混沌理科数学

发布日期: 2022-04-05

更新日期: 2022-10-05

文章字数: 7.9k

阅读时长: 43 分

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混沌的计算实验与分析

混沌（chaos）是指确定性动力学系统因对初值敏感而表现出的不可预测的、类似随机性的运动。
动力学系统的确定性是一个数学概念，指系统在任一时刻的状态被初始状态所决定。虽然根据运动的初始状态数据和运动规律能推算出任一未来时刻的运动状态，但由于初始数据的测定不可能完全精确，预测的结果必然出现误差，甚至不可预测。运动的可预测性是一个物理概念。一个运动即使是确定性的，也仍可为不可预测的，二者并不矛盾。牛顿力学的成功，特别是它在预言海王星上的成功，在一定程度上产生误解，把确定性和可预测性等同起来，以为确定性运动一定是可预测的。20世纪70年代后的研究表明，大量非线性系统中尽管系统是确定性的，却普遍存在着对运动状态初始值极为敏感、貌似随机的不可预测的运动状态——混沌运动。
混沌是指现实世界中存在的一种貌似无规律的复杂运动形态。共同特征是原来遵循简单物理规律的有序运动形态，在某种条件下突然偏离预期的规律性而变成了无序的形态。混沌可在相当广泛的一些确定性动力学系统中发生。混沌在统计特性上类似于随机过程，被认为是确定性系统中的一种内禀随机性。

一、混沌介绍

1.1迭代

1.1.1迭代

迭代是重复反馈过程的活动，其目的通常是为了逼近所需目标或结果。每一次对过程的重复称为一次“迭代”，而每一次迭代得到的结果会作为下一次迭代的初始值

迭代的定义

例题1.1

fortran实现

module diedai
    implicit none
contains
    real function complex_x(x) result(ans)  !创建迭代符合函数，参数为变量值
        real :: x,a=1.0
        integer,parameter :: number = 100 !迭代次数
        integer :: i
        ans=x
        do i=1,number,1
            ans = 2*(ans)+a
        end do
        return
    end function complex_x
end module diedai
program main
    use diedai
    implicit none
    real :: answer,x
    write(*,*) "请输入x的初始值："
    read(*,*) x
    answer=complex_x(x)
    write(*,*) "第100次迭代后，结果为：",answer
    stop
end program main

例1.2

module diedai
    implicit none
contains
    real function complex_x(x) result(ans)  !创建迭代符合函数，参数为变量值
        real :: x
        integer,parameter :: number = 100000000 !迭代次数
        integer :: i
        ans=x
        do i=1,number,1
            ans = sin(ans)
        end do
        return
    end function complex_x
end module diedai
program main
    use diedai
    implicit none
    real :: answer,x
    write(*,*) "请输入x的初始值："
    read(*,*) x
    answer=complex_x(x)
    write(*,*) "第100000000次迭代后，结果为：",answer
    stop
end program main

结果：n趋近于无穷大时，f(x)的值趋近于0，与初值无关。

练习题

module diedai
    implicit none
contains
    real function complex_x(x) result(ans)  !创建迭代符合函数，参数为变量值
        real :: x
        integer,parameter :: number = 10 !迭代次数
        integer :: i
        ans=x
        do i=1,number,1
            ans = ans*ans-1
        end do
        return
    end function complex_x
end module diedai
program main
    use diedai
    implicit none
    real :: answer,x
    write(*,*) "请输入x的初始值："
    read(*,*) x
    answer=complex_x(x)
    write(*,*) "第10次迭代后，结果为：",answer
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_sin(x,number)
        implicit none
        real :: x,ans
        integer :: number,i
        ans=x
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai.txt',status='old')
            write(fileid,"('迭代次数-----------迭代后值-----初值',F6.4)") ans
            do i=1,number,1
                ans=sin(ans)+cos(ans)
                write(fileid,"(I3,'  ',F6.4)") i,ans
            end do
            close(fileid)
        end if
        return
    end subroutine complex_sin

end module diedai

program main
    use diedai
    implicit none
    real :: x
    integer :: n
    write(*,*) "请输入迭代初值:"
    read(*,*) x
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_sin(x,n)
    write(*,*) "over!"
    stop
end program main

结果：通过不同的初值，经过迭代后都收敛到1.2597，图片可以看出初值越大，收敛速度越快。

1.1.2二元二次迭代

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(a,b,number)
        implicit none
        real :: a,b,ans_x,ans_y,ans_xi,ans_yi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary.txt',status='old')
            write(fileid,"('迭代次数---X----Y----a,b值',F6.2,F6.2)") a,b
            ans_x=0.3
            ans_y=0.5
            do i=1,number,1
                !迭代函数
                ans_xi=1-a*ans_x*ans_x+ans_y
                ans_yi=b*ans_x
                ans_x=ans_xi
                ans_y=ans_yi!这里需要用到ans_x，所以需要区分一下ans_yi与ans_y，ans_x与ans_xi
                write(fileid,"(I4,'  ',F6.2,'  ',F6.2)") i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代a,b的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

