I am trying to draw the backward and forward points of the slow invariant manifold (SIM) from the equilibrium point using the following code of Intrinsic low dimensional manifold (ILDM), but didn’t get all backward points. could anyone identify the mistake.

by using below code i am getting following graph which is not plotting the backward points

i am trying to get following result, but unfortunately there is a mistake in my code.

code:

% A + Z <=====> AZ
% AZ + B <=====> AB + Z
% Let A=c1, B=c2, Z=c3, AZ=c4 and AB=c5
% c1 + c3 <=====> c4 % c4 + c2 <=====> c5 + c3
clc;
clear
syms c1 c2 c3 c4 c5 x1 x2
syms k1p k2p k1n k2n %positive and negative k’s
%Stoichiometric Vectors
r1 = [-1; 0; -1; 1; 0];
r2 = [0; -1; 1; -1; 1];
%Parameters
k1p=1; k2p=0.5; k3p=1 ;
EP=[.5 .1 .1 .4 .1]; %Equilibrium Point
C1=EP(1); C2=EP(2); C3=EP(3); C4=EP(4); C5=EP(5);
%Finding k1n, k2n (((((((((((((((((((((((((((((((( %Reaction Rates
w1 = k1p*C1*C3 – k1n*C4;
w2 = k2p*C4*C2 – k2n*C5*C3;
%solving for k1n and k2n
k1n=solve(w1==0, k1n);
k2n=solve(w2==0, k2n);
%)))))))))))))))))))))))))))))))))))))))))))))))))
%Molecular Matrix[1 0 0 1 1; 0 1 0 0 1; 0 0 1 1 0]
bA = C1+C3+C4; %for A
bB = C2+C5; %for B
bZ = C3+C4; %for Z
% Reducing the system for species c1=A, c4=AZ;
c3 = bZ – c4;
c5 = bA – c1 – c4;
c2 = bB – c5;
%Reaction Rates
w1 = k1p*c1*c3 – k1n*c4;
w2 = k2p*c4*c2 – k2n*c5*c3;
%Using Law of Conservation of Mass
G = r1*w1 + r2*w2;
%Reduced System
F = [G(1); G(4)];
F = subs(F,[c1,c4],[x1,x2]);
EP = [C1; C4];
m=20; n=17; %no of points forward and backward to the equilibrium point
h=0.03; %constant increment
%Finding points above EP
P=EP; r=P;
for forward = 1:m
J = ILDM1D_jacobian(F,r);
b = ILDM1D_slowest(J);
r = P + h*b;
Q = ILDM1D_projection(J);
P = ILDM1D_manifold(Q,F,r);
allForward(forward,:) = P; % all Forward matrix stores all forward points
end
%Finding points below EP
P=EP; r=P;
for backward =1:n
J= ILDM1D_jacobian (F,r);
b = ILDM1D_slowest (J);
r = P + h*b;
Q= ILDM1D_projection (J);
ILDM1D_manifold (Q,F,r);
allBackward(backward,:) = P;
%allBackward matrix stores all backward points
end
%Plotting
plot (EP (1),EP (2), ‘ks’);hold on;
plot (allForward(:,1),allForward(:,2),’+k-‘);
plot (allBackward(:,2),allBackward(:,2),’*k-‘);
grid on;
axis ([0 1 0 .7]);
xlabel (‘A’); ylabel (‘AZ’);
legend(‘Eguilibrium point’, ‘Forward direction’, ‘Backword direction’)I am trying to draw the backward and forward points of the slow invariant manifold (SIM) from the equilibrium point using the following code of Intrinsic low dimensional manifold (ILDM), but didn’t get all backward points. could anyone identify the mistake.

by using below code i am getting following graph which is not plotting the backward points

i am trying to get following result, but unfortunately there is a mistake in my code.

code:

