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Plotting the Muller-Brown Potential and Trajectory Along It

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Jacob Andrzejczyk
Jacob Andrzejczyk 2019 年 7 月 31 日
Commented: Walter Roberson 2019 年 12 月 11 日
Hey everyone,
I've been trying to plot the Muller-Brown potential using a contour plot and then simulation a trajectory along it. Here is the entirety of my code currently.
%known constants
A = [-200, -100, -170, 15];
a = [-1, -1, -6.5, 0.7];
b = [0, 0, 11, 0.6];
c = [-10, -10, -6.5, 0.7];
x0 = [1, 0, -0.5, -1];
y0 = [0, 0.5, 1.5, 1];
%commands to recreate the mapping of the PES
xx = linspace(-1.5, 1, 100);
yy = linspace(-0.5, 2, 100);
[X,Y] = meshgrid(xx,yy);
%Muller-Brown Potential Equation
W = A(1).*exp(a(1).*((X-x0(1)).^2) + b(1).*(X-x0(1)).*(Y-y0(1)) + c(1).*((Y-y0(1)).^2)) + A(2).*exp(a(2).*((X-x0(2)).^2) + b(2).*(X-x0(2)).*(Y-y0(2)) + c(2).*((Y-y0(2)).^2)) + A(3).*exp(a(3).*((X-x0(3)).^2) + b(3).*(X-x0(3)).*(Y-y0(3)) + c(3).*((Y-y0(3)).^2)) + A(4).*exp(a(4).*((X-x0(4)).^2) + b(4).*(X-x0(4)).*(Y-y0(4)) + c(4).*((Y-y0(4)).^2));
Z = W - min(W, [], 'all');
Z(Z >= 180) = NaN;
%plotting commands for the original PES
colormap('default')
contourf(X, Y, Z, 29)
hold on
%gradient equation
[DX, DY] = gradient(Z);
quiver(X, Y, DX, DY)
plot(1,0,'.', 'MarkerSize', 20) %plots starting point of the trajectory
hold off
%beginning of forward euler method to map trajectory on PES
h = 2*10^-5; %time step size
t = 2*10^4; %length of simulation
%N = t/h; %number of steps to be taken per simulation
N = 100; %shorter number of runs for practice
XX = zeros(1,100); %second value is the value of N, must be entered as an integer
YY = zeros(1,100);
YY(1) = 0; %initial condition for Y
XX(1) = 1; %initial condition for X
%equations for adjusting trajectory in x and y dimensions
%included for reference purposes currently
%dx = -DX + noise*sqrt(beta);
%dy = -DY + noise*sqrt(beta);
for i = 1:(N-1)
XX(i+1) = XX(i) + h*f(XX(i));
YY(i+1) = YY(i) + h*g(YY(i));
end
figure
colormap('default')
contourf(X, Y, Z, 29)
hold on
plot(XX,YY, 'o', 'MarkerSize', 5S)
hold off
function dx = f(DX)
noise = 1;
beta = 40;
dx = -DX + noise*sqrt(beta);
end
function dy = g(DY)
noise = 1;
beta = 40;
dy = -DY + noise*sqrt(beta);
end
The issue I am having is in the implementation of the forward Euler method. For some reason the values of XX and YY increase constantly by only the time step and I am really struggling to figure out how to fix that.
Thanks in advance if anyone can get this.

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件の回答 (1)

Kavya Vuriti
Kavya Vuriti 2019 年 8 月 9 日
編集済み: Kavya Vuriti 2019 年 8 月 9 日
The values “XX” and “YY” are not increasing constantly according to your code. There is a slight difference in the increments. Set the output format to long fixed-decimal format to view this difference.
format long
diff1=XX(3)-XX(2);
diff2=XX(2)-XX(1);
The values diff1 and diff2 are different.

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Walter Roberson
Walter Roberson 2019 年 12 月 11 日
this didnt help

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