Related Experiment Video
Updated: Jan 28, 2026

The Microfluidic Probe: Operation and Use for Localized Surface Processing
Published on: June 4, 2009
Strange attractor existence for non-local operators applied to four-dimensional chaotic systems with two equilibrium
1Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa.
Abstract:
Not every chaotic system has the particularity of displaying attractors with a fractal structure. That is why strange attractors remain enthralling not only for their fractal structure, but also for their amazing chaotic and multi-scroll dynamics. In this work, we apply the non-local and non-singular kernel operator to a four-dimensional chaotic system with two equilibrium points and show the existence of various types of attractors, including the butterfly type and strange type. Recently, there have been virulent communications related to the validity or not of the index law in fractional differentiation with non-local operators. These discussions resulted in pointing out many important features of the Mittag-Leffler function used as kernel and suitable to describe more complex real world problems. This paper follows the same momentum by pointing out another important feature of the non-local and non-singular kernel operator applied to chaotic models. We solve the model numerically and discuss the bifurcation and period doubling dynamics that eventually lead to chaos (in the form of butterfly attractor). Lastly, we provide related numerical simulations which prove the existence of a chaotic fractal structure (strange attractors).
Related Concept Videos
First Law: Particles in One-dimensional Equilibrium
First Law: Particles in Two-dimensional Equilibrium
Newton's first law tells us about...
Dynamic Equilibrium
Free Energy and Equilibrium
Recall that Q is the numerical value of the mass action...
Calculating the Equilibrium Constant
For example, gaseous nitrogen dioxide forms dinitrogen tetroxide according to this equation:
Solution Equilibrium and Saturation

