I have an autonomous system of nonlinear equations of the form:

$$Mx'' + C(\omega)x' + K(\omega)x + F_{nl}(x) = 0$$

where $M$ is the mass, $C$ the damping and $K$ the stiffness matrix. The nonlinear force term $F_{nl}$ depends on the solution and is of a piecewise linear shape, where the nonsmoothness occurs at $x=0$

$$F_{nl}(x) = \begin{cases} 0 & \text{if }x < 0 \\ 10x & \text{if }x \ge 0 \end{cases}$$

The system results from two rings rotating relative to each other at $\omega$. For certain rotational speeds, energy can be transferred from the rotating ring to the stationary ring, resulting in a divergence of the zero solution.

I am trying to study the stability of the zeros solution fixed point ($x=0$). Floquet multipliers won't work as the linearization at the fixed point will only take into account the left or right sided derivative.

Except for doing time simulation with a small perturbation around $x=0$, I am lost as to how to study stability in a more academically sound way. Any help is appreciated!


It seems there is a proof that this kind of system is stable using Lyapunov theory, see:

Shorten, R.N. and Narendra, K.S.: "On the stability and existence of common Lyapunov functions for stable linear switching systems", Proceedings of the 37th IEEE Conference on Decision and Control, http://dx.doi.org/10.1109/CDC.1998.761788.

In essence, the paper states that if the system matrices for each of the modes (i.e., the cases x<0 and x >= 0) are stable and simultaneously triangularizable (which is the case here), the resulting hybrid system is stable as well.

  • $\begingroup$ Thanks for the answer, but I still have some troubles. I end up with a system of the form $\dot{x}(t) = A_i(t)x(t)$ where the $A_i$ are my linear, time dependent and periodic coefficient matrices. Checking for the existence of quadratic Lyapunov matrices, seems to entail matrices with constant coefficents describing the different systems $i$. Is this idea still applicable to time dependent/periodic coefficient matrices? $\endgroup$ – Markus Mar 6 '12 at 17:22
  • $\begingroup$ I did not notice that your system had time-dependent matrices. In that case, I do not think the result I linked to is applicable. Nevertheless, you may be interested in Lyapunov functions for stability analysis, since they provide an approach that can deal with non-linearities. $\endgroup$ – Johannes Kloos Mar 9 '12 at 12:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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