Storm eyes in the Lorenz attractors

Fractals and Chaos


Bi-parametric screening uncovers the stunning complexity and universality of spiral fractal structures in three exemplary systems with the Lorenz attractor: the iconic Lorenz equation, a normal form (the Shimizu-Morioka model) and a laser model from nonlinear optics.
For more information, contact Andrey Shilnikov ( or Roberto Barrio (


The bifurcation diagrams of the Shimizu-Morioka model obtained by (left) the standard parameter continuation method for local and global homo- and hetero-clinic bifurcations [courtesy of A. L. Shilnikov, L. P. Shilnikov, and D. V. Turaev, J. Bifur. Chaos 3, 1123 (1993)] and by the biparametric symbolic screening revealing self-similar fractal structures organized by cod-2 T(erminal)-points.


Biparametric chaos due to tornado eyes-shaped patterns around the T-points separated by the saddle structures." The biparametric sweeping revels a plethora of the T-points within a fractal mapping of the Lorenz equation in the symbolic bifurcation diagram: dark (brown) region corresponding to the simple Morse-Smale dynamics; lighter (blue) stripe corresponding to the chaotic saddle due to the homoclinic explosion after the formation of the primary homoclinic butterfly on the boundary between the two regions. Bright (reddish) region of complex dynamics are due to homo- and hetero-clinic connections between the saddle equilibria.


Biparametric mapping of the laser model featuring the T-points and saddles organizing the complex fractal structure, common for the Lorenz-like systems, in the (electric-field-amplitude, omission-frequency)-diagram of the 6D optically pumped far-infrared-red laser model obtained using the proposed technique. Solid windows and fractal region correspond to trivial and chaotic dynamics generated by the laser.

Author Institutional AffiliationAndrey Shilnikov, Neuroscience Institute, GSU, GA, 30303; Roberto Barrio, Universidad de Zaragoza
Author Email
Author Postal Mail100 Piedmont Ave SE Atlanta, GA 30303
Shilnikov A., Shilnikov L. and Barrio R, Symbolic dynamics and spiral structures due to the saddle-focus bifurcations, in “Chaos, CNN, Memristors and Beyond”, 2011
KeywordsLorenz equation, Lorenz attractor, bifurcations, kneading invariants, bi-parametric sweeping

Please login or register to post comments.