A Simple Yet Complex One-parameter Family of Generalized Lorenz-like Systems

This paper reports the finding of a simple one-parameter family of three-dimensional quadratic autonomous chaotic systems. By tuning the only parameter, this system can continuously generate a variety of cascading Lorenz-like attractors, which appears to be richer than the unified chaotic system that contains the Lorenz and the Chen systems as its two extremes. Although this new family of chaotic systems has very rich and complex dynamics, it has a very simple algebraic structure with only two quadratic terms (same as the Lorenz and the Chen systems) and all nonzero coefficients in the linear part being −1 except one −0.1 (thus, simpler than the Lorenz and Chen systems). Surprisingly, although this new system belongs to the family of Lorenz-type systems in some existing classifications such as the generalized Lorenz canonical form, it can generate not only Lorenz-like attractors but also Chen-like attractors. This suggests that there may exist some other unknown yet more essential algebraic characteristics for classifying general three-dimensional quadratic autonomous chaotic systems.