结果

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,number)
        implicit none
        real :: ans_x,ans_y,ans_xi,ans_yi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.3,F6.3)") ans_x,ans_y
            do i=1,number,1
                !迭代函数
                ans_xi=0.0164-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_y
                ans_yi=0.1-ans_x+ans_x*ans_y
                ans_x=ans_xi
                ans_y=ans_yi
                write(fileid,"(I4,'  ',F6.4,'  ',F6.4)") i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

常数0.0165

常数为0.0164

x0=0.215,yo=0.215,常数项0.0164

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,number)
        implicit none
        real :: ans_x,ans_y,ans_xi,ans_yi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.2,F6.2)") ans_x,ans_y
            do i=1,number,1
                !迭代函数
                ans_xi=-0.2-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+sna_y*ans_y
                ans_yi=0.1--ans_x+ans_x*ans_y
                ans_x=ans_xi
                ans_y=ans_yi
                write(fileid,*) i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

1.1.3线性随机IFS迭代

IFS是英文iterated function systems(迭代函数系统)的缩写，下面是一种常用的线性随机迭代算法

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,number)
        implicit none
        real :: ans_x,ans_y,ans_xi,ans_yi,random0_1
        integer :: number,i,e,f,n=9876
        real :: a(4,6)
        DATA((a(e,f),e=1,4),f=1,6) /0.00,0.85,0.20,-0.15,0.00,0.04,-0.26,0.28,0.00,-0.04,0.23,0.26,0.16,0.85,0.22,&
        0.24,0.00,0.00,0.01,0.00,0.00,1.60,1.60,0.44/
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.2,F6.2)") ans_x,ans_y
            do i=1,number,1
                !迭代函数
                n=mod(8121*n+28411,134456)
                random0_1=real(n)/134456
                write(*,*) random0_1
                if(random0_1<=0.01) then
                    ans_xi=a(1,1)*ans_x+a(1,2)*ans_y+a(1,5)
                    ans_yi=a(1,3)*ans_x+a(1,4)*ans_y+a(1,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.01.and.random0_1<=0.86) then
                    ans_xi=a(2,1)*ans_x+a(2,2)*ans_y+a(2,5)
                    ans_yi=a(2,3)*ans_x+a(2,4)*ans_y+a(2,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.86.and.random0_1<=0.93) then
                    ans_xi=a(3,1)*ans_x+a(3,2)*ans_y+a(3,5)
                    ans_yi=a(3,3)*ans_x+a(3,4)*ans_y+a(3,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else
                    ans_xi=a(4,1)*ans_x+a(4,2)*ans_y+a(4,5)
                    ans_yi=a(4,3)*ans_x+a(4,4)*ans_y+a(4,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                end if
                write(fileid,*) i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,number)
        implicit none
        real :: ans_x,ans_y,ans_xi,ans_yi,random0_1
        integer :: number,i,e,f,n=9876
        real :: a(8,6)
        DATA((a(e,f),e=1,8),f=1,6) /0.08,0.0801,0.75,0.9430,-0.4020,0.2170,0.2620,0.2200,-0.0310,0.0212,0.00,&
        0.00,0.00,-0.0520,-0.1050,0.00,0.084,-0.08,0.00,0.00,0.00,0.075,0.1140,0.00,0.0306,0.0212,0.5300,0.4740,&
        0.4020,0.1500,0.2410,0.4300,5.17,6.62,-0.3570,-1.98,15.5130,3.000,-0.4730,14.60,7.9700,9.400,1.1060,&
        -0.6500,4.5880,5.7400,3.0450,4.2860/
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.2,F6.2)") ans_x,ans_y
            do i=1,number,1
                !迭代函数
                n=mod(8121*n+28411,134456)
                random0_1=real(n)/134456
                if(random0_1<=0.03) then
                    ans_xi=a(1,1)*ans_x+a(1,2)*ans_y+a(1,5)
                    ans_yi=a(1,3)*ans_x+a(1,4)*ans_y+a(1,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.03.and.random0_1<=0.055) then
                    ans_xi=a(2,1)*ans_x+a(2,2)*ans_y+a(2,5)
                    ans_yi=a(2,3)*ans_x+a(2,4)*ans_y+a(2,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.055.and.random0_1<=0.275) then
                    ans_xi=a(3,1)*ans_x+a(3,2)*ans_y+a(3,5)
                    ans_yi=a(3,3)*ans_x+a(3,4)*ans_y+a(3,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.275.and.random0_1<=0.520) then
                    ans_xi=a(4,1)*ans_x+a(4,2)*ans_y+a(4,5)
                    ans_yi=a(4,3)*ans_x+a(4,4)*ans_y+a(4,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.520.and.random0_1<=0.730) then
                    ans_xi=a(5,1)*ans_x+a(5,2)*ans_y+a(5,5)
                    ans_yi=a(5,3)*ans_x+a(5,4)*ans_y+a(5,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.730.and.random0_1<=0.80) then
                    ans_xi=a(6,1)*ans_x+a(6,2)*ans_y+a(6,5)
                    ans_yi=a(6,3)*ans_x+a(6,4)*ans_y+a(6,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else if(random0_1>0.80.and.random0_1<=0.9) then
                    ans_xi=a(7,1)*ans_x+a(7,2)*ans_y+a(7,5)
                    ans_yi=a(7,3)*ans_x+a(7,4)*ans_y+a(7,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                else
                    ans_xi=a(8,1)*ans_x+a(8,2)*ans_y+a(8,5)
                    ans_yi=a(8,3)*ans_x+a(8,4)*ans_y+a(8,6)
                    ans_x=ans_xi
                    ans_y=ans_yi
                end if
                write(fileid,*) i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

