- #1
nacho-man
- 171
- 0
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.
$$x'' + \kappa x' + x - x^3 = \varGamma \cos(\omega t) \ \ \cdots \ \ (1)$$
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.