Limit Cycles, Part 1: Introduction & Examples

[@moslemmolaie3506]

I have followed your channel since two month ago, your method to teach is great and clear. Is it possible for us to access to the whole courses?

[@ProfessorRoss]

All of the lectures on Nonlinear Dynamics and Chaos are available in the KZbin playlist, kzbin.info/www/bejne/mIDTqYRtnayjo68 If you want to take the class for credit, you need to be enrolled as a student at Virginia Tech.

[@oluremifalowo3921]

Amazing video! How does a double limit cycle differ to 2 single limit cycle?

[@ProfessorRoss]

Thank you. But what do you mean by a "double limit cycle"?

[@CalculusCraze1]

kzbin.info/www/bejne/pGKmnZanpLiLp7c In this series, dynamical systems will be taught in details, subscribe,like and share to learn Dynamical Systems.

[@CalculusCraze1]

kzbin.info/www/bejne/pGKmnZanpLiLp7c In this series, dynamical systems will be taught in details, subscribe,like and share to learn Dynamical Systems.

[@internationalpopsta1]

love the way you explain things! thanks!

[@ProfessorRoss]

Thank you Kostis! I love the subject and like to share.

[@CalculusCraze1]

kzbin.info/www/bejne/pGKmnZanpLiLp7c In this series, dynamical systems will be taught in details, subscribe,like and share to learn Dynamical Systems.

[@triton62674]

Fantastic examples thanks

[@floatingbonesjournal]

Bravo. I asked a question in a recent online math lecture; a mathematician in the chat pointed me to Limit Cycles. I appreciate Dr. Ross's examples of heart beats and walking. About 10 years ago, I stumbled across the interesting lateral rope technique: David Weck's "Dragon Roll" kzbin.info/www/bejne/gGOYkGdno7V5gtE . The Dragon Roll orbit is clearly a Limit Cycle; it follows Viviani's Curve over and around the body. When you change to orbit in the opposite direction, there are 3-4 cycles where you home in on the new stable orbit. The same thing happens with walking or running: a change in terrain or an obstacle will perturb the cycle, but you will rapidly find the nominal orbit. I abstractly understood the principle, but I couldn't get any further information until I got the name of the term and searched on it. Thanks for publishing this video, Dr. Ross.

[@karthiramachandran1257]

hello professor, assume a limit cycle oscillator which are linearly coupled(diffusive coupling). On some conditions each oscillator oscillates at different amplitudes. They typically look like linear periodic orbits around origin like the conservative system mentioned in video. How to classify that?

[@hewadsaad3014]

Those are marginal oscillations and not limit Oscillation(limit cycle) A marginal ossilation: is a phenomena which usually occurs in pure linear systems and totally depend on initial conditions which results in different amplitudes, and are not robust to system uncertainties and disturbances. A limit cycle or limit oscillation:is a phenomena which only accures in some nonlinear systems and independent of initial conditions and are of fixed amplitude and fixed frequency and are strongly robust against system uncertainty and disturbances Here is an example: If you start at a different initial point and get the same closed loop again, then you have a stable limit cycle, However if you change the initial condition and get every time a compeletly different and new closed loop then that is not a limit cycle, as it is not unique and depends on initial condition, those are marginal oscillations and can accrue only in ideal cases of linear systems, because with slight disturbances or uncertainty such oscillations will usually lose energy and come to a point(resting point or in most cases origin), I.e. in LC electric circuit theoretical there is an endless marginal ossilations, however such ossilations are not possible realistically because in real world there is internal resistor in each LC circuit which will lead to losses and such ossilations will eventually come to origin. Resulting in a stable spiral 🌀. Therefore a marginal ossilation is not robust (or is sensitive) to model uncertainty (in this case resistor) and disturbances. Such ossilations can be seen in simulations, Phase-Portrait is a good Matlab library to simulate such oscillations, can be accessed through Pplane open library of Matlab. Hope this answered your question 😊

[@sarbasthusseinmikael7670]

Could you write down the link of book, please

[@ProfessorRoss]

Strogatz, Nonlinear Dynamics and Chaos www.stevenstrogatz.com/books/nonlinear-dynamics-and-chaos-with-applications-to-physics-biology-chemistry-and-engineering

[@Jo-ce6gd]

Amazing tutorial

[@ProfessorRoss]

Glad you liked it

[@brendawilliams8062]

Thankyou