线性随机IFS迭代也是迭代的一种，由于其是线性的，所以目前的研究成果相对比较多。一般把上面的似树叶，山的点集叫做（近似的混沌）吸引子

1.1.4三元二次迭代

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,ans_z,number)
        implicit none
        real :: ans_x,ans_y,ans_z,ans_xi,ans_yi,ans_zi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y,ans_z
            do i=1,number,1
                !迭代函数
                ans_xi=-0.3-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_y
                ans_yi=0.2-ans_x+ans_x*ans_y
                ans_zi=-ans_x*ans_y-ans_y*ans_z-ans_x*ans_z
                ans_x=ans_xi
                ans_y=ans_yi
                ans_z=ans_zi
                write(fileid,*) i,ans_x,ans_y,ans_z
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b,c
    integer :: n
    write(*,*) "请输入迭代x,y,z的初值:"
    read(*,*) a,b,c
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,c,n)
    write(*,*) "over!"
    stop
end program main

1.1.5三角函数迭代

三角函数迭代一般绘制出的吸引子比较复杂，理论分析也比多项式迭代困难。
Clifford系统

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,ans_z,number)
        implicit none
        real :: ans_x,ans_y,ans_z,ans_xi,ans_yi,ans_zi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y,ans_z
            do i=1,number,1
                !迭代函数
                ans_xi=sin(2.24*ans_y)-ans_z*cos(0.43*ans_x)
                ans_yi=ans_z*sin(-0.65*ans_x)-cos(-2.43*ans_y)
                ans_zi=1.0*sin(0.43*ans_x)
                ans_x=ans_xi
                ans_y=ans_yi
                ans_z=ans_zi
                if(number>3000) then
                write(fileid,*) i,ans_x,ans_y,ans_z
                end if
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b,c
    integer :: n
    write(*,*) "请输入迭代x,y,z的初值:"
    read(*,*) a,b,c
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,c,n)
    write(*,*) "over!"
    stop
end program main

练习：把例1.9迭代表达式中的正弦、余弦函数都使用泰勒公式展开，然后代替三角函数进行迭代，观察实验结果

1.1.6小波函数对二元二次函数迭代的控制

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(k,u)
        implicit none
        real :: ans_x=-5,ans_y=0
        real :: k,u
        open(unit=fileid,file='data_diedai_wavelet.txt',status='new')
        write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.2,F6.2)") ans_x,ans_y
        do while(.true.)
            !迭代函数
            if(ans_x>5) exit
            ans_y=k(1-ans_x*ans_x)*exp(-u*(ans_x-0.5)*(ans_x-0.5))
            write(fileid,*) ans_x,ans_y
            ans_x=ans_x+0.01
        end do
        close(fileid)

        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: k,u
    write(*,*) "请输入迭代k,u值:"
    read(*,*) k,u
    call complex_binary(k,u)
    write(*,*) "over!"
    stop
end program main

k=1,u=1

k=1,u=0.2

k=1,u=0.8

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,number)
        implicit none
        real :: ans_x,ans_y,ans_xi,ans_yi,k=1,u=1
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.2,F6.2,F6.2)") ans_x,ans_y
            do i=1,number,1
                !迭代函数
                ans_xi=(2-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_Y)*k*(1-ans_x*ans_x)*&
                exp(-u*(ans_x-0.5)*(ans_x-0.5))
                ans_yi=(0.2-ans_x+ans_x*ans_y)*k*(1-ans_x*ans_x)*exp(-u*(ans_x-0.5)*(ans_x-0.5))
                ans_x=ans_xi
                ans_y=ans_yi
                write(fileid,*) i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

x0=6,y0=6,k=1,u=1

x0=6,y0=6,k=1.73,u=1

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,number)
        implicit none
        real :: ans_x,ans_y,ans_xi,ans_yi,k=1.73,u=1
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----X0,Y0值',F6.2,F6.2,F6.2)") ans_x,ans_y
            do i=1,number,1
                !迭代函数
                ans_xi=(2-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_Y)*k*(1-ans_x*ans_x)*&
                exp(-0.5*ans_x*ans_x)
                ans_yi=(0.2-ans_x+ans_x*ans_y)*k*(1-ans_x*ans_x)*exp(-0.5*ans_x*ans_x)
                ans_x=ans_xi
                ans_y=ans_yi
                write(fileid,*) i,ans_x,ans_y
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,n)
    write(*,*) "over!"
    stop
end program main

