How to solve and plot system of nonlinear differential equations?

I'm trying to recreate graphs from a modeling paper by plotting a system of differential equations in MatLab. Unfortunately, I don't have much MatLab experience if any. I've found other questions on systems of nonlinear equations asked in MatLab answers and have managed to produce a plot for my own system, but this plot is not the same as the one in the paper I'm using.

https://imgur.com/a/hBQ3z This should show the differential equations I'm using, the parameter values, the graph I'm trying to recreate, and the plot I ended up with.

Here is the code I used to set up my system and plot the equations.

syms p(t) m(t) l(t) T Y

Eqns = [diff(p(t),t) == (3*p(t)*(1-p(t)))-(30*(p(t)*m(t))); diff(m(t),t) == ((25*p(t)+l(t))*m(t)*(1-m(t)))-m(t); diff(l(t),t) == (15*(1+tanh(2*m(t)-2)))-l(t)]

[DEsys,Subs] = odeToVectorField(Eqns);

DEFcn = matlabFunction(DEsys, 'Vars',{T,Y});

tspan = [0,25];

y0 = [0.01 0.05 0.539];

[T,Y] = ode45(DEFcn, tspan, y0);

figure(1)

plot(T,Y)

legend('p(t)','m(t)','l(t)')

grid

These are the same equations shown in the imgur album which I tried to recreate through diff(p(t),t), diff(m(t),t), and diff(l(t),t)

Here are the parameter values and graph I'm trying to recreate

And this is the graph I ended up with

I based my code off of a response provided to a question asked here (https://www.mathworks.com/matlabcentral/answers/365203-how-to-solve-a-system-of-nonlinear-differential-equations), but I would really appreciate anyone that could help me solve my plotting problems and show me what I did incorrectly and how to get the right plot.

Thanks.

[Rahiti Arapari Hargous]

Hi everyone,

I have a similar problem with the next governing equation, with y(t) and x(t). The system of equation is the next one:

ay''' - b*x*y' + w*cos(y) = 0

x' - w*L*sin(y)=0

If anyone can help thanks

Answer 1

[Daniel Jovin]

    % code
kp = 3;
kpm = 30;
kmp = 25;
klm = 15;
kl = 1;
f = @(t,y) [(kp*y(1)*(1-y(1)))-(kpm*(y(1)*y(2))); ((kmp*y(1)+y(3))*y(2)*(1-y(2)))-y(2);(klm*(1+tanh(2*y(2)-2)))-(kl*y(3))];
tspan = [0, 25];
xinit = [0.01, 0.05, 0.539];
ode45(f, tspan, xinit)
legend('p(t)', 'm(t)', 'l(t)')

[Torsten]

And what was the problem with your first version from above ?

Best wishes

Torsten.

[Daniel Jovin]

I'm new to Matlab, so I don't really understand what I did incorrectly and what differentiates my failed solution from the correct solution. But, the problem was that the plot I was generating, Figure 1, was incorrect- the values from the graph were not in the correct range and lacked the periodic nature of the graph from the modeling paper, Fig. 1.

[Vikrant Nitin Joshi]

How exactly do you add initial conditions to these equations?

Answer 2

[thanos tria]

Hello, i am trying to solve a non linear system but when i plot the solutions only a straight line in one of the variables appears on the graph, and all the variables just can't get away from the initial conditions. That's my code:

syms c1(t) c2(t) c3(t) c4(t) T Y
Eqns = [diff(c1(t),t) == -0.015*c1+1.8*10^(-14)*c2*c4;
           diff(c2(t),t) ==  0.015*c1-1.8*10^(-14)*c2*c4;
           diff(c3(t),t) == -6*10^(-34)*c3*0.051765*10^(20)*0.2465*10^(20)+0.015*c1;
           diff(c4(t),t) ==  6*10^(-34)*c3*0.051765*10^(20)*0.2465*10^(20)-1.8*10^(-14)*c2*c4];
[DEsys,Subs] = odeToVectorField(Eqns);
DEFcn = matlabFunction(DEsys, 'Vars',{T,Y});
tspan = [0,50000];
y0 = [0.040923*10^(12) 0 0 0];
[T,Y] = ode45(DEFcn, tspan, y0);
plot(T,Y)

And that's my graph:

It sould have had some curves because it describes how ozon is formulated in the atmosphere. Any suggestions??

[Walter Roberson]

I agree that when I use those equations with Maple's numeric ODE solver, that I get distinct curves for each of the functions, for all of Maple's numeric methods that I have tried.

I also find in Maple that using stiff methods improves performance a lot without changing the solution. That would correspond to using routines such as ode23s() in MATLAB --- which produces the same (not so good) result as ode45 except faster.

[thanos tria]

I tried ode23s and ode15s and still straight line