How can I find equilibrium points in a non linear ODE

Hello everyone, kinda new to Matlab and trying some excercises. Have some probleme here. I hope someone can help me. #

I have the non linear ODE:

And i want to find the equilibrium points. How do I find them?

[Torsten]

That's not a MATLAB question, it's pure math.

[Karl-JR]

yeah, i know but it´s an excercise in my book and i´m still not really familiar with Matlab.

[Torsten]

As said: the exercise has nothing to do with MATLAB (and cannot be solved with MATLAB).

Answer 1

[Sam Chak]

Hi @Karl-JR

If you unsure of how to analytically find the equilibrium point for the unforced case, then try look for "how to find the equilibrium point" in the calculus textbooks or online materials. Else, you can also simulate the nonlinear ODE a dozen times for a range of initial values x0. I usually use this method for forced cases (non-zero u).

If the states converge to some steady values after some time t, then you can empirically say that set of values is the equilibrium point of the system.

% Define the input signal
u = @(x) 0;     % unforced 

% Define the system dynamics
f = @(t, x) [x(2); x(3); (u(x).^2 - 10*sin(x(3)) - x(2)./(x(2).^2 + 1) - x(1))/3]; 

% Define the initial conditions
% x0 = [1 0 0]; % test 1
% x0 = [0 1 0]; % test 2
% x0 = [1 1 0]; % test 3
% x0 = [0 0 1]; % test 4
% x0 = [1 0 1]; % test 5
% x0 = [0 1 1]; % test 6
x0 = [1 1 1]; % test 7

% Define the time interval
tspan = [0 300];

% Solve the system using the ode45 solver
[t, x] = ode45(f, tspan, x0);

% Plot the responses of the states
plot(t, x); grid on
xlabel('Time');
ylabel('System states');
legend({'$x$', '$\dot{x}$', '$\ddot{x}$'}, 'interpreter', 'latex', 'fontsize', 14);

% Values of the states at the end of simulation time
x(end, :)

ans = 1×3

1.0e-03 * 0.4310 0.0174 -0.0450

From the responses and the steady-state values, we can intuitively say that the states will eventually converge to zero, as .