What is the best software for the analysis of chaos in electric circuits.
Most SPICE packages could be used. Are you familiar with any SPICE packages?
Yes,I am familiar with them.But isn't chaos analysis done by simulating the differential equation or difference equations for a system.May be SPICE analysis is also possible but for that,may be some value is varied such as inductance and the system has a closed loop voltage or current control.Why is chaos analysis done with circuits which have closed loop control?I have simulated circuits in SPICE.It is an easy software,but how to do chaos analysis on it?
I'm not that familiar with Chaos Theory https://en.wikipedia.org/wiki/Chaos_theory
but it looks like you either need non-linearities or feedback to help get the chaotic behaviors (I could be wrong about that).
And SPICE does use differential equations for part of the modeling. The disadvantage is that it takes a long time to accurately simulate many time-domain waveforms, and you might be able to use a different type of simulation to speed things up (like using a State Space model)...
Attached is a spice simulation of Chua's circuit.
Remove the .txt extensions to run the .asc file in LTspice.
Which software can be used for analysis if I make state space representation of the circuit?
There is some code here that might help.
See; pdf links at start of post #1, see; FBasic code in post #5.
My recommendation: VISSIM (Visual Solutions Inc.).
This package is optimized for real time simulations - however, it is block-oriented and not based on parts level (like SPICE)
Yes, since the differential equation is the mathematical model that describes the dynamics of the circuit.
Here follow some other remarks that I hope you will find helpful.
In addition to pure simulation, it is often useful to perform a numerical continuation analysis. This typically begins by selecting a system parameter and watching how simple invariant sets (such as equilibria and limit cycles) (dis)appear or change their stability at bifurcation points. A rather user friendly MATLAB package for this purpose that has seen frequent use in research is MatCont. The corresponding literature reference is:
Dhooge, A., Govaerts, W. and Kuznetsov, Yu. A. MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Software 29 (2003), no. 2, 141–164.
While simulation can show you chaos, continuation and bifurcation analysis can provide insight into how a system may become chaotic in the first place.
For the Chua circuit in particular, you could have a look at the end-of-chapter project (Section 16.5) in the second edition of Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsch, Smale and Devaney. (Meanwhile, there is a third edition, but as far as I could see, the section numbering did not change for this particular section.) This project - together with the material in that chapter - will help you understand how chaos in Chua's circuit relates to Shil'nikov's homoclinic bifurcation.
In case you cannot (or: don't want to) get the book, here are the two literature references that the project cites on Chua's circuit:
Chua, L., Komuro, M., and Matsumoto, T. The double scroll family. IEEE Trans. on Circuits and Systems. 33 (1986), 1073.
Khibnik, A., Roose, D., and Chua, L. On periodic orbits and homoclinic bifurcations in Chua’s circuit with a smooth nonlinearity. Int. J. Bifurcation and Chaos. 3 (1993), 363.
This and related systems have been the subject of intense study, so the above is just a start. A lot more can be found if you look for it. Have fun!
For future readers of this thread, I would also like to mention the book Linear and Nonlinear Circuits by Chua, Desoer and Kuh. The original (1987) hardcover is out of print and expensive to buy used, but in 2015 an affordable softcover came out. Among other things, it is a bridge between circuit analysis and dynamical systems, so it could also be interesting for certain mathematicians.
Thanks to @jasonRF for briefly discussing this book in a couple of old threads in the textbook forum.
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