While studying dynamical systems with piecewise linear (PWL) systems I found the article of Silva [1]. In which by means of the homoclinic and heteroclinic Shil'nikov´s theorem and the Poincare maps, chaotic behavior is conclude.
My question is:
Is it valid for any continuous R^3 PWL system with either homoclinic or heteroclinic orbits to conclude it is chaotic if it satisfies the definition on the Theorems 2.1 and 2.1 of [1]?
In case this is correct, what relationship or value does the Lyapunov exponent one must expect?
Thanks in advance.
[1] Silva, C. P. (1993). Shil'nikov's theorem-a tutorial. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40(10), 675-682.
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=246142&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D246142
Dear Sir
Please see the attached file which can help you.
regards
Salahuddin
I did something similar (for a continuos time system):
P.GRABOWSKI, An application of Shilnikov’s theorem to linear systems with piecewise linear
feedback. Structure, Coherence and Chaos in Dynamical Systems. 473 - 479. Eds. P.L.Christiansen,
P.L. Parmentier. Manchester University Press. 1989.
The Shil'nikov bifurcation supposes (among others) that the governing ODE has a continuously differentiable right hand side. This is not the case for a piecewise linear system. On the other hand, experience says that suppositions of many theorems can be weakened as long as they hold at the critical places. In your case, the slope of the linear functions must not change at the system' s fixpoint (where the homoclinic orbit starts and ends), then you may expect the same behavior as in Shil'nikov's original theorem. But don't forget, this is still a heuristic argument.
Thank you all for your time and valuable comments.
Professor Uddin. I wasn't aware of this publication that you shared. It is a very interesting collaboration by Ushio and Hirai. They addressed the discrete case of piecewise linear systems (PWL). Although the systems that I want to study are on the continuous time, there must be some reference to the work related to the case.
Professor Grabowski. It was very difficult to me to find your publication. However I found a piece of it on Google books as I attached in the link. You describe a very interesting contribution on PWL systems. One question from the example 2. Due to the type of orbits generated on the system (2) with the condition marked in the example, does the system can be described as chaotic?
Professor Jansen. Thank you very much for your insight on the Shil'nikov theorem. Indeed my system's slope doesn't change near the equilibria. However if the system presents heteroclinic orbits do this consideration needs to satisfy for both equilibria?
https://books.google.com.mx/books?id=2C68AAAAIAAJ&pg=PA473&lpg=PA473&dq=P.+GRABOWSKI,+An+application+of+Shilnikov+theorem+to+linear+systems+with+piecewise+linear+feedback.&source=bl&ots=NDcteHsHPz&sig=6LY6FcYuOe0YXxoY-4o2wiUhKLk&hl=en&sa=X&ved=0ahUKEwjpp6W1yvDNAhWm44MKHR8mB4gQ6AEIGjAA#v=onepage&q=P.%20GRABOWSKI%2C%20An%20application%20of%20Shilnikov%20theorem%20to%20linear%20systems%20with%20piecewise%20linear%20feedback.&f=false
Sorry, I really don't know any extensions of Shil'nikov's work to heteroclinic orbits (this may not state very much, I left the field of dynamical systems about 15 years ago). Anyway, if there is such an extension then I would guess that the ODE must be smooth around any equilibrium.
Dear Juis Javier,
Enclosed pages 476 and 477 (done using a smartphone) which are not available from Google's.
Systems satisfying the assumptions of Shilnikov's theorem are normally called chaotic in a Shilnikov's sense, however you are correct that naming them as chaotic is not precise.
- Badges
- Science topic
- Similar topics
- Engineering
- Control Theory