x0=6,y0=6,k=-0.675

x0=6,y0=6,k=-7.25

x0=6,y0=6,k=0.68

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,ans_z,number)
        implicit none
        real :: ans_x,ans_y,ans_z,ans_xi,ans_yi,ans_zi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z0----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y,ans_z
            do i=1,number,1
                !迭代函数
                ans_xi=(2-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_Y)*k*(1-ans_x*ans_x)*&
                exp(-0.5*ans_x*ans_x)
                ans_yi=(0.2-ans_x+ans_x*ans_y)*k*(1-ans_x*ans_x)*exp(-0.5*ans_x*ans_x)
                ans_zi=2-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_Y
                ans_x=ans_xi
                ans_y=ans_yi
                ans_z=ans_zi
                write(fileid,*) i,ans_x,ans_y,ans_z
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b,c
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b,c
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,c,n)
    write(*,*) "over!"
    stop
end program main

1.2混沌与分岔

1.2.1混沌的描述性定义

  module diedai
      implicit none
      integer,parameter :: fileid=10
  contains
      subroutine complex_binary(ans_x)
          implicit none
          real :: ans_x,ans_y,u=4.0
          real,parameter :: h=0.045

          open(unit=fileid,file='data_diedai_binary2.txt',status='old')
          write(fileid,"('迭代次数---X----,X0值',F6.2)") ans_x
          do while(.true.)
              !迭代函数
              ans_y=u*ans_x*(1-ans_x)
              write(fileid,*) ans_x,ans_y
              ans_x=ans_x+h
              if(ans_x>1) exit
          end do
          close(fileid)
          return
      end subroutine complex_binary

  end module diedai

  program main
      use diedai
      implicit none
      real :: x
      write(*,*) "请输入迭代x的初值:"
      read(*,*) x
      call complex_binary(x)
      write(*,*) "over!"
      stop
  end program main

x0=0.100001

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,ans_z,number)
        implicit none
        real :: ans_x,ans_y,ans_z,ans_xi,ans_yi,ans_zi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y,ans_z
            do i=1,number,1
                !迭代函数
                ans_xi=-0.3-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_y
                ans_yi=0.2-ans_x+ans_x*ans_y
                ans_zi=-ans_x*ans_y-ans_y*ans_z-ans_x*ans_z
                ans_x=ans_xi
                ans_y=ans_yi
                ans_z=ans_zi
                write(fileid,*) i,ans_x,ans_y,ans_z
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b,c
    integer :: n
    write(*,*) "请输入迭代x,y,z的初值:"
    read(*,*) a,b,c
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,c,n)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,ans_z,number)
        implicit none
        real :: x,y,z
        real :: ans_x,ans_y,ans_z,ans_xi,ans_yi,ans_zi
        real :: ans_x1=0.2001,ans_y1=0.2,ans_z1=0.2,ans_xi1,ans_yi1,ans_zi1
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y,ans_z
            do i=1,number,1
                !迭代函数
                ans_xi=-0.3-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_y
                ans_yi=0.2-ans_x+ans_x*ans_y
                ans_zi=-ans_x*ans_y-ans_y*ans_z-ans_x*ans_z
                ans_x=ans_xi
                ans_y=ans_yi
                ans_z=ans_zi
                ans_xi1=-0.3-ans_x1+ans_x1*ans_x1-ans_x1*ans_y1+ans_y1+ans_y1*ans_y1
                ans_yi1=0.2-ans_x1+ans_x1*ans_y1
                ans_zi1=-ans_x1*ans_y1-ans_y1*ans_z1-ans_x1*ans_z1
                ans_x1=ans_xi1
                ans_y1=ans_yi1
                ans_z1=ans_zi1
                x=ans_x1-ans_x
                y=ans_y1-ans_y
                z=ans_z1-ans_z
                write(fileid,*) i,x,y,z
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b,c
    integer :: n
    write(*,*) "请输入迭代x,y,z的初值:"
    read(*,*) a,b,c
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,c,n)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,number)
        implicit none
        real :: ans_x
        real :: ans_y
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
            do i=1,number,1
                !迭代函数
                ans_y=4*ans_x*(1-ans_x)
                write(fileid,*) i,ans_y
                ans_x=ans_x+0.01
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,n)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y,ans_z,number)
        implicit none
        real :: ans_x,ans_y,ans_z,ans_xi,ans_yi,ans_zi
        integer :: number,i
        if(number<0) then
            write(*,*) "迭代次数有误，stop"
            stop
        else
            open(unit=fileid,file='data_diedai_binary2.txt',status='old')
            write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y,ans_z
            do i=1,number,1
                !迭代函数
                ans_xi=-0.3-ans_x+ans_x*ans_x-ans_x*ans_y+ans_y+ans_y*ans_y
                ans_yi=0.2-0.99999*ans_x+ans_x*ans_y
                ans_zi=-ans_x*ans_y-ans_y*ans_z-ans_x*ans_z
                ans_x=ans_xi
                ans_y=ans_yi
                ans_z=ans_zi
                write(fileid,*) i,ans_x,ans_y,ans_z
            end do
            close(fileid)
        end if
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b,c
    integer :: n
    write(*,*) "请输入迭代x,y,z的初值:"
    read(*,*) a,b,c
    write(*,*) "请输入迭代次数:"
    read(*,*) n
    call complex_binary(a,b,c,n)
    write(*,*) "over!"
    stop
end program main

