Determining critical values which cause multi-solutions and instability

In summary, to investigate the stability of the solutions of equation (1) with forcing period $T = \frac{2\pi}{\omega}$ and asymptotic radius $R_0(\kappa,t,\omega)$, you can vary the parameters $\kappa$ and $\varGamma$ and use a numerical solver to obtain numerical solutions. By comparing these solutions to the asymptotic radius, you can determine the critical values of $\varGamma$ at which the response curve becomes multivalued and the solutions become unstable.
  • #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.
 

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  • #2
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.
 

FAQ: Determining critical values which cause multi-solutions and instability

What is the significance of determining critical values in a scientific study?

Determining critical values helps researchers identify the threshold at which a phenomenon or system undergoes a significant change or becomes unstable. This information is crucial in understanding the behavior of complex systems and predicting potential outcomes.

How do critical values cause multi-solutions in a problem?

In many scientific problems, there may be multiple solutions that satisfy the given conditions. Critical values play a role in determining these solutions by defining the boundary between different regions in which the solution may lie. If the critical value falls within this boundary, it can lead to the existence of multiple solutions.

Can critical values also cause instability in a system?

Yes, critical values can cause a system to become unstable. This can happen when the critical value falls within a range where the system's behavior shifts abruptly, leading to sudden changes and unpredictable outcomes. This instability can have significant consequences in fields such as climate science or economics.

How do scientists determine critical values in their studies?

Scientists use various methods such as mathematical calculations, experiments, and simulations to determine critical values. These values may also be derived from existing data or through advanced statistical techniques. The chosen method depends on the specific problem and available resources.

What are some potential challenges in determining critical values?

Determining critical values can be a complex and challenging task as it involves analyzing large amounts of data and accounting for various factors that may influence the outcome. Additionally, different methods of determining critical values may yield different results, and it is essential to carefully select the most appropriate approach for each study.

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