Intermediate Axis Theorem - Lyapunov's Indirect Method

[wmrunner24]

So I'm investigating the stability properties of the following nonlinear system of equations:
$$\frac{dx}{dt}=-\rho k_2 \cos(x) \cos(y) \sin(y)$$
$$\frac{dy}{dt}=-\rho \sin(x)[k_1+k_2\sin^2(y)]$$
where $\rho > 0 \text{ and where } k_1 \text{ and } k_2$ are real constants.

In particular, I'm looking at the critical point $(x,y) = (0,0)$. I'm expecting to find that this is an at least stable critical point when $k_1k_2 < 0$ and an unstable critical point when $k_1k_2 > 0$ (in fact, I've already shown that this is true in the latter case using Lyapunov's indirect method, but unfortunately the almost-linear system corresponding to these equations gives a stable center for $k_1k_2 < 0$, which is an inconclusive result). I'm now looking instead to use Lyapunov's direct method to diagnose the stability of the system, but I'm spinning my wheels as far as determining a suitable auxiliary function goes. I'm seeking advice/suggestions for determining this function.

Some things that I've tried:
$$V(x,y) = 1-\cos(x)\cos(y)$$
$$V(x,y) = \sin(x)\sin(y)$$
$$V(x,y) = k_1 - [k_1\cos(x)+\frac{k_2}{2}\sin^2(y)]$$
And many variations of these. I've yet to find one that is useful as an auxiliary function. Because I'm expecting $k_1 \text{ and } k_2$ are directly related to the stability of this point, I'm also expecting that a suitable auxiliary function will include these constants somewhere in it. Any advice for constructing said function or an example of one would be much appreciated.

As some background, this particular system of equations is a transformation of Euler's equations for the torque-free rotation of a rigid body with respect to a set of body-fixed principal axes. The transformation to a spherical coordinate system to parameterize the state space was needed for the application of Lyapunov's methods, as they rely on the consideration of isolated equilibrium points. As an added bonus, one of the equations becomes redundant following the transformation, since the rotational kinetic energy of the system is conserved, and thus $\frac{d\rho}{dt} = 0$. The coefficients $k_1 \text{ and } k_2$ are:
$$k_1 = \frac{I_2 - I_3}{\sqrt{I_1I_2I_3}}$$
$$k_2 = \frac{I_3 - I_1}{\sqrt{I_1I_2I_3}}$$
where $I_1, I_2, \text{ and } I_3$ are the moments of inertia about the principal axes of the body. These equations represent rotation purely about the $x_3$-axis for $(x,y) = (0,0)$, and so by the intermediate axis theorem, if $I_2>I_3>I_1 \text{ or } I_1>I_3>I_2$, then this critical point should be unstable and $k_1k_2 > 0$, and the opposite should be true if $I_3$ is the maximum or minimum principal moment of inertia.

My plan is to continue attempting to find a useful auxiliary function, but in the meantime, I thought I'd also seek the advice of others. Again, any help provided is much appreciated!

 

[jvicens]

Thank you for sharing your research on this interesting nonlinear system of equations. Based on your findings, it seems that there is still some uncertainty regarding the stability of the critical point (x,y) = (0,0) when k_1k_2 < 0. In order to use Lyapunov's direct method, it is important to find a suitable auxiliary function that can help determine the stability of the critical point.

As you mentioned, the coefficients k_1 and k_2 play a crucial role in the stability of the system. Therefore, it may be helpful to incorporate these coefficients into the auxiliary function. One approach could be to consider a function of the form V(x,y) = k_1f(x) + k_2g(y), where f(x) and g(y) are functions that satisfy certain properties (such as being positive definite and decreasing along trajectories of the system). This could potentially lead to a Lyapunov function that is dependent on the coefficients k_1 and k_2.

Another approach could be to consider a function of the form V(x,y) = \alpha x^2 + \beta y^2, where \alpha and \beta are positive constants. This type of function has been used in the analysis of nonlinear systems and has shown to be effective in determining stability properties.

Overall, finding a suitable auxiliary function may require some trial and error and creative thinking. It may also be helpful to consult with other experts in the field or to look at similar systems of equations for inspiration.

Best of luck in your research and I hope you are able to find a suitable auxiliary function to determine the stability of the critical point (x,y) = (0,0).