Most of the continuous chaotic systems (chaotic flows) like Lorenz, Rössler, Sprott systems (cases B-S) have a Kaplan–Yorke dimension slightly greater than 2:
1. Is there any 3D dissipative chaotic flow with a Kaplan-Yorke dimension near 3?
2. Which 3D dissipative chaotic flow has the highest Kaplan-Yorke dimension?
Regards,
Sajad
http://sprott.physics.wisc.edu/pubs/paper410.pdf is a preprint of a paper I coauthored describing a dissipative 3-D system of ODEs whose KY dimension can be changed continuously from 2.0 to 3.0.
Dear Professor Sprott,
Thank you very much. That was very nice and helpful. However, is there any other famous system with large (near 3) Dky?
Regards,
Sajad
Sajad -- I suppose it depends on how you define "famous." The generalized Nose-Hoover system (with a temperature gradient) has been studied a lot and has a KY dimension close to 3.0 (and a capacity dimension of exactly 3.0). There are a number of 4-D thermostated systems that have a capacity dimension of 4.0 and a KY dimension close to 4.0 as described in http://sprott.physics.wisc.edu/pubs/paper421.pdf. The 3-D Burke-Shaw attractor, described on page 347 of my Chaos and Time-Series Analysis book also has a capacity dimension of exactly 3.0, but perhaps only a moderately high KY dimension. You might also look at the paper http://sprott.physics.wisc.edu/pubs/paper289.pdf which has some 3-D systems with Dky as high as 2.99. Nonautonomous systems like the damped, driven pendulum, the Duffing oscillator, the Ueda oscillator, etc. can usually be made to have an arbitrarily large Dky by reducing the damping. But you are right that most of the common autonomous 3-D dissipative chaotic systems that people study typically have Dky in the range 2.0 to 2.2 or so.
Another fun game to play is to fix the form of the equations and then adjust the parameters to give the maximum Dky. For example, you can tune the Lorenz system to have a KY dimension of 2.235, and the Rossler system to have a KY dimension of 2.158. See http://sprott.physics.wisc.edu/pubs/paper309.pdf.
In a dissipative chaotic flow of 3D system, the highest value of KY dimension is depends on the system. Generaly, KY dimension is changed continuously between 2 to 3 in a 3D ode system.
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