Determining critical values which cause multi-solutions and instability

[nacho-man]

I have the motion of a forced spring:

$$x'' + \kappa x' + x - x^3 = \varGamma \cos(\omega t) \ \ \cdots \ \ (1)$$
and I am investigating the stability of its solutions with forcing period
$T = \frac{2\pi}{\omega}$. I am given the asymptotic radius of the solutions:
$$R_0(\kappa,t,\omega) = \lim_{t \to \infty} (x^2+(x')^2)^\frac{1}{2}$$
How can I determine critical values for $\varGamma$ at which stage the response curve becomes multivalued, resulting in unstable solutions?

I will investigate $(1)$ over $0.5 \leq \omega \leq 1.5$ and $\kappa = 0.05,0.1,0.2$ as $\varGamma$ is varied.How would I investigate numerical solutions $\omega = (\omega_1, . . . , \omega_N )$ over a large enough time frame where I can observe a periodic solution, whilst holding $\varGamma, \kappa$ fixed, like what has been done in the attached image.

 

[gtoguy97]

To investigate numerical solutions, you can use a numerical solver such as MATLAB's ode45 function. You will need to provide the initial conditions (the values of $x$ and $x'$ at some initial time), the time range over which you want the solution, and the right hand side of equation (1). The numerical solution will then be calculated for the given parameters. You can then plot the results of the numerical solution over the desired time frame and compare it to the asymptotic radius of the solution. For each $\omega$ value, you can then calculate the critical values of $\varGamma$ at which stage the response curve becomes multivalued, resulting in unstable solutions.