1.2.2Logistic映射

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do i=1,100,1
            !迭代函数
            ans_x=2*ans_x*(1-ans_x)
            write(fileid,*) i,ans_x
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do i=1,100,1
            !迭代函数
            ans_x=4.0*ans_x*(1-ans_x)
            write(fileid,*) i,ans_x
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do i=1,1000,1
            !迭代函数
            ans_x=3.8*ans_x*(1-ans_x)
            write(fileid,*) i,ans_x
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

x0=0.001

增加0.0001，u=3.8001,x0=0.001

继续增加，u=3.801,x0=0.001,出现稳定点

1.2.3不动点与周期点

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do i=1,1000,1
            !迭代函数
            ans_x=(1+ans_x)**(1.0/3.0)
            write(fileid,*) i,ans_x
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do i=1,1000,1
            !迭代函数
            ans_x=3.2*ans_x*(1-ans_x)
            write(fileid,*) i,ans_x
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do i=1,1000,1
            !迭代函数
            ans_x=3.5*ans_x*(1-ans_x)
            write(fileid,*) i,ans_x
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

1.2.4分岔图

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x,u=3
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do while(.true.)
            !迭代函数
            u=u+0.001
            ans_x=0.01
            if(u>4) exit
            do i=1,100,1
                ans_x=u*ans_x*(1-ans_x)
                write(fileid,*) u,ans_x
            end do
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x)
        implicit none
        real :: ans_x,u=-2
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x
        do while(.true.)
            !迭代函数
            u=u+0.001
            ans_x=0.001
            if(u>1.5) exit
            do i=1,100,1
                ans_x=2.0*(1.0-ans_x**2)*exp(-0.5*(ans_x+u)**2)
                write(fileid,*) u,ans_x
            end do
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a
    integer :: n
    write(*,*) "请输入迭代x的初值:"
    read(*,*) a
    call complex_binary(a)
    write(*,*) "over!"
    stop
end program main

module diedai
    implicit none
    integer,parameter :: fileid=10
contains
    subroutine complex_binary(ans_x,ans_y)
        implicit none
        real :: ans_x,ans_y,a=-2,b=1.0
        real :: ans_xi,ans_yi
        integer :: i
        open(unit=fileid,file='data_diedai_binary2.txt',status='old')
        write(fileid,"('迭代次数---X----Y----Z----X0,Y0,Z0值',F6.2,F6.2,F6.2)") ans_x,ans_y
        do while(.true.)
            !迭代函数
            a=a+0.01
            ans_x=0.01
            ans_y=0.01
            if(a>2.0) exit
            do i=1,5,1
                ans_xi=1-a*ans_x**2+ans_y
                ans_yi=b*ans_x
                ans_x=ans_xi
                ans_y=ans_yi
                write(fileid,*) a,ans_y
            end do
        end do
        close(fileid)
        return
    end subroutine complex_binary

end module diedai

program main
    use diedai
    implicit none
    real :: a,b
    integer :: n
    write(*,*) "请输入迭代x,y的初值:"
    read(*,*) a,b
    call complex_binary(a,b)
    write(*,*) "over!"
    stop
end program main

1.3混沌的数学特征

1.3.1关联维数

1.3.2Lyapunov指数

module Lyapunov
    integer,parameter :: fileid = 10
    integer,parameter :: N = 500
    contains
    subroutine sub1(x)
        implicit none
        integer :: i
        real x,y,u,yy
        real p(N)
        open(10,file="data.txt")
        do u=3.5,3.9,0.001
            do i=1,N,1
                y=u*x*(1-x)
                p(i) = u-2*u*x
                x=y
            end do
            p=abs(p) !一维函数，取模后即是特征值
            p=log(p) !取对数，以10为底
            yy=sum(p)/N !对数组p求和
            write(10,*) u,yy
        end do

        return
    end subroutine sub1
end module Lyapunov

program main
    use Lyapunov
    implicit none
    real :: x = 0.1
    call sub1(x)
    stop
end program main

