The simplest chaotic Lotka-Volterra system with reflection, rotation, and inversion symmetries

^a Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, 1591634311, Iran
^b Health Technology Research Institute, Amirkabir University of Technology, Tehran, 1591634311, Iran
^c Center for Research, Easwari Engineering College, Chennai, India
^d Center for Cognitive Science, Trichy SRM Medical College Hospital and Research Center, Trichy, India
^e Center for Research, SRM TRP Engineering College, Trichy, India
^f School of Artificial Intelligence, Nanjing University of Information Science and Technology, Nanjing, 210044, China
^g University of Wisconsin, 1150 University Avenue, Madison, WI, 53706-1390, USA
* Corresponding author at: Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, 1591634311, Iran. E-mail address: sajadjafari@aut.ac.ir (S. Jafari).
 
This study presents the simplest known three-species Lotka–Volterra system capable of exhibiting chaotic dynamics. The model is constructed with nonlinear growth and mortality terms defined as products of population densities and quadratic functions of species concentrations, capturing essential ecological nonlinearities in a minimal framework. Unlike many classical three-species Lotka–Volterra models, which typically exhibit only stable or periodic behavior, this system displays rich dynamical behaviors, including chaos, under specific parameter regimes and seven terms. Bifurcation analysis and Lyapunov exponent calculations confirm transitions between periodic oscillations and chaotic attractors. Notably, the chaotic attractor possesses a rare com bination of reflection, rotation, and inversion symmetries, despite the system’s structural simplicity. These results demonstrate that even the most minimal Lotka–Volterra formulations can generate multiple symmetric chaotic attractors, establishing a new benchmark in the study of simple yet chaotic ecological models.