Old Treasure
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C Program to compute basin of attraction of
C a Forced Damped Pendulum
C using 4th order Runge-Kutta Method
C A number of steps using RK4 is performed and the sign
C of angular velocity is printed on stdout
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
program fdp_solve
implicit none
integer P
parameter(P=1010000)
real*8 T0,TF,X10,X20
integer STEPS,PSIZE
real*8 T(P),X1(P),X2(P)
real*8 Energy
real*8 omega_0,omega,gamma,a_0,omega_02,omega2
common /params/omega_0,omega,gamma,a_0,omega_02,omega2
integer i,Nstart
real *8 PI2,PI4,PI
parameter(PI2=6.283185307179D0,PI4=2.0D0*PI2,PI=PI2/2.0D0)
omega_0 = 1.0D0
omega = 2.0D0
omega_02 = omega_0*omega_0
omega2 = omega *omega
gamma = 0.2D0
STEPS = 10000
T0 = 0.0D0
TF = 200.0D0
print *, '# Enter A and no of initial conditions:'
read(5,*) a_0, Nstart
C The Calculation:
PSIZE=P
do i = 1, Nstart
C Random initial conditions with -PI<theta<PI, -2 PI<dtheta/dt<2 PI
X10 = PI2*(rand()-0.5D0)
X20 = PI4*(rand()-0.5D0)
C X10 = PI *rand()
C X20 = PI2*rand()
call RK(T,X1,X2,T0,TF,X10,X20,STEPS,PSIZE)
if( X2(STEPS) .GT. 0.0D0)then
print *,X10,X20,1
else
print *,X10,X20,-1
endif
enddo
end
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C The functions f1,f2(t,x1,x2) provided by the user
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
real*8 function f1(t,x1,x2)
implicit none
real*8 t,x1,x2
f1=x2 !dx1/dt= v = x2
end
real*8 function f2(t,x1,x2)
implicit none
real*8 omega_0,omega,gamma,a_0,omega_02,omega2
common /params/omega_0,omega,gamma,a_0,omega_02,omega2
real*8 t,x1,x2
f2=-(omega_02+2.0D0*a_0*dcos(omega*t))*dsin(x1)-gamma*x2
end
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C RK(T,X1,X2,T0,TF,X10,X20,STEPS,PSIZE) is the driver
C for the Runge-Kutta integration routine RKSTEP
C Input: Initial and final times T0,T1
C Initial values at t=T0 X10,X20
C Number of steps of integration STEPS
C Size of arrays T,X1,X2
C Output: real arrays T(PSIZE),X1(PSIZE),X2(PSIZE) where
C T(1) = T0 X1(1) = X10 X2(1) = X20
C X1(i) = X1(at t=T(i)) X2(i) = X2(at t=T(i))
C T(STEPS+1)=TF
C Therefore we must have PSIZE>STEPS
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
subroutine RK(T,X1,X2,T0,TF,X10,X20,STEPS,PSIZE)
implicit none
integer STEPS,PSIZE
real*8 T(PSIZE),X1(PSIZE),X2(PSIZE),T0,TF,X10,X20
real*8 dt
real*8 TS,X1S,X2S !values of time and X1,X2 at given step
integer i
C Some checks:
if(STEPS .le. 1 )then
print *,'rk: STEPS must be >= 1'
stop
endif
if(STEPS .ge. PSIZE)then
print *,'rk: STEPS must be < ',PSIZE
stop
endif
C Initialize variables:
dt = (TF-T0)/STEPS
T (1) = T0
X1(1) = X10
X2(1) = X20
TS = T0
X1S = X10
X2S = X20
C Make RK steps: The arguments of RKSTEP are replaced with the new ones!
do i=2,STEPS+1
call RKSTEP(TS,X1S,X2S,dt)
T(i) = TS
X1(i) = X1S
X2(i) = X2S
enddo
end
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C Subroutine RKSTEP(t,x1,x2,dt)
C Runge-Kutta Integration routine of ODE
C dx1/dt=f1(t,x1,x2) dx2/dt=f2(t,x1,x2)
C User must supply derivative functions:
C real function f1(t,x1,x2)
C real Function f2(t,x1,x2)
C Given initial point (t,x1,x2) the routine advnaces it
C by time dt.
C Input : Inital time t and function values x1,x2
C Output: Final time t+dt and function values x1,x2
C Careful!: values of t,x1,x2 are overwritten...
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
subroutine RKSTEP(t,x1,x2,dt)
implicit none
real*8 t,x1,x2,dt
real*8 f1,f2
real*8 k11,k12,k13,k14,k21,k22,k23,k24
real*8 h,h2,h6,pi,pi2
parameter(pi =3.14159265358979324D0)
parameter(pi2=6.28318530717958648D0)
h=dt !h =dt, integration step
h2=0.5D0*h !h2=h/2
h6=0.166666666666666666666666D0*h !h6=h/6
k11=f1(t,x1,x2)
k21=f2(t,x1,x2)
k12=f1(t+h2,x1+h2*k11,x2+h2*k21)
k22=f2(t+h2,x1+h2*k11,x2+h2*k21)
k13=f1(t+h2,x1+h2*k12,x2+h2*k22)
k23=f2(t+h2,x1+h2*k12,x2+h2*k22)
k14=f1(t+h ,x1+h *k13,x2+h *k23)
k24=f2(t+h ,x1+h *k13,x2+h *k23)
t =t+h
x1=x1+h6*(k11+2.0D0*(k12+k13)+k14)
x2=x2+h6*(k21+2.0D0*(k22+k23)+k24)
if( x1 .gt. pi) x1 = x1 - pi2
if( x1 .lt. -pi) x1 = x1 + pi2
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