图1.22

module package
    implicit none
    integer,parameter :: fileid = 10
    integer,parameter :: N = 50
contains
    subroutine sub1(x,y)
        implicit none
        real(kind=8) :: x,y,x1,y1
        real(kind=8) :: a(2,2),b(2,2),e(2,2),f(2,2)
        real(kind=8) :: xx(N),yy(N),ans,nabuda1=0.4,nabuda2=-1.6
        integer(kind=4) :: i,c,d
        DATA((a(c,d),c=1,2),d=1,2) /-0.56,0.3,1.0,0/
        DATA((b(c,d),c=1,2),d=1,2) /0,0.3,1.0,0/
        DATA((e(c,d),c=1,2),d=1,2) /1,0,0,1/
        xx(1)=x
        yy(1)=y
        do i=2,N,1
            x1=1.0-1.4*x**2+y
            y1=0.3*x
            x=x1
            y=y1
            xx(i)=x
            yy(i)=y
            b(1,1) = -2.8*xx(i)
            a=matmul(a,b) !矩阵相乘
            !write(*,*) abs(f(1,1)*f(2,2)-f(1,2)*f(2,1))
            !write(*,*) nabuda
        end do
        !ans=log(abs(a(1,1)*a(2,2)-a(1,2)*a(2,1)))
        !write(*,*) ans/50.0
        !do nabuda=-2.0,1.0,0.1
            !f = nabuda*e-a
            !write(*,*) abs(f(1,1)*f(2,2)-f(1,2)*f(2,1))
            !if(abs(f(1,1)*f(2,2)-f(1,2)*f(2,1))<0.09) then
                !write(*,*) log(nabuda)
            !end if
        !end do
        write(*,*) abs((a(1,1)*nabuda1*a(2,2)*nabuda2-a(1,2)*nabuda1*a(2,1)*nabuda2))
        write(*,*) log(0.4)/50.0
        return
    end subroutine sub1
end module package

program Lyapunov
    use package
    implicit none
    real(kind=8) :: x,y
    x=0.2
    y=0.3
    call sub1(x,y)
    stop
end program Lyapunov

module Lyapunov
    integer,parameter :: fileid = 10
    integer,parameter :: N = 500
    contains
    subroutine sub1(x)
        implicit none
        integer :: i
        real x,y,u,yy
        real p(N)
        open(10,file="data.txt")
        do u=-1.5,1.5,0.001
            do i=1,N,1
                y=2.0*(1.0*x**2)*exp(-0.5*(x+u)**2) ! 原函数用来递归
                p(i) = -4.0*x*exp(-0.5*(x+u)**2)-2.0*(1.0*x**2)*exp(-0.5*(x+u)**2)*(x+u)
                x=y
            end do
            p=abs(p) !一维函数，取模后即是特征值
            p=log(p) !取对数，以10为底
            yy=sum(p)/N !对数组p求和
            write(10,*) u,yy
        end do

        return
    end subroutine sub1
end module Lyapunov

program main
    use Lyapunov
    implicit none
    real :: x = 0.1
    call sub1(x)
    stop
end program main

练习题1

练习题2

module Lyapunov
    integer,parameter :: fileid = 10
    integer,parameter :: N = 5
    integer,parameter :: number = 2
contains
    subroutine sub1()
        implicit none
        integer :: i,c,d
        real :: x=1.0,y=2.0,x1,y1,xx(N),yy(N)
        real :: a(2,2),b(2,2)
        real :: p(N)
        open(10,file="data.txt")
        x1=x
        y1=y
        DATA((a(c,d),c=1,2),d=1,2) /-1,1,5,0/
        DATA((b(c,d),c=1,2),d=1,2) /0,0,0,0/
        do i=2,N,1
            x1=0.0165-x1+x1**2-x1*y1+y1+y1**2
            y1=0.1-x1+x1*y1
            x=x1
            y=y1
            xx(i)=x
            yy(i)=y
            b(1,1)=-1+2*xx(i)-yy(i)
            b(1,2)=1+2*yy(i)
            b(2,1)=-1+yy(i)
            a=a*b
            write(*,*) a
        end do

        return
    end subroutine sub1
end module Lyapunov

program main
    use Lyapunov
    implicit none
    call sub1()
    stop
end program main

1.3.3混沌的数学定义

module Lyapunov
    integer,parameter :: fileid = 10
    integer,parameter :: N = 100
contains
    subroutine sub1()
        implicit none
        integer :: i,j
        real(kind=4) :: x,x1
        real(kind=4) :: y(N)
        do x=0.737,0.999,0.00001
            x1=x
            open(10,file="data.txt")
            do i=1,N,1
                y(i)=3.8*x1*(1.0-x1)
                x1=y(i)
            end do
                do j=1,N,1
                    if(abs(x-y(j))<0.000001) then
                        write(10,*) 'j=',j,'x=',x
                    end if
                end do
        end do
        return
    end subroutine sub1
end module Lyapunov

program main
    use Lyapunov
    implicit none
    call sub1()
    stop
end program main