% A + Z <=====> AZ
% AZ + B <=====> AB + Z
% Let A=c1, B=c2, Z=c3, AZ=c4 and AB=c5
% c1 + c3 <=====> c4 % c4 + c2 <=====> c5 + c3
clc;
clear
syms c1 c2 c3 c4 c5 x1 x2
syms k1p k2p k1n k2n %positive and negative k’s
%Stoichiometric Vectors
r1 = [-1; 0; -1; 1; 0];
r2 = [0; -1; 1; -1; 1];
%Parameters
k1p=1; k2p=0.5; k3p=1 ;
EP=[.5 .1 .1 .4 .1]; %Equilibrium Point
C1=EP(1); C2=EP(2); C3=EP(3); C4=EP(4); C5=EP(5);
%Finding k1n, k2n (((((((((((((((((((((((((((((((( %Reaction Rates
w1 = k1p*C1*C3 – k1n*C4;
w2 = k2p*C4*C2 – k2n*C5*C3;
%solving for k1n and k2n
k1n=solve(w1==0, k1n);
k2n=solve(w2==0, k2n);
%)))))))))))))))))))))))))))))))))))))))))))))))))
%Molecular Matrix[1 0 0 1 1; 0 1 0 0 1; 0 0 1 1 0]
bA = C1+C3+C4; %for A
bB = C2+C5; %for B
bZ = C3+C4; %for Z
% Reducing the system for species c1=A, c4=AZ;
c3 = bZ – c4;
c5 = bA – c1 – c4;
c2 = bB – c5;
%Reaction Rates
w1 = k1p*c1*c3 – k1n*c4;
w2 = k2p*c4*c2 – k2n*c5*c3;
%Using Law of Conservation of Mass
G = r1*w1 + r2*w2;
%Reduced System
F = [G(1); G(4)];
F = subs(F,[c1,c4],[x1,x2]);
EP = [C1; C4];
m=20; n=17; %no of points forward and backward to the equilibrium point
h=0.03; %constant increment
%Finding points above EP
P=EP; r=P;
for forward = 1:m
J = ILDM1D_jacobian(F,r);
b = ILDM1D_slowest(J);
r = P + h*b;
Q = ILDM1D_projection(J);
P = ILDM1D_manifold(Q,F,r);
allForward(forward,:) = P; % all Forward matrix stores all forward points
end
%Finding points below EP
P=EP; r=P;
for backward =1:n
J= ILDM1D_jacobian (F,r);
b = ILDM1D_slowest (J);
r = P + h*b;
Q= ILDM1D_projection (J);
ILDM1D_manifold (Q,F,r);
allBackward(backward,:) = P;
%allBackward matrix stores all backward points
end
%Plotting
plot (EP (1),EP (2), ‘ks’);hold on;
plot (allForward(:,1),allForward(:,2),’+k-‘);
plot (allBackward(:,2),allBackward(:,2),’*k-‘);
grid on;
axis ([0 1 0 .7]);
xlabel (‘A’); ylabel (‘AZ’);
legend(‘Eguilibrium point’, ‘Forward direction’, ‘Backword direction’) I am trying to draw the backward and forward points of the slow invariant manifold (SIM) from the equilibrium point using the following code of Intrinsic low dimensional manifold (ILDM), but didn’t get all backward points. could anyone identify the mistake.

by using below code i am getting following graph which is not plotting the backward points

i am trying to get following result, but unfortunately there is a mistake in my code.

code:

% A + Z <=====> AZ
% AZ + B <=====> AB + Z
% Let A=c1, B=c2, Z=c3, AZ=c4 and AB=c5
% c1 + c3 <=====> c4 % c4 + c2 <=====> c5 + c3
clc;
clear
syms c1 c2 c3 c4 c5 x1 x2
syms k1p k2p k1n k2n %positive and negative k’s
%Stoichiometric Vectors
r1 = [-1; 0; -1; 1; 0];
r2 = [0; -1; 1; -1; 1];
%Parameters
k1p=1; k2p=0.5; k3p=1 ;
EP=[.5 .1 .1 .4 .1]; %Equilibrium Point
C1=EP(1); C2=EP(2); C3=EP(3); C4=EP(4); C5=EP(5);
%Finding k1n, k2n (((((((((((((((((((((((((((((((( %Reaction Rates
w1 = k1p*C1*C3 – k1n*C4;
w2 = k2p*C4*C2 – k2n*C5*C3;
%solving for k1n and k2n
k1n=solve(w1==0, k1n);
k2n=solve(w2==0, k2n);
%)))))))))))))))))))))))))))))))))))))))))))))))))
%Molecular Matrix[1 0 0 1 1; 0 1 0 0 1; 0 0 1 1 0]
bA = C1+C3+C4; %for A
bB = C2+C5; %for B
bZ = C3+C4; %for Z
% Reducing the system for species c1=A, c4=AZ;
c3 = bZ – c4;
c5 = bA – c1 – c4;
c2 = bB – c5;
%Reaction Rates
w1 = k1p*c1*c3 – k1n*c4;
w2 = k2p*c4*c2 – k2n*c5*c3;
%Using Law of Conservation of Mass
G = r1*w1 + r2*w2;
%Reduced System
F = [G(1); G(4)];
F = subs(F,[c1,c4],[x1,x2]);
EP = [C1; C4];
m=20; n=17; %no of points forward and backward to the equilibrium point
h=0.03; %constant increment
%Finding points above EP
P=EP; r=P;
for forward = 1:m
J = ILDM1D_jacobian(F,r);
b = ILDM1D_slowest(J);
r = P + h*b;
Q = ILDM1D_projection(J);
P = ILDM1D_manifold(Q,F,r);
allForward(forward,:) = P; % all Forward matrix stores all forward points
end
%Finding points below EP
P=EP; r=P;
for backward =1:n
J= ILDM1D_jacobian (F,r);
b = ILDM1D_slowest (J);
r = P + h*b;
Q= ILDM1D_projection (J);
ILDM1D_manifold (Q,F,r);
allBackward(backward,:) = P;
%allBackward matrix stores all backward points
end
%Plotting
plot (EP (1),EP (2), ‘ks’);hold on;
plot (allForward(:,1),allForward(:,2),’+k-‘);
plot (allBackward(:,2),allBackward(:,2),’*k-‘);
grid on;
axis ([0 1 0 .7]);
xlabel (‘A’); ylabel (‘AZ’);
legend(‘Eguilibrium point’, ‘Forward direction’, ‘Backword direction’) manifold, plot, intrinsic low dimensional manifold, chemical kinetics MATLAB Answers — New Questions

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