0
$\begingroup$

I need help with this problem: I have this ODE systementer image description here

For this system, manually, I calculated the stability points, eigenvalues, and Jacobian matrix for a range of ALPHA values (I considered those alpha values when doing this process: Table[\[Alpha], {\[Alpha], N[1.0 Degree], N[89 Degree]}]}.

Is there an option where I can obtain a phase portrait of this system, for the range of alpha values that I showed above?

The sketch of the code (until now):

$Assumptions = \[Alpha] > 0 && k > 0 && d0 > 0 && d > 0 && x > 0 && \[Omega]1 > 0 && [Omega]2 > 0;
Sys = {x'[t] == y[t], y'[t] == -1*(\[Omega]1^2 *x[t] + \[Omega]2^2 *x[t]^3)*Csc[\[Alpha]]};
jacobian = Grad[{y, -Csc[\[Alpha]]*(\[Omega]1^2 x + \[Omega]2^2 x^3)}, {x, y}]
Roots[0 == Csc[\[Alpha]] * (\[Omega]1^2 x + \[Omega]2^2 x^3), x];
{\[Alpha] -> Table[\[Alpha], {\[Alpha], N[1.0 Degree], N[89 Degree]}]};

After the results of the roots were obtained, I replaced them in the Jacobian matrix and obtained 3 different Jacobian matrices (1 for each angle that I used).

After this point, I calculated every eigenvalue for the specific angles I wanted (alpha== 1 Degree, alpha== 63.0254 Degree and alpha== 89 Degree).

My problem starts after this point because I can't find a way to connect the alpha value in phase portrait. Is there any option to do that?

$\endgroup$
2

1 Answer 1

3
$\begingroup$

You have this system

\begin{align*} x'(t)&=y(t)\\ y'(t)&=-(\omega_1^{2} x + \omega_2^{2} x^3) \csc(\alpha) \end{align*}

Convert to state space. Let $x_1=x,x_2=y$ then

\begin{align*} \dot{x_1}&=x_2\\ \dot{x_2}&=-(\omega_1^{2} x_1 + \omega_2^{2} x_1^3) \csc(\alpha) \end{align*}

Now use StreamPlot. Here is a basic Manipulate

Manipulate[
 StreamPlot[{x2, -(omega1^2*x1 + omega2^2*x1^3)*
    Csc[alpha  Degree]}, {x1, -4, 4}, {x2, -4, 4}],
 {{omega1, 1, " omega 1"}, 0, 10, .1, Appearance -> "Labeled"},
 {{omega2, 1, " omega 2"}, 0, 10, .1, Appearance -> "Labeled"},
 {{alpha, 50, " alpha"}, 40, 70, .1, Appearance -> "Labeled"},
 TrackedSymbols :> {omega1, omega2, alpha}
 ]

enter image description here

$\endgroup$
2
  • $\begingroup$ Thank you so much for the help you provided to me. $\endgroup$ Commented Feb 17 at 11:37
  • 1
    $\begingroup$ @VictorPintoMscStudent If this answer solved your problem, you can accept it by clicking the checkmark sign! $\endgroup$
    – Chris K
    Commented Feb 19 at 6:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.