1.4Lorenz系统族

1.4.1从微分方程到迭代

 module Lyapunov
     integer,parameter :: fileid = 10
     integer,parameter :: N = 10000
     real,parameter :: dt = 0.005
 contains
     subroutine sub1(x,y,z)
         implicit none
         integer :: i
         real :: x,y,z,x1,y1,z1
         real :: a=10.0,b=8.0/3.0,c=28.0
         open(10,file="data.txt")
         do i=1,N,1
             x1=x+a*(y-x)*dt
             y1=y+(c*x-x*z-y)*dt
             z1=z+(x*y-b*z)*dt
             x=x1
             y=y1
             z=z1
             write(10,*) i,x,y,z
         end do
         return
     end subroutine sub1
 end module Lyapunov

 program main
     use Lyapunov
     implicit none
     real :: x,y,z
     write(*,*) "请输入x,y,z的初值："
     read(*,*) x,y,z
     call sub1(x,y,z)
     write(*,*) "over"
     stop
 end program main

1.4.2Lorenz系统的简单分析(洛伦兹系统)

module Lyapunov
    integer,parameter :: fileid = 10
    integer,parameter :: N = 10000
    real,parameter :: dt = 0.005
contains
    subroutine sub1(x,y,z)
        implicit none
        integer :: i
        real :: x,y,z,x1,y1,z1
        real :: a=10.0,b=8.0/3.0,c=28.0
        open(10,file="data.txt")
        do i=1,N,1
            x1=x+a*(y-x)*dt
            y1=y+(c*x-x*z-y)*dt
            z1=z+(x*y-b*z)*dt
            x=x1
            y=y1
            z=z1
            write(10,*) i,x,y,z
        end do
        return
    end subroutine sub1
end module Lyapunov

program main
    use Lyapunov
    implicit none
    real :: x,y,z
    write(*,*) "请输入x,y,z的初值："
    read(*,*) x,y,z
    call sub1(x,y,z)
    write(*,*) "over"
    stop
end program main

x,y平面

x，z平面

y，z平面

1.4.3陈系统与吕系统

练习题1

module Lorenz
    implicit none
    integer,parameter :: fileid = 10
    integer,parameter :: N = 10000
    integer,parameter :: size = 3
contains
    subroutine sub1(x,fx,size,a)
        implicit none
        integer :: i
        real :: x(size),fx(size)
        real :: a
        integer :: size
        fx(1)=(25.0*a+10.0)*(x(2)-x(1))
        fx(2)=(28.0-35.0*a)*x(1)-x(1)*x(3)+(29.0*a-1.0)*x(2)
        fx(3)=x(1)*x(2)-((a+8.0)/3.0)*x(3)
        return
    end subroutine sub1

    subroutine rk(x,h,size,a)
        implicit none
        real :: x(size),xx(size),h,fx(size)
        real :: kx1(size),kx2(size),kx3(size),kx4(size)
        real :: a
        integer :: size
        xx=x
        call sub1(xx,fx,size,a)
        kx1=h*fx
        xx=x+0.5*kx1
        call sub1(xx,fx,size,a)
        kx2=h*fx
        xx=x+0.5*kx2
        call sub1(xx,fx,size,a)
        kx3=h*fx
        xx=x+kx3
        call sub1(xx,fx,size,a)
        kx4=h*fx
        x=x+(kx1+2.0*kx2+2.0*kx3+kx4)/6.0
        return
    end subroutine rk
end module Lorenz

program main
    use Lorenz
    implicit none
    real :: x(size),h=0.001,a
    integer :: i
    write(*,*) "请输入x,y,z的初值："
    read(*,*) x(1),x(2),x(3)
    write(*,*) "请输入a的值："
    read(*,*) a
    open(10,file="data.txt")
    do i=1,N,1
        call rk(x,h,size,a)
        if(i>1000) then
            write(10,*) x
        end if
    end do
    write(*,*) "over"
    stop
end program main

x,y,z=1,a=0.5

练习题2

module Lorenz
    implicit none
    integer,parameter :: fileid = 10
    integer,parameter :: N = 10000
    integer,parameter :: size = 3
contains
    subroutine sub1(x,fx,size)
        implicit none
        integer :: i
        real :: x(size),fx(size)
        real :: a=10.0,b=3.0,c=28.0
        integer :: size
        fx(1)=a*(x(2)-x(1))
        fx(2)=(c-a)*x(1)-x(1)*x(3)+c*x(2)
        fx(3)=x(1)*x(2)-b*x(3)
        return
    end subroutine sub1

    subroutine rk(x,h,size)
        implicit none
        real :: x(size),xx(size),h,fx(size)
        real :: kx1(size),kx2(size),kx3(size),kx4(size)
        integer :: size
        xx=x
        call sub1(xx,fx,size,a)
        kx1=h*fx
        xx=x+0.5*kx1
        call sub1(xx,fx,size,a)
        kx2=h*fx
        xx=x+0.5*kx2
        call sub1(xx,fx,size,a)
        kx3=h*fx
        xx=x+kx3
        call sub1(xx,fx,size,a)
        kx4=h*fx
        x=x+(kx1+2.0*kx2+2.0*kx3+kx4)/6.0
        return
    end subroutine rk
end module Lorenz

