi9,phi10,phi11,phi13,phi14'); var('b1,b2,b3,b4'); phi1=0;phi2=0;phi3=0;phi4=0;phi7=1/9;phi8=0;phi11=4/9;phi13=1/9;phi14=4/9; b1=sin(pi/3); b2=sin(2*pi/3) ;b3= 0 ; b4=0; solve([10*phi5-4.5*phi6-0*phi9+0*phi10==b1,-4.5*phi5+10*phi6+0*phi9+0*phi10==b2,-phi5+0*phi6+10*phi9 -4.5*phi10==b3,0*phi5-phi6-4.5*phi9+10*phi10==b4],phi5,phi6,phi9,phi10) 

[[phi5 == sqrt(3)/11, phi6 == sqrt(3)/11, phi9 == 2*sqrt(3)/121, phi10
== 2*sqrt(3)/121]]

[[phi5 == sqrt(3)/11, phi6 == sqrt(3)/11, phi9 == 2*sqrt(3)/121, phi10 == 2*sqrt(3)/121]]

R=RealField(20); R(sqrt(3)/11) ;R(sqrt(3)/11) ; R( 2*sqrt(3)/121 );R(2*sqrt(3)/121 ) 

0.15746
0.15746
0.028629
0.028629

0.15746
0.15746
0.028629
0.028629