Definition of a neutral/indifferent orbital stable systems

Question

Dear Physicsoverflow members,

I'm currently working on a thesis about the COT, merit of walking, stability and robustness of little walking machines called passive dynamic walkers. For this thesis need to explain different kinds of stability. First, I described static stability followed by orbital or cyclic stability. For static stability, a system can be stable, unstable or neutral or indifferent. When translating these concepts to orbital stability, I, however, run into a problem. It was not so hard to define the meaning of systems that are orbitally stable or unstable, but a neutral or indifferent orbital system was a little harder to imagine. In static stability an indifferent stable system can be visualised as a ball on a levelled surface when you perturb it by applying a force it simple moves a little en then stops. How can I understand this in the context of orbital or periodic motions?

After a search on google I found this definition:

"Neutral orbital stability of an isolated equilibrium point for an autonomous system of differential equations obtains if all solutions sufficiently close to the equilibrium point remain close to that equilibrium point as
"t."

What does this mean when a neutral orbital stable is perturbed?

Thanks a lot in advance,

Greetings Rick,

Comments

When a neutral orbital stable point in 2D is perturbed it may becomes a small periodic motion around the stable point.

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Thanks for your answer %@ArnoldNeumaier. After a long search, I found a beautiful syllabus of MIT explaining the stability of different orbital cycles.

https://www.google.nl/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0ahUKEwiUpt7vq6rNAhXKNxQKHf1oDJwQFggyMAI&url=http%3A%2F%2Focw.mit.edu%2Fcourses%2Fmathematics%2F18-03sc-differential-equations-fall-2011%2Funit-iv-first-order-systems%2Flimitations-of-the-linear-limit-cycles-and-chaos%2FMIT18_03SCF11_s38_3text.pdf&usg=AFQjCNEzYJXyNb3Ir0Vmnf3NAiMDrMFgsg