program main
    use Lorenz
    implicit none
    real :: x(size),h=0.001
    integer :: i
    write(*,*) "请输入x,y,z的初值："
    read(*,*) x(1),x(2),x(3)
    open(10,file="data.txt")
    do i=1,N,1
        call rk(x,h,size)
        if(i>1000) then
            write(10,*) x
        end if
    end do
    write(*,*) "over"
    stop
end program main

x,y,z=1

练习3

module Lyapunov
    integer,parameter :: fileid = 10
    integer,parameter :: N = 10000
    real,parameter :: dt = 0.005
contains
    subroutine sub1(x,y,z)
        implicit none
        integer :: i
        real :: x,y,z,x1,y1,z1
        real :: a=10.0,b=8.0/3.0,c=28.0
        open(10,file="data.txt")
        do i=1,N,1
            x1=x+a*(y-x)*dt
            y1=y+(-x*z+c*y)*dt
            z1=z+(x*y-b*z)*dt
            x=x1
            y=y1
            z=z1
            write(10,*) i,x,y,z
        end do
        return
    end subroutine sub1
end module Lyapunov

program main
    use Lyapunov
    implicit none
    real :: x,y,z
    write(*,*) "请输入x,y,z的初值："
    read(*,*) x,y,z
    call sub1(x,y,z)
    write(*,*) "over"
    stop
end program main

二、混沌的相关研究领域

2.1混沌系统

2.1.1常微分方程组

2.1.2动力系统

2.1.3人工神经网络

2.2人工神经网络中的混沌

2.2.1反馈神经网络中的混沌

2.2.2细胞神经网络中的混沌

2.2.3一种混沌神经元网络模型

2.2.4图像作为混沌神经网络的权值矩阵

2.3分行与图形设计

2.3.1Julia集合与Mandelbrot集合

2.3.2牛顿迭代

2.3.3三维吸引子设计

2.4混沌加密

2.5智能模拟中的混沌

2.5.1混沌人工脑模型

2.5.2IFS分形图覆盖相交交点变化曲线

2.5.3交点变化曲线变化率的研究

三、混沌本质的探索

3.1多种迭代形式

3.1.1交叉迭代

3.1.2串行迭代

3.1.3迭代过程中表达式发生变化的情况

3.2混沌吸引子的分布

3.2.1混沌吸引子的重心位置

3.2.2Lorenz系统的边界估计

3.3分岔现象

3.3.1三角函数的分岔现象

3.3.2小波函数的分岔现象

3.4混沌同步

3.4.1线性耦合反馈同步

3.4.2参数调节子自适用同步

3.5系统的混沌化

3.5.1迭代式乘以倍数后产生混沌

3.5.2离散时间系统反馈混沌化

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混沌的计算实验与分析

混沌的计算实验与分析

一、混沌介绍

1.1迭代

1.1.1迭代

1.1.2二元二次迭代

1.1.3线性随机IFS迭代

1.1.4三元二次迭代

1.1.5三角函数迭代

1.1.6小波函数对二元二次函数迭代的控制

1.2混沌与分岔

1.2.1混沌的描述性定义

1.2.2Logistic映射

1.2.3不动点与周期点

1.2.4分岔图

1.3混沌的数学特征

1.3.1关联维数

1.3.2Lyapunov指数

1.3.3混沌的数学定义

1.4Lorenz系统族

1.4.1从微分方程到迭代

1.4.2Lorenz系统的简单分析(洛伦兹系统)

1.4.3陈系统与吕系统

二、混沌的相关研究领域

2.1混沌系统

2.1.1常微分方程组

2.1.2动力系统

2.1.3人工神经网络

2.2人工神经网络中的混沌

2.2.1反馈神经网络中的混沌

2.2.2细胞神经网络中的混沌

2.2.3一种混沌神经元网络模型

2.2.4图像作为混沌神经网络的权值矩阵

2.3分行与图形设计

2.3.1Julia集合与Mandelbrot集合

2.3.2牛顿迭代

2.3.3三维吸引子设计

2.4混沌加密

2.5智能模拟中的混沌

2.5.1混沌人工脑模型

2.5.2IFS分形图覆盖相交交点变化曲线

2.5.3交点变化曲线变化率的研究

三、混沌本质的探索

3.1多种迭代形式

3.1.1交叉迭代

3.1.2串行迭代

3.1.3迭代过程中表达式发生变化的情况

3.2混沌吸引子的分布

3.2.1混沌吸引子的重心位置

3.2.2Lorenz系统的边界估计

3.3分岔现象

3.3.1三角函数的分岔现象

3.3.2小波函数的分岔现象

3.4混沌同步

3.4.1线性耦合反馈同步

3.4.2参数调节子自适用同步

3.5系统的混沌化

3.5.1迭代式乘以倍数后产生混沌

3.5.2离散时间系统反馈混沌化

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